C*-ALGEBRAS OF THE PLANAR CRYSTAL GROUPS AND THEIR

C*-ALGEBRAS OF THE PLANAR CRYSTAL GROUPS

AND THEIR IRREDUCIBLE *-REPRESENTATIONS

A Thesis

Submitted to the Faculty of Graduate Studies and Research

in Partial Fulfillment of the Requirements

for the Degree of

Master of Science

in the

Department of Mathematics

University of Saskatchewan

Saskatoon

by

Eric M. Pohorecky

August, 1990

@1990 E. M. Pohorecky

In presenting this thesis in partial fulfilment of the requirements for a Postgraduate degree from the

University of Saskatchewan, I agree that the Libraries of this University may make it freely available for

inspection. I further agree that permission for copying of this thesis in any manner, in whole or in part, for

scholarly purposes may be granted by the professor or professors who supervised my thesis work or, in their

absence, by the Head of the Department or the Dean of the College in which my thesis work was done. It is

understood that any copying or publication or use of this thesis or parts thereof for financial gain shall not

be allowed without my written permission. It is also understood that due recognition shall be given to me

and to the University of Saskatchewan in any scholarly use which may be made of any material in my thesis.

Requests for permission to copy or to make other use of material in this thesis in whole or part should

be addressed to:

Head of the Department of Mathematics

University of Saskatchewan

Saskatoon, Saskatchewan S7N OWO

I would like to express my gratitude to Professor Keith Taylor for the invaluable assistance he provided

in the preparation of this thesis. I would also like to thank the Natural Sciences and Engineering Research

Council for their financial assistance.

Since this page is mine and not read by anyone, except, perhaps, by my mother and wife, I will now

thank miscellaneous people. Thanks to my wife Leslie Jean Walter for her understanding during the past

months: it has been a long summer. Thanks, of course, to Margaret Pohorecky (mom) without whom

nothing would be possible. Wally Polsom took hold of the rudder just as I was sinking away and kept me

afloat (with coffee) during that 40th hour; Art Raymond and Manuel Cabral provided invaluable advice and

lent me a darn fine pair of scissors for my cutting and pasting: thanks to all.

11

z R c T

Page Symbol

5 L1 (G), ll·lh 4 L2(S), ll·ll2 15 vx)(A), (L1(A))# 3 1l 3 K,l.

3 '}{,(f)K,

4 8(1t), 8(11,, K,) 4 T* 5 u 7 p 7 P.LQ 7 P$.Q 8 P- Q (in A) 6 A 8 A' 12 11",ll 12 11"-u

12 .A 5 f*g 5 AG

6 Co(X) 10 B 10 Bx 11 r(B), ro(B) 50 r~(E) 14 c•(G) 14 Ct(G) 14 11·11· 15 A 15 j 15 A( A) 48

A D Mn(Co(A))

51 Da 18 ISOM(En) 18 TRANS( En)

List of Notation

Integers Reals Complex 1-torus (z E C, lzl = 1)

Description

L1 (G)-space L2 ( S)-space continuous linear dual Hilbert space orthogonality Hilbert direct sum bounded operators Hilbert adjoint unitary operator projection in 8(11,) two orthogonal projections subprojection, minimal projection two equivalent projections C*-algebra commutant of C* -algebra *-representation, irreducible representation two equivalent representations structure set of a C*-algebra convolution left-regular representation C*-algebra of functions bundle, C*-bundle fibre in a bundle sections of a bundle equivariant sections group C* -Algebra reduced group C*-Algebra C*-norm on L1 (G) Pontryagin dual of abelian group fourier transform on L1(A) fourier algebra

fixed point algebra isotropy subgroup group of Euclidean isometries subgroup of translations

TABLE OF CONTENTS

Chapter 1: Introduction 1

Chapter 2: C*-Algebras and C*-Bundles 3

I. Hilbert spaces and C*-Algebras 3

1. Hilbert spaces 3

2. Bounded operators on Hilbert spaces 4

3. C*-algebras 5

4. Projections in B(H) 7

II. C*-Bundles 9

Chapter 3: Representation Theory 12

I. *-Representations of C*-algebras 12

II. Group C*-algebras 13

1. General locally compact groups 14

2. Abelian locally compact groups 14

III. Irreducible representations of C*-bundles 17

Chapter 4: Crystallographic Groups and the 17 planar crystal groups 18

I. Crystal groups 18

II. The 17 planar crystal groups 20

1. Preliminaries 20

2. Descriptions 23

Chapter 5: The C*-Algebra of a Group with Abelian Normal Subgroup of Finite Index 39

I. The group C*-algebra C*(G) 39

II. The identification of r o(E) and C*(G) 49

1. The tilde bundle E 49

2. r o(E) ~ C*(G) 56

Chapter 6: The C*-Algebras of Planar Crystal Groups and their Irreducible *-Representations 58

References

I. Description of A 58

II. D-action on A and A, and description of C*(G) 59

III. Deter~nation and description of A/ D

IV. Alternate description of C*(G)

V. Irreducible representations of C*(G)

60

61

61

63

Apoendix A

lll

Chapter 1 Introduction

The main result of this thesis is the explicit construction of the group C*-Algebra C*(G) for

each of the seventeen planar crystal groups G. A secondary result is the explicit description of all

the irreducible *-representations of these C*-Algebras. Two further applications are described in

the concluding paragraph below.

The study of generalized representation theory arose from the study of unitary representations

of groups. Quantum theory, for example, is where representation theory of groups finds its most

important physical application ([6], p.29). Moving up to the general representation theory on *

algebras can often solve a problem which yields a solution in the group case by restriction. Thus,

one finds, for example, titles such as "Operator Algebras and Quantum Statistical Mechanics", and

there is a strong connection between representations of groups and *-representations of *-algebras.

Chapter two is basic in nature. The definitions and results for Hilbert spaces, C*-Algebras, and

C*-Bundles are laid down. The major result is the Gelfand-Neumark theorem which says, basically,

that C*-Algebras are not more general than, and, in fact, are subsumed by, operator *-algebras on

Hilbert spaces. A large portion of the chapter is devoted to finite-dimensional C*-Algebras since

this case is required in later chapters for finding the structure set of C* (G) .

Chapter three introduces representations of *-algebras. One of the long range objectives of rep

resentation theory is to classify all the *-representations of a given *-algebra ([5], p.416). Any

*-algebra gives rise to a C*-algebra, and the *-representations of each have a natural one to one

correspondence. So, "from a representation theoretic viewpoint", *-algebras are not more general

than C*-algebras. In the finite dimensional case, a C*-Algebra has a corresponding finite condition

on its irreducible representations.

Chapter three also gives a development for the result that the group C*-Algebra for a locally

compact abelian group A is the space C 0 (A) for the pontryagin dual A. This result is extended in

chapter five (see [14]) to a general locally compact group G that has an abelian normal subgroup of

finite index. In fact, C* (G) is tied to a wonderfully explicit formulation via [14].

Chapter five also ties together C*(G) with the section C*-Algebra of a C*-bundle. Evans' ([3])

analysis of compact symmetry groups is modified for this result. Then, with this bundle identifica

tion, the irreducible representations of C* (G) are readily accessible in the case of a crystal group's

C*-Algebra.

1

Chapter four essentially lists the planar crystal groups for use in the sequel. Symmetry, in general,

is one of the most fascinating areas in the known universe and finds its expression in various forms.

Hilbert's eighteenth problem, which deals with bounded symmetry and was solved by Bieberbach

around 1910, shows the restrictive nature of symmetry. Yet symmetry, balance and uniformity seem

to be universal characteristics of all systems, in some sense.

In chapter six, this thesis concretely carries out part of the program of representation theory. The

C*-Algebra C*(G) is constructed for each of the seventeen planar crystal groups G. For each C*(G),

the irreducible *-representation set is described. These results form an important set of examples of

non-trivial, non-commutative C*-Algebras of infinite dimension and also provide interesting exam

ples of the structure sets IRR(C*(G)). The methods employed are not restricted to planar groups,

but could be applied to any crystal group (although the computation time required would grow quite

quickly).

A few concluding remarks will now be made. The structure set and the description of C* (G)

lend themselves quite readily to calculation of the topology on the structure space, which is a further

objective of representation theory. Lastly, there is a natural embedding of the group G itself into

L 1( G)- by point-mass functions x ~ flz- and then the unitary representations of G can be deduced

from the structure set of C* (G) . So there are at least these two further applications which could

be carried out from the information provided in this thesis.

2

Chapter 2 C*-Algebras and C*-Bundles

Complex Hilbert spaces are extremely nice and "square" geometric objects. It turns out that

C*-Algebras and Hilbert spaces are intimately connected by the Gelfand-Neumark Theorem. The

first few sections of this chapter deal with Hilbert spaces and C*-Algebras and list results which are

needed in the sequel. The last section sets up preliminary notation and conventions for C*-bundles.

! Hilbert Spaces and C*-algebras.

The bulk of the material in this section comes from (9]. The Hilbert space section is gleaned from

[9], 2.1.

1.1 Hilbert Spaces.

A positive definite inner product on a (complex) vector space 1{, is a map < ·, · >: 1{, x 1{, -+ C

that is sesquilinear and positive definite. That is,

(1) <ax+ by, z >=a< x, z > +b < y, z >for all a, bE C and x, y, z E 1£,

(2) < y,x >= < x,y >for all x,y E ?i,

(3) < x, x >~ 0 for all x E ?i, and < x, x >= 0 iff x = 0.

In this case, the inner product defines a norm on 1£: llxll =< x, x > ~. A positive definite inner

product space which is complete with respect to the norm defined by the inner product is called a

Hilbert space. So any Hilbert space is a Banach space (that is, a complete normed linear space).

Orthogonality plays a major role in the study of Hilbert spaces. Let 1i be a Hilbert space. Two

elements x, y E 1i are called orthogonal if< x, y >= 0. Two subsets X, Y ~ 1i are called orthogonal

if< x, y >= 0 for each pair x EX andy E Y. The set y.L is defined by y.L := {x E 1i 1:< x, y >=

0 \lyE Y} which turns out to be a closed subspace of 1i . It is routine to verify that if X ~ Y ~ 1i

then y.L ~ X.L and that Y n y.L = {0}. So ?i.L = 1i n ?i.L = {0}. If Y is a closed subspace of 1i

then Y .L is called the orthogonal complement of Y. One has the following important and elementary

result for a Hilbert space 1i : given any closed subspace Y, then 1i ~ Y $ Y .L. It then makes sense

to define the Hilbert direct sum of two Hilbert spaces 1i and JC by usual component-wise operations

and norm determined by ll(x, Y)II~$K := llxll~ + IIYII~ for (x, y) in 1i $ IC.

An orthogonal subset B of a Hilbert space is just a subset of 1i whose elements are pairwise

orthogonal. As usual, an orthonormal set means an orthogonal set each of whose elements have

unit-norm (that is, !lxll = 1).

3

THEOREM ([9], 2.2.10). Every Hilbert space has an orthonormal basis. Any orthonormal set in a

Hilbert space 1l is contained in some orthonormal basis of 1(, . Finally, any two orthonormal bases

for 1l have the same cardinality.

Thus, one can define the dimension of a Hilbert space to be the cardinality of any one of its

orthonormal bases, since they all share the same cardinality. In fact, there is, in essence, only one

Hilbert space of any given dimension.

THEOREM ([9], 2.2.12). Two Hilbert space are isomorphic iff they have the same dimension.

So Hilbert spaces are determined (up to isomorphism) by their dimension. Thus, any finite dimen

sional Hilbert space of dimension n is isomorphic to C" - the space of all complex n-tuples with

inner product defined by < (x1, ... , Xn), (y1, ... , Yn) >:= L:~ XiYi, which yields the (usual) norm .1

ll(xl, ... , xn)ll = (L:~ lxil 2)

2 on C". When dealing with Hilbert spaces, isomorphism will mean

norm-preserving (or isometric) isomorphism.

Given au-finite measure m on au-algebraS of subsets of a set S, one defines the space

L2(S,S, m) :={I: S-+ C 1: I is measurable, lllll2 < oo},

1

where llllb := (fs ll(s)l2 dm(s))". (In fact, each I is actually a representative from its equivalence

class of all functions which differ from it on a measure zero set, as usual.) Then L 2 ( S, S, m) (or just

L2(S) if all else is understood) is a Hilbert space ([9], 2.1.14). The inner product on L2(S) which

yields the above norm is defined for two elements I, g by< l,g >= fs l(s)g(s)dm(s).

1.2 Bounded Operators on Hilbert Spaces ([9],2.4).

Let 1l, K, be two Hilbert spaces. A linear operator T : 1(, -+ K, turns out to be continuous if

and only if there is a real constant c such that for all x E 1l, IITxllx: 5 cllxll-x ([9], 1.5.5). Such

an operator T is called bounded and 8(11,, K.) denotes the linear space of all such bounded linear

maps. The operator norm for an element T E 8(11,, K.) is defined to be the infimum over all such

bounds c. Equivalently, the operator norm can be defined by IITII := sup{JITxlll: llxll = 1}. Then

(8(1l, K.), II ·II) is a Banach space, essentially since the range of each (continuous, linear) operator T

is in a Banach space ([9], 1.5.6). In the case that K, = 1l, 8(1l, 1l) will be denoted simply by B(1l).

For an operator T E B(1l, K.) there is a unique operator T* E 8( K,, 1l) such that < T* y, x >-x = < y,Tx >IC for all x E 1l andy E K.. This operator T* is called the adjoint ofT. If C, is yet another

Hilbert space and S, T E 8(1l, K.) and R E 8(/C, C.), then the following conditions hold:

(1) (aS+ bT)* =aS* + bT* for any a, bE C;

(2) (RS)* = S* R*;

4

(3) (T*)* = T;

(4) IIT*TII = IITII2;

(5) IIT*II = IITII.

In the case of B('H), the adjoint defines an involution.

Let l1t and I~e be the identity operators on 1l and JC respectively. An operator U E B('H, JC) is

called unitary if U*U = I1t and UU* = I~e.

THEOREM ([9], 2.4.5 AND 2.4.6). A surjective operator U: 1i-+ JC is an isomorphism if and only

if u- 1 exists and u-1 = U*. Further, V : 1l -+ JC is a unitary operator if and only if V is a

norm-preserving (or, equivalently, inner product preserving) map of'H onto /C.

So unit aries are exactly the (isometric) isomorphisms of two Hilbert spaces.

An important example of a set of bounded linear operators on a Hilbert space is L1(G, m) acting

on L 2(G,m) where m is a Haar measure on the locally compact group G. For simplicity of notation,

assume G is unimodular so that the Haar measure is both left and right invariant. Denote by L1 (G)

the Banach space

L 1(G,m) := {/: G-+ C 1: f is measurable, llfll1 < oo},

where 11/lh := fs lf(s)ldm(s) and, as usual, functions are identified that differ on a null set ([9],

p.53). L 1(G) becomes a Banach *-algebra (see definitions below) with convolution as the product,

which is defined for two elements/, g E L1(G) by the function f * g, where

f * g(x) := L f(y)g(y- 1x)dm(y) (for almost all x E G).

Cauchy-Schwarz shows this integral equation convergent for almost all x E G, and then f * g is

shown absolutely integrable over G, and so f * g E L1(G). The involution on L 1(G) is defined, for

f E L 1(G), by the L 1(G) function f*(x) := f(x- 1) (for almost all x E G). The Banach *-algebra

L1(G) acts on the Hilbert space L2(G) by the left-regular representation >..0 as follows:

>..J h := j * h (for f E L1 (G), and all hE L2 (G)),

where convolution is defined as above. Moreover, >..J can be shown to be linear and bounded by

11!111· Thus, for each f E L 1(G), one has >..J E B(L2(G)). Chapter three expands on this important

representation.

1.3 C*-algebras.

5

A Banach algebra A is defined to be an algebra (over C ) which is a complete normed linear

space over C such that for any two elements A, B E A, IIABII ::=; IIAIIIIBII ([9], 3.1). An involution

on a Banach algebra A is defined to be a map T 1-+ T* :A~ A such that the following conditions

hold for any two elements S, T E A:

(1) (aS+ bT)* =aS*+ bT* for any a, bE C;

(2) (TS)* = S*T*;

(3) (T*)* = T.

If A has a continuous involution, then it is a Banach *-algebra. A Banach *-algebra A IS a

C*-algebra if it satisfies the C*-condition:

IIT*TII = IITW (for all TEA).

A *-homomorphism of two C*-Algebras (or Banach *-algebras) is an algebra homomorphism that

preserves involution. *-homomorphisms of C*-Algebras are necessarily continuous. Further, if a

*-homomorphism is an isomorphism, then it is necessarily norm-preserving ([9], 4.1.8). Thus, a

C*-map or C*-isomorphism will refer to an (isometric) *-isomorphism of C*-Algebras.

For a Hilbert space 1l , B(1l) equipped with adjoint is a Banach *-algebra and, as noted above,

the C*-condition is met. So B(1l) is a C*-algebra. B(1l) (or any C*-subalgebra of B(1l)) is often

called a (C*-)algebra of operators on 1l . It turns out that these are essentially the only examples

of C*-algebras.

THEOREM: GELFAND-NEUMARK ((9], 4.5.6). Any C*-algebra A is isometrically *-isomorphic to

a C*-subaJgebra of B(1l) for some Hilbert space 1l .

So it makes sense to define a C*-algebra to be finite dimensional if it is embedded in B(C") for

some finite integer n. Further, the following alternative definition of a C*-algebra can now be shown

equivalent to the above: a norm-closed subalgebra A of B(1l) which is closed under involution is

called a C*-algebra.

Finally, one defines the C*-direct sum of C*-Algebras. For a finite number of C*-Algebras,

At, ... Ak, let A be the set of all k-tuples (A1J ... ,Ak) E A 1 x ··· x Ak. Define addition, scalar

multiplication, involution, and multiplication coordinatewise on A . Define the norm on A by

II( At, ... , Ak)ll := max{IIAdl, ... , IIAkll}. Then A is a C*-Algebra, denoted At 61· · · El1 Ak.

An important example of a C*-algebra is C0 (X). If X is a locally compact Hausdorff space, then

C0 (X) denotes the space {I : X ~ C 1: I is continuous, vanishing at oo }, with pointwise scalar

multiplication, addition, multiplication. Further, if involution is defined for each I E Co(X) by

6

J*(x) := ](x), then C0 (X) is an involutive algebra. If Co(X) is given the supremum norm ll·lloo

defined for f E Co(X) by 11/lloo := sup{l/(x)l 1: x EX}, then it is a C*-Algebra ([12], p.250).

1.4 Projections in 8(1£).

The study of irreducible representations of finite dimensional C*-Algebras depends largely on

the study of projections in 8(1l) . The ultimate decomposition theorem for finite dimensional

C* -Algebras is expressible in terms of minimal projections. The following sub-section will define

projections in 8(1£) and list results needed in the sequel. Most of the results follow from (9], 2.5.

Definition. An element A in 8(1£) is called a projection if A is a self-adjoint idempotent. (That

is, A= A2 and A= A*.)

Let P be a projection in 8(1l) and then let Y = P1l. Then Y is a closed subspace. On the other

hand, recall from the Hilbert space subsection above that if Y is a closed subspace of a Hilbert space

1£ , then 1£ e:! Y Ef7 Y .L. So any element v E ?{, is uniquely expressible as v = y + z where y E Y

and z E y.L, and so the equation Py( v) := y defines a (bounded) linear operator Py on 1l called

the projection onto Y parallel to Y .L. It is easily seen that Py = P~ and Py = Py. Thus, there

is a one-to-one correspondence between closed subspaces in 1{. and projections in 8(1£). Further,

the projection Py~ onto y.L is seen to be Py~ =I- Py. Using direct summation, one has that

Y = {x E 1l 1: Pyx= x} and y.L = {y E 1{. 1: Py~y = y}, and so Py is called the orthogonal

projection of 1£ onto Y. Henceforth, projection will mean orthogonal projection.

The projections in 8(1l) form a lattice which determine practically all of the structure of any

C*-subalgebra.

Definitions.

(1) (Order Structure) Define the partial-ordering on the set of projections in 8(?-l) as follows:

for two projections P and Q, P ::; Q iff P1l ~ Q1l. In the case of P ::; Q, P is called a

subproiection of Q.

(2) For a *-subalgebra A of 8(1l), let P be a non-zero projection in A. P is called a minimal

projection in A if P has no subprojections in A other than 0 and P.

(3) A set X of projections is called orthogonal if PQ = 0 whenever P "# Q are both elements of

X.

PROPOSITION ((9], 2.5.2). Let P and Q be two projections in 8(1l) that project onto (closed)

subspaces Y and Z respectively. The following are equivalent:

(1) Y ~ Z (or, equivalently, P ::; Q)

7

(2) PQ = P

(3) QP = P

( 4) IIPxll $ IIQxll (Vx E 1£)

The following theorem describes the connection between minimal projections and C*-Algebras.

PROPOSITION ([1], 1.4.1). Let A be a finite dimensional C*-algebra.

(1) A (non-zero) projection Pin A is minimal if and only if PAP= CP. In this case, there is a

(continuous linear) functional f E A# such that PAP= f(A)P for all A EA.

(2) A is generated by its minimal projections.

(3) Every projection Pin A is the finite sum of an orthogonal set of minimal projections in A .

Suppose A is a finite-dimensional C*-Algebra containing the identity I. I is trivially a projection,

so I= P1 + · · · + Pr for some set of minimal and pairwise orthogonal projections P1, ... , Pr E A.

Since Pi1lnPj1l = {0} whenever i ::fi j, one can break 1l up into the direct sum 1l ~ P11l$· · ·$Pr1l.

To facilitate the connection between irreducible representations and minimal projections which will

be made in chapter three, we will now define a certain equivalence relation on the set of projections

in a C*-Algebra. Let A be a C*-subalgebra of 8(1£) . Two projections P and Q in a C*-Algebra

A are called equivalent (in A), written P "' Q (in A), if there is an element V in A such that

P = VV* and Q = V*V. In the case that P and Q are minimal projections in A, then P "'Q (in

A) is equivalent to the condition that PAQ ::fi {0}. Finally, equivalent projections have isomorphic

ranges.

As noted above, there is a strong connection between projections in a C*-Algebra and its decom

position into a finite sum of simple C*-Algebras. The following definitions and results are used to

decompose a finite dimensional C*-Algebra A into simple C*-Algebras.

Definitions.

(1) A simple C*-Algebra is one which is C*-isomorphic to 8(~), the full algebra of operators on

some Hilbert space ~ .

(2) Let A be a C*-subalgebra of 8(1£) . The commutant of A in 8(1£) is defined to be (the

C*-algebra with unit I) A':= {BE 8(1£) 1: BA = AB VA E A}.

(3) For a subset X ~ 8(1£), a subspace ~ ~ 1l is called X -invariant (or sometimes called X

stable) if X~~~.

( 4) Let A be a *-subalgebra of 8(1£) .

(a) A is called irreducible on 1l if {0} and 1l are the only A-invariant subspaces of 1£.

8

{b) An .A-invariant subspace JC ~ 1l is called irreducible (for .A) if .A k is irreducible on IC .

(This is equivalent to JC having no proper, non-trivial, .A-invariant subspaces.)

PROPOSITION. Let .A be a *-subalgebra of 8(1£) containing the identity I. Then one has the

following properties:

(1) ({1}, 1.4.2) If .A is a C*-Algebra in 8(/C) and .A is irreducible on IC , then .A= 8(/C).

{2) For any e E 1£, Ae := {Ae 1: A E .A} is an .A-invariant subspace of 1l .

{3) Suppose Pis a minimal projection in .A and e E P1l (e ;f; 0). Then the projection PA{ onto

.Ae is a minimal projection in the commutant .A' .

( 4) If JC is an irreducible subspace for .A , then the projection PIC onto JC is a minimal projection

in the commutant .A' . Conversely, if P is a minimal projection in .A' , then P1l is an

irreducible subspace for .A .

We are now in a position to decompose a finite dimensional C*-subalgebra of 8(1£) . Let .A be a

finite dimensional C*-subalgebra of 8(1£) (for a finite dimensional Hilbert space 1l ). Recall I is

in the commutant .A' of .A . Let P1 , ... , Pr be a set of mutually orthogonal, minimal projections in

.A' such that P1 + · · · + Pr = I. Then let 1-li := Pi1l (for each i, 1 :5 i :5 r). These r Hilbert spaces

are pairwise orthogonal, .A-stable, and .A-irreducible subspaces of 1l . Further, 1l = 1£1 E!J • · · E!J 1-lr,

since the projections sum to I. Since each 1ti is irreducible for .A, one has .APi =.A I'Hi~ 8(1-li)·

Now, one can routinely verify that .APi n .APj = {0} whenever i :/; j (1 :5 i,j :5 r) by using the

fact that each P~~; commutes with all .A , and the fact that the projections are mutually orthogonal.

The set { P1, ... , Pr} is divided into equivalence classes by the previously mentioned equivalence

relation for projections in .A. Let {Pi11 ••• , Pi,} be a set of representatives, one from each of these

equivalence classes. Then .A decomposes as the C*-direct sum .A~ 8(1li 1 ) $ · · · $ 8(1li, ). So one

has the following theorem:

THEOREM ([1], 1.4.5). Any finite-dimensional C*-Algebra is C*-isomorphic to a direct sum of

finitely many simple C*-Algebras.

II C*-bundles.

Bundle and C*-algebra theory tie together when a C*-Algebra .A is isomorphic to r o(C) , the set

of sections on some bundle C which vanish at infinity. It will be seen in chapter five for a certain

type of group that the group algebra C*(G) is isometrically isomorphic to r o(E), the continuous

sections of a certain C*-bundle E which vanish at infinity. Further, chapter three will show that

the irreducible *-representations on fo(E)- hence, on C*(G) as well- have a concrete description.

9

But for now some preliminary notions about bundles are required.

The definitions below follow from [5] and [2]. Note that all topologies are Hausdorff.

( 1) A Bundle over X is a triple B = (p, B, X) where B, X are Hausdorff spaces and p : B -+ X

is an open and continuous surjection.

X is called the Base Space,

B is called the Bundle or Total Space,

pis called the Bundle Projection

of B.

For each x EX, p-1(x) is called the Fibre over x and is denoted B:r (or B(x)) when there is

no ambiguity concerning the projection being used. Notice that the total space, as a set, is

the disjoint union of all the fibres, B 8~ U:rex B:z:.

(2) A Banach bundle B over X is a bundle B = (p, B, X) with each fibre B:r equipped with

some given (complex) Banach space structure such that

(a) b 1-+ llbll: B-+ R is continuous (where llbll := llbiiB~, the norm in b's fibre)

(b) addition in B defined fibrewise is continuous from {(b, c) E B x B 1: p(b) = p(c)} to B.

(c) scalar multiplication on B defined fibrewise is continuous from C x B to B.

(d) (The "zero-limit" property) If x E X and {bi};eJ is any net in B such that llbill -+ 0 and

p(bi)-+ x (in X), then bi-+ OB~ (the 0 in banach space B:z:; also denoted O:z:).

Note that the last condition has an equivalent form which gives a base of neighborhoods for

O:r in B:

( d ') Let x E X. Then the collection of all sets of the following form establishes a base of neigh

borhoods for O:r in B:

N(x: U,t) :={bE B 1: p(b) E U and llbll < t}

where U is an X -neighborhood of x and f > 0.

(3) A C*-Algebra Bundle (or just C*-Bundle) is a banach bundle B = (p, B, X) with each fibre

B:r having a given C*-Algebra structure such that

(a) multiplication (defined fibrewise on B) is continuous from {(b,c) E B x B 1: p(b) = p(c)} to

B,

(b) involution defined fibrewise on B is continuous.

( 4) For a Banach space B and Hausdorff X, the Trivial Bundle is defined by B = (p, B x X, X)

where p projects onto the second factor, B x X has the product topology, and the fibres

B(x) := B x {x} have the Banach space structure derived from the bijection b 1-+ (b,x)

10

(bE B, fixed z EX). It is routine to verify B is a banach bundle.

(5) Maps Between Bundles and Cross-sections.

For two bundles B = (p, B, X), C = ( q, C, Y), we have the following definitions:

(a) A Bundle Map denoted t.p: B ~Cis a pair of continuous maps t.p: B ~ C, f: X~ Y such

that the following commutes:

B __!___. C

X ------+ y I

t.p is called over f. In the case X = Y and f = idx, then t.p is called over X. Notice that in

general q(t.p(b)) = f(p(b)) by commutativity, so t.p(b) E CJ(x) where b E Bx (x = p(b)), so <p

takes Bx ~ CJ(x) (that is, carries a fibre into a fibre). So for each z EX, denote by 1./)x the

fibre restriction map: 1./)x := 'PIBz : Bx ~ CJ(x) .

(b) A Banach Bundle Map between two banach bundles B and C is a bundle map t.p : B ~ C

over, say,/, such that each fibre restriction map 1./)x: Bx ~ CJ(x) is bounded and linear.

(c) A C*-Bundle Map between two C*-Bundles B and C is a bundle map t.p: B ~Cover,

say, /, such that each fibre restriction map 1./)x : Bx ~ CJ(x) is a C*-map. Since C*-maps are

bounded and linear, then a C*-Bundle map is a Banach bundle map.

(6) A Cross-section of a bundle B = (p,B,X) is any functions: X~ B such that po s = idx.

That is, s maps each z E X to an element in the fibre over X.

A Section of B is a cmitinuous cross-section. r(B) denotes the space of all sections s on

the bundle B . s is said to "pass through" each b E Range( s). If each b E B has a section

passing through it, then B "has enough sections".

fo(B) denotes the subspace of f(B) whose sections vanish at infinity (that is, Vc > 0, there

is a compact K ~X such that s(z) < f. for all z EX\ K). Of course if X is compact, then

f(B) = r o(B).

(7) The C*-Algebra of sections ((FD], p.581). For a C*-bundle B over a locally compact Hausdorff

space, equip r o(B) with supremum-norm and pointwise operations for multiplications and

addition. For s E r o(B), define the involution for s pointwise: s*(z) := (s(x))* (for each

x EX). Then, by the definition of a C*-bundle, each of these operations defines continuous

cross-sections which vanish at infinity. The supremum-norm is a C*-norm on this space.

(r o(B), II · lloo) is, thus, a C*-Algebra called the cross-sectional C*-Al~ebra of B ((FD],

p.581).

11

Chapter 3 Representation Theory

As noted in the introduction, part of the program of analysis is the classification of representations.

This chapter lists representation theory results for C*-Algebras which are required in the sequel,

especially in chapter five. Since this thesis deals only with *-algebras, the words "*-representation"

and "representation" are freely interchanged.

! *-Representations of C*-algebras.

Recall all the definitions and results from 2.1. This section will develop the representation theory

needed in later chapters for C*-Algebras. The first part of this section deals with general C*-Algebras

of operators while the second part specializes to finite dimensional C*-Algebras.

Definitions. Let .A be a C*-algebra in 8(1i) for some Hilbert space 1i.

(1) A *-representation (or just representation) of .A is a *-homomorphism 1r : .A--+- 8(1i1r) for

some Hilbert space 1i1r. 1r is called a finite dimensional representation if the space on which

1r(.A) acts, namely 1i1r, is finite dimensional. Note, too, that this definition applies if .A is

just a Banach *-algebra.

(2) An irreducible representation of .A is a non-zero representation 71" such that 1r(.A) is irreducible

for 1i1r (that is, 1i1r has no non-trivial, closed subspaces which are 1r(.A)-invariant).

(3) Two representations 71", u of .A are (unitarily) equivalent (written 1r ,_ u) if there is an isometry

U of 1i1r onto 1i(1 that "intertwines" their actions:

U1r(A) = u(A)U (for each A E .A),

or, equivalently, U1r(A)U~ = u(A) for each A E .A. Note that any such U is a unitary

operator.

One recalls from section 2.1.2 that unitaries on Hilbert spaces are really just "changes of basis", and

so the ranges of equivalent *-representations are indistinguishable as far as geometric properties are

concerned ([1], p.13). The spectrum of a C*-Algebra .A is defined to be the set of equivalence classes

of all the irreducible *-representations on .A (partitioned by unitary equivalence), often denoted by

A. The description of A for a given C*-Algebra .A is a major objective of representation theory.

We now specialize to the case where .A is a finite dimensional C*-Algebra.

12

THEOREM. Let A be a C*-algebra in 8(1£) for a finite-dimensional Hilbert space 1£. Then any

irreducible representation of A is equivalent to one obtained by restricting A to an A-irreducible

subspace K, ~ 1£.

PROOF: From 2.1.4, A is generated by its minimal projections. So if 1r is an irreducible represen

tation of A, then there is a minimal projection E E A such that 1r(E) i= 0. Let f E A# be a

continuous linear functional as in 2.1.4. So EAE = f(A)E for all A E A (since E is minimal). Now

1r(E) is a (non-zero) projection in 8(1l1r) and so one may pick 71 E 1r(E)1l1r with 111111 = 1. On the

other hand, choose e E E1l with 11e11 = 1. Then 1r(A)71 and K, := Ae are irreducible subspaces of 1lr

and 1l respectively. Now, since 1r is a representation and Eisa projection, one has, for any TEA,

ll1r(T)77W = ll1r(T)1r(E)7JII 2 = < 1r(TE)7J,1r(TE)71 >=< 1r(ET*TE)71,7J >= f(T*T) < 1r(E)71,71 >= f(T*T) = /(T*T) < Ee,e >=<

J(T*T)Ee,e >=< ET*TEe,e >=< rEe,rEe >=< re,re >= urell2• sou : 1£1r ~ x:. by

1r(T)77 ~--+ Te is an isometry. Finally, the following equation intertwines 1r with the representation of

A defined by restricting A to K.:

1r(S) = U*(SI.~:)U (for all SEA).

To see this, let S E A. Then for any 1r(T)71 E 1l.1r we have U*(SI,~:)U1r(T)77 = U*(SI.~:)Te = U* STe = 1r(ST)71 = 1r(S)1r(T)1J as required.

Now recall from 2.1.4 that any irreducible subspace K, ~ 1{. for A is the range F1l for some

minimal projection F E A'. So along with the above theorem one now has that for any irreducible

representation 1r of A there is a minimal projection Fin the commutant A' of A such that 1r l"oJ IF?-l

(restriction of A to F1i). One also recalls that for any FE A', F1i is an irreducible subspace for A

and so IF?-l defines an irreducible representation of A. Thus, there is a strong connection between

irreducible representations and minimal commutant projections for a C *-Algebra A .

From 2.1.4, one has that the identity I is decomposable as P1 + · · · + Pr for a set of mutually

orthogonal and minimal projections in the commutant of A . For any two minimal projections in the

commutant, say P and Q, one can also show that P- Q (in A') iff the irreducible representations

they define are equivalent. Since equivalence of projections partitions the set of minimal projections

in a finite dimensional C*-Algebra into a finite number of classes, we now have that A is a finite

set. Thus, it will be sufficient to find a set of mutually orthogonal, irreducible projections with sum

I in A' in order to get a representative for each irreducible representation of A .

II Group C*-algebras.

13

In this section the C*-algebra associated with a locally compact group G is defined. We specialize

quickly to the case where the group is locally compact and abelian to produce results which will be

needed in the case that G is the extension of a finite group by an abelian group (chapter 5). The

main reference for the locally compact abelian case is [12].

11.1 General locally compact groups.

For a general unimodular locally compact group G with measure m, the group C* -algebra C* (G) is

formed from L1(G) . From 2.1.2, L1(G) is a Banach *-algebra. By [5] (p.387), any *-representation

1r of L 1(G) is continuous and ll1rll :5 1, where ll1rll := sup{ll7r(/)lll: 11/lh = 1}. Define the C*-norm

on L1(G) as follows, for an element f E L1(G):

11/11. := sup{ll7r(/)IIB(7t,.) 1: 1r is a *-representation of L 1(G)}.

Notice that for any *-representation 1r, one has 1r(j* *f) = 1r(j)*7r(j) and these two elements have

the same norm in the range space, so one easily verifies the C*-condition II/** /II. = 11111;. However,

in general, L 1 (G) fails to be complete with respect to this new norm, and, thus, falls short of being

a C*-Algebra. The group C*-Algebra is defined to be the completion of L 1 (G) with respect to the

C*-norm II· II• and is denoted by C*(G) .

Recall the left-regular _AG representation of L1(G) on L2(G) from 2.1.2. In the case that G is

an amenable group, the C*-norm is determined by this left-regular representation. See (6] for the

definition and basic properties of amenable groups. So the C*-Algebra of an amenable group G is

isomorphic to the reduced~ C*-Algebra, denoted C~(G), which is the closure of _AG(L 1 (G))

in the C*-algebra B(L2(G)) ([14]). Chapter five will use the fact that any locally compact abelian

group is amenable, as is any extension of a finite group by a (locally compact) abelian group ([6],

p.8).

11.2 Abelian locally compact groups.

The rest of the current section deals with a locally compact and abelian group A. It is shown that

the group C*-algebra is particularly simple: C*(A) £I! C0 (A), where A is the pontryagin dual of A.

Rudin ([12], ch.1) is relied upon heavily for all of the results.

For the rest of the section fix a locally compact abelian group A. A has a non-negative regular

measure which is not identically zero and is translation-invariant (that is, a Haar measure). To prove

this, one constructs a positive translation-invariant linear functional Ton Coo(A) (the continuous

functions on A with compact support)([12], p.1). Then the Riesz representation theorem gives a

measure mas required ([11], p.42).

14

The Pontryagin Dual of A is defined to be

A== {x: A-+ T 1: X is continuous, x(ab) = x(a)x(b)\t'a, bE A}'

where T is the 1-torus: {c E C 1: lei = 1}. Each x E A is called a character. Note that this

definition implies that lx(a)l = 1 (X e .A, a E A) and that 1 = x(1A), since each character is

a homomorphism. A itself becomes an abelian group when equipped with pointwise product: for

two characters x, 1/J, xt/J is defined by x'f/J(a) := x(a),P(a). So xt/J is also a character. Note that

the identity in A is defined by the homomorphism 1 A (a) := 1 (for each a E A). Inverse is easily

calculated: 1 = xx-1(a) := x(a)x- 1(a) (Va E A), so x- 1 (a) = x(a) since we are working in the

1-torus.

A topology is introduced on A with respect to which it, too, is a locally compact (and abelian)

group. The topology will be defined by embedding A into L 00 (A). Each x E A is a bounded and

continuous (hence Borel) function on A, so

X E L00 (A) := {g :A-+ C 1: g is m-measurable and ll9lloo < oo}

(with the usual identifications of zero-measure sets), and, in fact, A is in the unit sphere of L 00 (A).

It is a standard result of analysis that L00 (A) ~ [L1{A)]# = {continuous linear functionals g :

L1(A)-+ C}, the continuous linear dual of L1{A) , via g(l) := JA l(x)g(x)dx {[9], p.55). For each

IE L1{A) define the Fourier transform function f on the set A :

f(x) := L l(a)x(a)da,

convergent for each x E A since x is bounded and continuous on A and I E L 1{A). For a fixed

X E A the map I ..-.. f(x) (=x(l)) is a linear functional on L 1 (A) corresponding to x in L 00 (A).

Further, A corresponds to multiplicative linear functionals in L 00 {A) . Now, give A the weak

topology with respect to {fl: IE L 1(A)} ;that is, the weakest topology such that each J: A-+ C

is continuous. This is just the weak-* topology on L 00 (A) restricted to A, and A is in the unit ball,

which is weak-* compact by Alaoglu-Bourbaki ([9], p.45). It can be shown that A is locally compact

{[12], 1.2.6).

Each fis in C0 (A), which is a locally compact space. Let A(A) := {fl: IE L1{A)} £; Co(A), the

"fourier algebra" of A . A(A) is a *-algebra when equipped with pointwise operations and adjoint:

f•(x) := f(x).

THEOREM ([12], P.9). A(A) is a self-adjoint, separating subalgebra of Co(A), so is dense in

(Co(A), ll·lloo) by Stone-Weirstrass.

15

A has a left and right invariant Haar measure since it is locally compact and abelian. The following

Plancherel theorem allows this measure to be chosen nicely.

PLANCHEREL THEOREM ([12], P.26). Haar measure on A can be normalized so that the map

I 1-+ f: L1(A) n L2 (A) ----+ L2(A) is an isometry with dense image, and so it is uniquely extendable

to an isometry P: L2(A)--+ L2(A) (since L1(A) n L2 (A) is dense in L2(A)).

The unitary P establishes the isomorphism B(L2(A)) ~ B(L2(A)). .A(A) and L1(A) will be

connected via this isomorphism.

With the isometry P, one has the property that for each I E L1 (A) and h E L2 (A) then P(l *h) = f. P(h) (where product is pointwise). This can be seen by first restricting h to L 1(A) n L2 (A) (so

I* h E L1(A) n L2 (A), too, by the left-regular action), and so one calculates the action of P: for

x eA.,

P(l * h )(x) = 1--;-h(x)

: = i I* h(a)x(a)da

= i x(a)x(b- 1 )x(b) L l(b)h(b- 1a)dbda

= i x(a)x(b- 1)x(b) i l(b- 1a)h(b)dbda (let b' = b- 1a)

= i h(b)x(b) i l(ab- 1 )x(ab- 1 )dadb (using Fubini)

= i h(b)x(b) i f(a)x(a)dadb (by translation-invariance)

== icxfh(x) = iP(h) (x)

The property is established for all L2(A) by extension (again, L1(A) n L2 (A) is dense in L2(A) and

P is a linear isometry).

Define M : C0 (A)--+ B(L2(A)) as follows for an element g E C0 (A):

M(g) h := g · h (pointwise product, for all hE L2(A)).

It is routine to verify that M is a representation of the C*-Algebra (C0 (A), II · lloo)· M is easily

shown injective, and so M embeds C0 (A) onto a C*-subalgebra of B(L2(A)).

Define the "left regular" *-representation of L1(A) in B(L2(A)) for each IE L1 (A) by

>.j h := I* h (for all hE L2(A)).

A is abelian and locally compact, so it is amenable, and so C*(A) is isomorphic to the reduced

C*-Algebra C~(A). That is, C*(G) is the closure in B(L2 (A)) of the image of L 1(A) under the

left-regular representation ).A.

16

Finally, one has M([) = 'P >tfP-1 for each IE L1(A). Thus, I 1-+ M-1(P>tfP-1 ) is an isometric

*-isomorphism of L1(A) and A(A). This implies that C*(A) ~ C0 (A) by extension, since A(A) and

L1(A) are dense in C0 (A) and c• (A) respectively.

III Irreducible Representations of C*-Bundles.

This section presents a major result which will give a method to find all the irreducible represen

tations of the group C*-algebra C*(G) for a crystal group G. This result follows [5] (p.582).

Throughout this section C will denote the C*-algebra of sections (r o(B), ll·lloo) defined in 2.11 for

a fixed C*-bundle B = (p, B, X) over a locally compact and Hausdorff space X. For a point z in the

base space X of B , let B; denote the set of (equivalence classes of) irreducible *-representations

of the C*-fibre Bx.

Fix x EX. If 1r E Bx then the following equation defines a *-representation II on C.

IIJ='lrJ(x) (VIEC).

Since B is a C*-bundle, it has enough sections ([5], 11.13.15). Thus, the range of II and 1r are

identical and so II is also an irreducible *-representation, but of C. Conversely, suppose II is an

irreducible *-representation of C. Then there is a unique z E X and 1r E B; such that II is given

by the above equation ([5], II.8.8). The proof of this converse consists of taking the kernel of II (an

ideal in C) and evaluating it at each x E X to produce ideals lx ~ Bx, and showing that lx is a

proper subset for exactly one x EX. Then II1 is shown to depend only on l(x) (IE C), so there

is a unique 1r E Bx such that the above equation holds ([5], p.583).

Thus, if one is working on a C*-bundle whose fibres are each finite dimensional, then the structure

space C of the (usually non-finite dimensional) C*-algebra C = r o(B) is simplified to the finite

dimensional case in some sense. This approach is taken in chapter five for c• (G) .

17

Chapter 4 Crystallographic Groups

and the Seventeen Planar Crystal Groups

This chapter will define crystallographic groups (or crystal groups for short), list some of their

properties, and give a detailed description of the seventeen planar crystal groups.

A crystallographic group arises quite naturally as the symmetry group of a repetitive cellular

{bounded) pattern in Euclidean n-space. In fact, crystallographers deal with physical representations

of such phenomena. In two dimensions, wallpaper and embroidery patterns often use such repetition.

One of Hilbert's famous problems (number 18) essentially was to show only a finite number

of distinct pattern-types (symmetry groups) exist for each given n-space. Bieberbach solved this

problem affirmatively and in general around 1910 although Leonardo Da Vinci apparently showed

as far back as the renaissance that only 17 such planar (n=2) pattern-types exist.

! Crystal Groups.

The notation used in this section to define crystal groups somewhat follows [8]. En will denote

Euclidean n-space: that is, R n with distance derived from inner product d{ x, y) = < x - y, x - y > ~

(and norm, llxll = < x, x >i in the case of a given fixed origin 0). ISOM{En) will denote the group

(with composition as product) of "rigid motions" of En : that is, distance-preserving permutations

of En , often called simply "isometries". It is a fundamental result of Euclidean geometry that each

u E ISOM(En) is uniquely the composition of an orthogonal linear isometry cp E e(En) (with a

fixed point 0 E En as origin) and a translation t" where v E En and tv(x) := v + x for all x E En.

That is, u = t" o cp {(10], p.101). TRANS(En) will denote the (normal, abelian) subgroup of

ISOM(En) consisting of pure translations. Identifying TRANS(En) with En, one writes u = (cp, v)

for elements of ISOM{En) . So ISOM(En) acts on En by (cp, v)x := tv(x) + cp(x). With this

notation, TRANS{En) = {(1,v) 1: v E En} where 1 is ida(E")· Product in TRANS(En) by

composition is then identified with addition in En. Product (composition) in the larger ISOM{En)

is determined by the action on a point x E En: (cp,v)(1P,w)x := v+cpw+cp1Px = (cp1P,cpw+v)x, and

so (cp, v)(1P, w) = (cp1P, cpw+v). Then ISOM{En) breaks up into the semi-direct product En XI e(En)

by using the natural action of e(En) on En (again, with fixed origin 0). Notice that {cp,v)- 1 =

(cp- 1 , -cp- 1v) and (1, 0) is the identity. Further, ISOM{En) is a topological group with the subspace

topology from GL{En) x En.

18

A subgroup G ~ ISOM(En) is defined to be a Crystal Group if G is discrete (in ISOM(En)) and

the quotient space En /G is compact. Note that this is the topological quotient space of G-equivalent

classes of points from En . Some elementary properties of crystal groups are now listed.

For a crystal group G ~ ISOM(En) let A denote G n TRANS(En) = {(1, v) E G}.

PROPOSITION. For a crystal group G ~ ISOM(En), A is a normal abelian subgroup of G and has

no (non-trivial) torsion elements.

PROOF: To see A is a subgroup, note (1, 0) e A and if (1, a), (1, b) E A, then (1, a)(1, b)-1 = (1, a- b) EGis of the right form to be in A. A is obviously abelian. For normality, let (<p, v) E G

and {l,a) EA. Then the conjugate (<p,v)(1,a)(<p,v)-1 = (<p,v)(1,a)(<p- 1,-<p-1v) = (1,<pa) is in

A. Now if (1, a) were torsion (¥ (1, 0)) of order k > 1, then a ¥ 0 but (1, 0) = {1, a)k = (1, ka), so

ka = 0, and thus a= 0, a contradiction.

Since A is normal, the quotient group D := G/A can be formed. It turns out that the quotient D

is exactly the projection of G into 9(En) , and D is finite.

PROPOSITION. For a crystal group G ~ ISOM(En), Dis isomorphic to P1(G) := {t.p 1: (<p,v) E G},

the projection of G into 9(En ).

PROOF: Note (1, 0) E G so P1(G) contains the identity transformation, and if we pick any two

elements <p,'I/J E P1(G) then (<p,v), (,P,w) E G and so (<p,v)(tP,w)-1 = (<p,P- 1 ,v- <p,P- 1w) shows

<p,P-1 E P1(G), which shows P1(G) is a group.

Now, if (<p,v), (<p,w) are in G, then (<p,v)(<p,w)-1 = (1,v- w) E A shows both these elements

must be in the same A-coset. On the other hand, if ( <p, v), ( 1/J, w) E G are in the same A-coset, then

( r.p, v) = (1, a)( 1/J, w) for some (1, a) E A, and so (r.p, v) = ( 1/J, a+ w). By uniqueness of decomposition

of ISOM(En) as En X1 9(En), r.p = tP· Thus, [(<p, v)] ...-+<pis easily an isomorphism.

PROPOSITION ((15]). For a crystal group G ~ ISOM(En),

(1) G is a closed subgroup of ISOM(En) ,

(2) Dis a finite group.

Thus any crystal group G fits the exact sequence

where A is abelian and D is finite (and t is inclusion, p is the natural projection). It can further be

shown that A is free abelian of rank n (when G ~ ISOM(En)) and A is maximal abelian in G. A

is called the "lattice" of G and D is called the "point group". Bieberbach showed that the above

19

definition for a crystal group is equivalent toG (~ ISOM(En)) being discrete and the projection of

the lattice A into En containing n linearly independent vectors ([8], p.770). (Thus, A can now be

identified with its projection in En.) The dimension of a crystal group is thus defined to be the rank

of A. Zassenhaus showed, on the other hand, that any group G is isomorphic to an n-dimensional

crystal group iff G has a normal, maximal and free abelian subgroup A of rank n with finite index

in G ([8], p.770).

If the above sequence splits (that is, there exists an injective homomorphism D --+- G), then G

splits as the semi-direct product G ~ A >4 D (where d 1-+ AUT(A) by pullback and conjugation)

and G is called a symmorph; otherwise G is non-symmorphic. Farkas solves Hilbert's 18th problem

with the following steps for each n EN:

(1) Only a finite number of (non-isomomorphic) n-dimensional symmorphic crystal groups exist.

(2) Each n-dimensional crystal group can be imbedded in ann-dimensional symmorphic crystal

group.

(3) Each n-dimensional crystal group admits only a finite number of (non-isomorphic) n-dimensional

crystal subgroups.

Hiller notes the implementation of an algorithm to enumerate the crystal groups of each dimension. ~

The numbers of groups for each n grow at least as fast as 2n . n = 2 has 17; n = 3 has 219; and

n = 4 has 4783 (!) ((8]).

II The Seventeen Planar Crystal Groups.

Following some preliminary notation and properties, a representative from each of the seventeen

planar crystal groups is listed below. Schwarzenberger's descriptions are relied upon heavily ([13]).

11.1 Preliminaries.

Schwarzenberger in [13] gives a very concrete accounting for the seventeen distinct classes of planar

crystal groups. In the case of n=2, a crystal group G has normal abelian subgroup A ~ Z2 and the

finite quotient D ~ 8(R2) consists of reflections and rotations of the plane R 2. The rotations in

D form a cyclic subgroup Do which must be of order 1,2,3,4,or 6 (known as the "crystallographic

restriction"). Then _a rotation u E D which generates Do may be taken as a rotation about the

origin 0 by an angle of Car~(Do). Fix any reflection p E D. Then the product of p with any other

reflection Pi E D is a rotation ui in D, and so the entire point group D is generated by u and p,

and so one need only find two matrices Mu and Mp to generate the point group D embedded in

M2(R). It is, in fact, consideration of two reflections which happen to generate the rotation u that

gives Schwarzenberger's accounting of the 17 planar groups. As noted above, Bieberbach showed

20

the projection of A into R 2 has 2 linearly independent vectors, say t and s, which may be pick~d

with minimal length since A is discrete and En /G is compact. Then A £!! {it+ js 1: i, j E Z}, the

projection of A into R 2 , which, of course, is isomorphic to Z2 • The following theorem enumerates

the 17 possibilities for plane crystal groups according to the number of reflections in their point

group D.

THEOREM ([13]). Let isomorphism partition the set of 2-dimensional crystal groups. Then

(1) There are exactly 5 equivalence classes of planar crystal groups whose point group D has no

reflections.

(2) There are exactly 3 equivalence classes of planar crystal groups whose point group D has

exactly one reflection.

(3) There are exactly 9 equivalence classes of planar crystal groups whose point group D has two

or more reflections.

In order to get a representative from each planar crystal class, the following characterization of

elements in a crystal group G is given ([13], p.126). Let (<p, v) E G and let <p have order k in D

(identifying D with its image in 6(R2) ). Notice that (<p, v)k = (<pk, v + <pv + ... + <pk-1v) = (1, v + <pv + · · · + <pk- 1v) by the definition of the product, so a := v + <pv + · · · + <pk- 1v is in (the

projection of) the lattice A. Since each pair of elements in <p's A-coset in G differ by an element of

A, this relation between <p, v and A gives valuable information. First notice that this a is fixed by

<p since <p( a) = <p( v + <pv + · · · + rpk - 1 v) = <pv + ... + <pk - 1 v + rpk v = a (since the order of <p is k). So

if <p is a (non-trivial) rotation, then a = 0 is the only point fixed by <p, and if <p is a reflection, say

p, then a E lp, the axis of reflection for pin R 2 • In the case of a reflection, two cases occur. Let a

be of minimal length in {lp nA} such that (p, v)2 = (1, a) (note that reflections are order 2). Either

vis in A (case 1), or it is not (case 2).

1(c) In the first case we have (1,v) E A (and (p,v) E G) and so (p,O) = (1,v)- 1(p,v) E G, so

v = 0 may be chosen, and so (M"', v) is a representative of <p's coset. This case is termed

"centered" .

2 Otherwise, it is termed "primitive", and there are two sub-cases:

2(m) a = v + pv may be chosen to be 0, in which case v E /~. This case is called "primitive with

mirror";

2(g) a= v + pv may NOT be chosen to be 0, in which case v is neither in/~ nor in A. This case

is called "primitive with glide reflection".

These situations are used to get a particular element from each A-coset of G (which will, in turn,

21

generate G) for a representative from each of the seventeen planar crystal groups. The standard

naming convention for the planar crystal groups uses the above cases and the size of the cyclic

subgroup Do of D. For example, p4mg is the group which has 4 elements in Do (so 8 elements in

D) and has both a mirror and glide mirror (hence, two of each). The names of the seventeen groups

are: pl, p2, p3, p4, p6, em, pm, pg, cmm2, pmm2, pmg2, pgg2, p31m, p3ml, p4mm, p4mg, p6mm.

The following information is presented below for a representative from each of the seventeen plane

crystal classes. Two (non-zero, minimum length) linearly independent vectors t and 8 are chosen

as generators for A (~ Z2). To determine the point group D, two matrices are determined for a

reflection p and the rotation u from the group with respect to the basis t, 8. Multlipication in the

point group is tabulated. It is noted here that if two matrices Md1 and Md2 are associated with two

elements d1 and d2, respectively, of D, then the products d1d2 and Md 1 Md2 are also associated; also,

d11 is associated with the matrix inverse Mi 1 (that is, product order is preserved in this notation).

Each element of G can be expressed as (Md, (:))for some dE D and (:) E R 2 (with respect

to t, 8). A cross-section of D is a function ; from D to G such that p o ; = idn. In the case that

; cannot be chosen to be a homomorphism, a set of generators for a fixed D cross-section will be

listed, for then G can be described by {'y( d)( 1, G ) ) I: d E D, i, j E Z} by A-coset decomposition.

Each cross-section generator will be listed in the form ;(d)= (Md, u) for some u E R 2 (with respect

to t,8). Without loss of generality, ;(lv) = (1,0) will always be chosen.

22

11.2 Descriptions of the planar crystal groups.

Group pl is absolutely trivial. It is isomorphic to its lattice: G ~ A.

Group p2.

This group has a parallelogram lattice, so t = (kol) and s = ( ::) may be taken as a basis for A for some

nonzero k1, k2 and k3.

Point Group:

D = {lv, u}

Generators for Mv (with respect tot, s):

[-1 0 ]

Mq = 0 -1

Multi:~ ·,ication in D: lv

lv

0' 0'

, ,... ,- : ,... r ,.,..

~-::_it,;:,:__ .-~.4~~ ...

I

,..-~;,..-~~

~l.A A~ p2 I

Diagram 4.2

0'

0'

D Multiplication Table

Cross-Section is a homomorphism.

23

Group p3.

Here, t = cl:~ ) may be chosen, and then s = <r( t) =

(-lkl) :J , and t, s form a basis for A.

2 kl

Point Group:

D = {In, u, u2 }

Generators for Mn (with respect tot, s):

[0 -1] Mf7 = 1 -1

Multi:,:ication in D:

1n

1n 1n

1n



Diagram 4.3

1n

I I I I

!~ l I

'

24

,.... t -~----~

Group p4.

In this group, there are no reflections, and the rotation

tr is through an angle of ~. So if t is chosen as t = ( ~1 )

then 8 = u(t) = ( ~1 ), and t, 8 is a basis for A.

Point Group:

D = {lD, u, u 2, u 3}

Generators for MD (with respect to t,s):

[0 -1] M(7 = 1 0

~ ~ .... l~ ~ ~ ~

_; __ ~_)]~ ~>~--;-, ~. , ~ ~t, ,. , p4 ~\~ .... ~ ....

I

Diagram 4.4

Multl · ication in D: dl. d2 1D u u2 u3

lD 1D u u2 u3

(T u u2 u3 1D

u2 u2 u3 ln u

u3 q3 ln (T u2



25

26

Group p6.

No reflections, ~nd u is a rotation of f. Thus, t = ( k1

) and s = ( !k1 ) forms a basis for A. 0 , ~k

2 1

Point Group:

D = {1D, u, u 2, u 3

, u4 , u 5 }

Generators for MD (with respect tot, s):

[0 -1] Mu = 1 1

Diagram 4.5

Multi: lication in D: dl-d2 1D (J' (1'2 (1'3 (1'4 (1'5

lD lD (J' (1'2 (1'3 (1'4 (1'5

(J' (J' (1'2 (1'3 (1'4 (1'5 1D

(1'2 (1'2 (1'3 (1'4 (1'5 1D (J'

(1'3 (1'3 (1'4 (1'5 lD (J' (1'2

(1'4 (1'4 (1'5 lD (J' (1'2 (1'3

(1'5 (1'5 1D (J' (1'2 (1'3 (1'4



Group em.

This group is case one. So 3! E 1,., say t = ( ~ ). and

3r E 1;, , say r = ( 2J . Then s = !t + !r is also in A,

and t, s form a basis for A.

Point Group:

D = {lv, Pt}


M,, = [~ !l]

Multi:~ :ication in D: lD

lv lD

Pt Pt

Diagram 4.6

Pt

Pt

lv



27

Group pm.

This is case 2(a) with only one reftection, Pt· t = ( ~1 )

and s = ( ~2 ) generate A for some nonzero k1 , k2 •

Point Group:

D = {lD, Pl}

Generators for MD (with respect to t, s):

M,, = [~ ~1]

Multi:~lication in D: ln

ln

P1 P1

Diagram 4.7

P1

P1

lD



28

Group pg.

This is case 2(b) with only one reflection. As in group

pm, t = ( ~ ) and s = ( 22

) generate A for some

nonzel'o k1 , k2.

For the cross-section, there is avE R 2 with v+p1(v) = t, and so v = ( !~1 ) for some 1:3 . Then, (p1, v) E G.

Point Group:

D = {lv, Pl}

Generators for MD (with respect to t, s):

Mp, = [~ ~1]

Multir lication in D:

Diagram 4.8

lv P1

P1 P1


Cross-Section Generators:

29

Group cmm2.

This group is case one. Since there is only one rotation,

u, through an angle of ?r, then lp1 J. lp,· So there is a

t E 1,., say t = ( ~ ) , and an r E 112 , say r = ( 22

) for

some nonzero k11 k2. Then s = !t + !r is also in A, and t,

s form a basis for A.

Point Group:

D = {lv, u, Pl, P2}


[ -1 0 ] [1 1 ] M (7 = 0 -1 , M Pl = 0 -1

"'--•

• • -:r---t I:-------.ef.

cmm2 • 4

Diagram 4.9

Multi\ 'ication in D: 1v

1D 1v

lD

Pl P1 P2

P2 P2 Pl



Pl

Pl

P2

lD

P2

P2

Pl

1D

30

Group pmm2.

This group is case 2(a) for both reflections. Since there

is only one rotation u through an angle of 1r, then l p 1 1.. l p2 •

So there is at E 1,., say t = ( ~), and an s E 1,,, say

s = ( ~2 ) for some nonzero k1o k2. Then t, s form a basis ·

for A.

Point Group:

D = {1v, u, Pt, P2}

Generators for Mv (with respect to t,s):

[-1 0 ] [1 0 ] M (1 = 0 -1 , M Pl = 0 -1

Diagram 4.10

Mult1"· tication in D: 1v Pt P2

,(f Pt P2

lv P2 P1

Pl Pl P2

P2 P2 Pl



31

Group pmg2.

This group is a combination of cases 2( a) and 2(b). Since

there is only one rotation rr through an angle of 1r, then

lp, .l lp,· So there is at E lp,. say t = ( ~). and an

s E 1,., says= (~2 ) for some nonzero .1:,,.1:2. Then t, s

form a basis for A.

One of the reflections, say p1, exhibits a glide (case 2(b)).

_So,_there is a vector u in R 2 such that u+p1(u) = t. Thus,

u = ( ~~1 ) works here. The reflection 1'2 is then case

2(a), so there is a vector v E R 2 such that v + P2(v) = 0,

and so v = ( ~) works here .. Hence, (p,, u), (p2, v) are

in G, and they generate a cross-section of D.

Point Group:

D = {lD, rr, Pll P2}


[-1 0 ] [1 0 ] Ma = 0 -1 ' MPl = 0 -1

.l ,f. I

) ):t (

'U (

-~------~-~ I

pmg2 I( .,

Diagram 4.11

Multi ication in D: dt·d2 lD (T Pl P2

lD lD (T P1 P2

(T (T lD P2 P1

Pl Pl P2 lD (T

P2 P2 P1 (f' lD



;(p,) = ( Mp,' 0)) ;(1'2) = ( M,., (~) )

32

) (

5 -----.f. ~

(

Group pgg2.

This group is case 2(b) for both reflections. Since there

is only one rotation u through an angle of 1r, then 1 p1 ..L 1 p2 •

So there is at E 1,., say t = ( ~), and an s E 1,,, say

8 = ( ~2 ) for some nonzero 1:, , l:2. Then t, 8 form a basis

for A.

Each of the reflections exhibits a glide (case 2(b )). So,

there are vectors u and v in R 2 such that u + p1 ( u) = t

and v+ p2(v) = s. In fact u = v = ( t:~) may be used

here. Hence, (Pt, u), (p2, v) are in G and they

generate a cross-section of D.

Point Group:

D = {1D, u, Pl, P2}


[-1 0 ] [1 0 ] Mu = 0 -1 , MPl = 0 -1

~ pgg2

Diagram 4.12

Multi ,oication in D: dt·d2 lD q Pl P2

lD lD (T P1 P2

q (T 1D P2 Pl

Pl P1 P2 lD (T

P2 P2 Pl (T lD



'Y(Pt) = MP11 ( (--2~1))

.ff.. 12. I I I

33

Group p31m.

As usual, pick t on 1,., say t = ( ~ ). Then r may

be chosen on l p2 as r = ( ~k1 ) • Then, similarly to the

2 kl centered case, one has s =it+ iris an element of A, and

t, s form a basis for A.

Point Group:

D = {lD, u, u2, Pl! P2, P3}

Generators for Mn (with respect tot, s):

[-2 -1] [1 1 ] M(T = 3 1 ' Mp1 = 0 -1

Diagram 4.13

MultJ;· ication in D: dt ·d2 1n (f (!2 Pt P2

lD In (f (!2 Pt P2

(f (f (!2 ln P2 P3

(!2 (!2 ln (f P3 P1

Pl Pt P3 P2 lD (!2

P2 P2 Pt P3 (f lD

P3 P3 P2 Pt (!2 (f



34

P3

P3

Pt

P2

(f

u2

ln

Group p3ml.

This group is case 2(a). Choose ton I,, say t = ( ~1 ).

Then 8 may be chosen on lp, as 8 = ( !k1

) , and t, 8 2 kl

form a basis for A.

Point Group:

D = {lv, 0', 0'2 , P1, P2, P3}

Generators for Mv (with respect tot, 8):

[ -1 -1] [1 1 ] Mu = 1 0 ' MPl = 0 -1

Diagram 4.14

Multi .. i ication in D: dl -d2 1v (J' (1'2 P1 P2

1v 1v (J' (1'2 P1 P2

(J' (J' (1'2 1v P2 P3

(1'2 (1'2 lv (J' P3 P1

Pl Pl P3 P2 1v 0'2

P2 P2 Pl P3 (J' lv

P3 P3 P2 P1 0'2 (J'



35

P3

P3

P1

P2

(J'

0'2

1v

Group p4mm.

This group is case 2(a). So there is at E lPtt say t = c=~ ). and s = ( ~J E 1 p~ for some nonzero k1. Then t,

s form a basis for A.

Point Group:

Generators for Mv (with respect tot, s):

[0 -1] [ 1 0 ] Mu = 1 0 ' MP1 = 0 -1

Diagram 4.15

Multi;<ication in D: dl ·d2 lv (f (!2 (!3 Pl P2

lv lv (f (!2 (!3 Pt P2

(f (f (!2 (!3 lv P2 P3

(!2 (!2 (!3 lv (f P3 P4

(!3 (!3 lv (f (!2 P4 Pt

Pl Pl P4 P3 P2 1D (!3

P2 P2 Pt P4 P3 (f lD

P3 P3 P2 Pl P4 (f2 (f

P4 P4 P3 P2 Pl ~ (!2



36

P3 P4

P3 P4

P4 Pt

Pl P2

P2 P3

(!2 (f

(!3 (f2

lD u3

(f 1v

Group p4mg.

This is case 2(b ), and A is a square lattice. So one can

choose the basis t = ( ~), and s = UJ for some

nonzero k~, and t, 8 are a basis for A.

Now, for the cross-section, suppose, without loss of gen-

erality, one of the glide reflections is on p1 , so we have c~e

2(b). Then 3v E R 2 such that v+p(v) = t, so v = ( t::) may be used. Then (p1, v) E G. Let r = 8 +t. Since rison . .e'r3

I

lp~, and this is not a glide-reflection axis, we have case 2( a). ' ..... ~ I I I I

s' So there is au E R 2 such that u + p2 (u) = 0, and thus

u E 1,, .L. So u = ( ~) wor~ here, _, and (P2, u) E G.

These two elements of G will generate a cross-section.

Point Group:

. ,,.,. f~,

. ' ' ',,, ~

37

..... ~ fe, ~ ,.,. /,/ 2.

//~ ..... "' ~,

~ ~--.,, __ _

------ ________ :::..JC-----~-- ft. ~ ~ /r', t . • ., ,.. / : ', , ' p4sm /, I ',,

Generators for MD (with respect to t,8):

[0 -1] [ 1 0 ] Mu = 1 0 , MP1 = 0 -1

Diagram 4.16

Multi:· ication in D: dl-d2 lD (f (/2 (/3 P1 P2 P3 P4

lD lD (f (/2 (/3 P1 P2 P3 P4

(f (f (/2 (7'3 lD P2 P3 P4 P1

(7'2 (7'2 (7'3 lD (7' P3 P4 P1 P2

(7'3 u3 lD (7' (7'2 P4 P1 P2 P3

P1 Pt P4 P3 P2 lD (7'3 (7'2 (7'

P2 P2 Pl P4 P3 (7' lD (7'3 (7'2

P3 P3 P2 Pt P4 (7'2 (7' lD (7'3

P4 P4 P3 P2 P1 (7'3 (7'2 (7' lD



7(Pt) = ( M,., (t)) 7(1'2) = ( M, •• (n)

38

. t .fr~ ..(~ '+ /

Group p6mm. ~~~~~~~~ "" -,~i~~i~ ~/ ~~ c) ck,) Case 2(a) applies. t = ~ and s = 4k

1 forms a -~~!~~~~-~~~-~f.

basis for A, for some nonzero k1.

~~~~~:~~~~~I Point Group: ---~ ~ r~:~~ ~ ,_ D _ {l 2 3 4 s } - D, u, u , u , u , u , Pl, P2, P3, P4, Ps, Ps . p6mm / 1 \

~~~l~~~~ Generators for Mv (with respect tot, s): \

Ma = [~ -1] [1 ~1] 1 , MPl = 0

Diagram 4.i7

Multip~ication in D: dl. d2 lD (j u2 (j3 (j4 (j5 Pl P2 P3 P4 Ps Ps

lv lv (j (j2 (j3 (j4 us Pl P2 P3 P4 Ps Ps

(j (j (j2 (j3 (j4 us lD P2 P3 P4 Ps Ps Pl

(j2 (j2 (j3 (j4 (j5 lD (j P3 P4 Ps Ps Pl P2

(7'3 (j3 (j4 us lv (7' u2 P4 Ps Ps P1 P2 P3

.,.4 (7'4 (j5 lD (7' u2 (7'3 Ps Ps P1 P2 P3 P4

.,.s us lD (j (7'2 (7'3 (7'4 P6 Pl P2 P3 P4 Ps

Pl Pl P6 Ps P4 P3 P2 lD .,.s (7'4 (j3 (j2 (7'

P2 P2 P1 Ps Ps P4 P3 (7' lv .,.s (7'4 (j3 (j2

Pa Pa P2 P1 P6 Ps P4 (7'2 (j lD .,.s (j4 .,.a

P4 P4 Pa P2 Pl P6 Ps .,.a u2 (7' lD .,.s (7'4

Ps Ps P4 Pa P2 Pl Ps (7'4 .,.a (7'2 (j lD .,.s

Ps P6 Ps P4 P3 P2 Pl .,.s (7'4 .,.a u2 (j lD



Chapter 5 The C*-Algebra of a Group

with Abelian Normal Subgroup of Finite Index

In this chapter we consider the group C*-Algebra C*(G) of a locally compact group G which has

an abelian normal subgroup A of finite index. There are two aims to this chapter. The first aim,

which is addressed in section I, is to write down an explicit description of c• (G) . This description

turns out to be a simple matrix formula which will be used in the next chapter to calculate the

group C*-algebra C*(G) . The second aim of this chapter is to reduce the problem of irreducible

representations of C*(G) to the finite dimensional C*-Algebra case. The reduction is accomplished

by identifying C* (G) with the full algebra of sections on a certain C*-Bundle and applying the

representation theory for bundles from chapter 3. The second aim is addressed in sections II and

III.

The major theorems and most of the (hard) body pertaining to the first aim of this chapter come

from a recent preprint of [14]. In the paper, an explicit matrix formulation for the C*-Algebra of

any locally compact group G which has an abelian normal subgroup A of finite index is derived. The

major result is the existence of an injective homomorphism {3 from the finite group D:=G I A into

the automorphism group AUT(Mn(Co(A))) (where A is the pontyagrin dual of the abelian subgroup

A, and M0 (Co(A)) is the nxn matrices with entries from C0 (A)) with respect to which C*(G) is

isomorphic to the fixed point algebra M0 (Co(A))0 :={FE M0 (C0 (A)) 1: {3(d)F = F, Vd ED}.

As noted in chapter four, any crystallographic group G fits the exact sequence

A~ G.!!.. D(= GIA)

with A maximal abelian and normal in G, and D finite, so the matrix formulation to be derived for

C* (G) will be applicable to any crystal group. Although the theorems and results would simplify

in the crystal group case (for example A~ zn for some integer n), this chapter will list the required

results in the generality suggested by the title.

! The group C*-algebra C*(G).

Throughout this section G will be a group, A an abelian normal subgroup of finite index in G, and

G I A .!!.. D the finite quotient group. Recall from chapter three the fixed Haar measure on abelian

A, the pontyagrin dual A , and the maps involved in the identification of C* (A) with C0 (A) . Fix

a cross-section 1: D-+ G (that is:p o 1 = idv) which, without loss of generality, fixes the identities

39

(;(ln) = la). Then, A-cosets in G may be denoted ;(d)A. Note that ; may not in general be

a homomorphism. {It is, in fact, the "distance" of; from a homomorphism that determines the

crystal group class of a crystal group G.)

Since A is normal in G, there is a natural action of D on A defined by "pull-back and conjugation":

for d E D, a E A,

d.a: = ;(d)a;{d)-1

= a';(d);(d)- 1 =a'

for some a' E A since A is normal. Note that the above action is independent of the particular

cross-section, for if 6 : D ~ G were another cross-section, then ford E D, 6(d) E ;(d)A by definition

of cross-section, so 6(d) = ;(d)a for some a E A, and so forb E A, 6(d)b6(d)- 1 = ;(d)aba-1;(d)-1 =

;(d)b;(d)- 1 (since A is abelian).

Since ; is a cross-section, ;(de) resides in the A-coset for de in G for each pair d, e E D. Thus

3a(d, c) E A such that ;(d);( c) = ;(dc)a(d, c). It will be shown that a : D x D ~ A is a 2-

cocycle ( cohomologically speaking) and that G ~ { ( d, a) I: d E D, a E A} when given group product

(d, a)(e, b) := (de, a(d, c)(e- 1 .a)b). Note that we already have G ~ {;(d)a 1: dE D, a E A} by coset

decomposition.

With the above action of Don A, the Haar measure on G will be built from that of A. Equipped

with an explicit connection between the measures on A and G, one can form L 1(G) , L 2(G) , and

ultimately connect C*(G) with M0 (C0 (A)) . But first, Gin coset-decomposition form is identified

with the above alternative form.

PROPOSITION. With the above product, let G' .- {(d, a) 1: d E D, a E A}. Then G' ~ G :=

{;(d)a 1: dE D,a E A} via t.p(d,a) := ;(d)a.

PROOF: Let (d,a),(e,b) E G'. Then for homomorphism note

(d,a)(c,b) = (de,a(d,c)(c- 1 .a)b)

.:!!.. ;(de)a(d, c)(c-1.a)b

= ;(de)a(d, e)a(c- 1 , e)- 1a(c-1 , e)(c- 1 .a)b

(A commutative)= ;(de)a(d, c)a(c-1 , c)- 1(e- 1 .a)a(e- 1 , e)b

(definition of a) = ;( d);(e)a( d, c)- 1a( d, c);{ c)-1;( c- 1 )- 1 (c- 1.a);(c-1 );(e)b

(defn of D-action) = ;(d);(e- 1)- 1;(e-1)a;(c- 1)- 1;(e- 1);(c)b

= ;(d)a;(e)b

= t.p( d, a )t.p( c, b) as required.

40

For injection note that if <p( d, a) = <p( c, b) then "Y( d) a = "Y( c )b, so by A-coset decomposition of G,

"'f(d) and "Y(c) are in the same coset, so d = c by projection, and then a=b. So (d, a) = (c, b).

Surjection is obvious.

PROPOSITION. The function a : D x D -+ A satisfies the 2-cocycle identity:

a(b, cd)a(c, d) = a(bc, d) d- 1.a(b, c)

for all b, c, dE D (using the above D-action on A).

PROOF: Simply calculate:

a(b,cd)a(c,d) := "'f(bcd)- 1r(b)i(cd)i(cd)- 1"Y(c)"Y(d)

= "'f(bcd)- 1"'1( bc)"'f(bc)- 1"Y(b )"Y( c)"Y( d)"Y( d-1 )"Y( d- 1 )-

1

= -y(bcd)- 1"Y(bc)a(b, c)a(d, d-1 )"Y(d- 1 )-

1

( commutivity in A) = i(bcd)- 1r(bc)a(d, d-1 )a(b, c)"Y(d- 1 )-

1

= "'f(bcd)- 1"Y(bc)"'f(d)"Y( d-1 )a(b, c)r(d- 1 )-

1

=a( be, d) d- 1 .a(b, c)

Thus A, D, the action of Don A, and the 2-cocycle a determine G (upto isomorphism). The no

tation used for G elements will be which ever is most convenient for the purpose. Before introducing

the Haar measure on G, a preliminary lemma is required.

LEMMA. The action of Don A does not affect the Haar measure on A. That is, if IE Coo( A) (the

continuous functions of compact support A-+ C) and dE D, then fA l(d.a)da =fA l(a)da.

PROOF: Define the function J : C00 (A) -+ C by J(l) := fA l(d.a)da (VI E Coo(A)). Then for

I E Coo(A) and any b E A, the translate by b is defined by /b(a) := l(ab) (Va E A). We have

lb E Coo(A), too, and

J(/b) := L t.(d.a)da = L f((d.a)b)da

= L l(d.(a(d- 1 .b)))da (by group-action,)

= L f(d.(a))da (by translation-invariance of Haar on A,)

= J(l)

so J is translation-invariant. J is obviously linear and positive, hence a positive invariant lin-

ear functional on C00 (A). So by the uniqueness of Haar measure on A, 36(d) > 0 such that

41

6(d) fA l(a)da = J(l) = fA l(d.a)da. 6 is a homomorphism of D into (R+, ·) since for any

I E Coo(A), 6(c)6(d) fA l(a)da = 6(c) fA l(d.a)da = fA l(cd.a)da = 6(cd) fA l(a)da. But since

Dis finite, 6(d) = 1 Vd ED. Thus fA l(d.a)da =fA l(a)da, as required.

For any I: G--+ C and dE D, define ld: A--+ C by ld(a) := I("Y(d)a). The Haar measure on G

is given by

f l(z)dz := L 1 ld(a)da. jG dED A

PROPOSITION. The above equation defines a left and right translation-invariant Haar measure on

G.

PROOF: Left-invariance is shown forgE G written g = 1(c)b for some c E D,b EA.

f l(gx)dz := L 1 I("Y(c)b"'f(d)a)da jG dED A

(defn of a)= L 1 l("'f(C)"'f(d)"'f(d- 1 )a(d,d-1)-

1b"'((d)a)da dED A

(commute in A)= L 1 l("'f(C)"'f(d)"Y(d-1 )ba(d,d-1)-

1"Y(d)a)da dED A

(defn of a)= L 1 I("Y(c)"'f(d)"'f(d- 1 )b"'f(d- 1)-

1a)da dED A

(defn of D-action and a)= L 1 I("Y(cd)a(c, d)(d- 1 .b)a)da dED A

(chng var: a'= a(c,d)(d- 1 .b)a) = L 1. I("Y(d)a)da dED A

=: Ll(z)dz

showing left-in variance. Right-invariance is similar.

Let U : L2 (G) --+ LdeD EBL2 (A) be defined by U(h) := (hd)deD for h E L2 (G). By above

definition of Haar measure on G, each hd E L2 (A).

PROPOSITION. U: L2(G)--+ LdeD EBL2(A) is an isometric isomorphism.

PROOF: Linearity, injection and surjection are trivial. To show U is unitary, let h E L2(G). Then

by definition of Hilbert direct-sum norm (2.1) IIU(h)ll~ = LdeD llhdll~ = LdeD fA I h(1(d)a) 12

da = fG I h(z) 12 dz =: llhll~ by definition of Haar measure on G.

In view of the above U and the isometry P : L2 (A) ~ L2(A) from 3.111, we now have the isometry

\II of L2 (G) and 1i := LdeD EBL2(A) given by \11(1) := (P(Id))deD for each IE L2 (G). Each element

of LdeD EBL2 (A) (or LdeD EBL2(A) ) can be considered as a vector b.:= (hd)deD for hd E L2 (A)

(or L 2 (A)).

42

As usual, let n be the number of elements of D, and let Mn(Co(A)) be matrices with entries from

Co(A) . It will now be shown that C*(G) is embedded onto a C*-subalgebra of Mn(Co(A)) . As

mentioned in chapter three, C* (G) is determined by the left regular representation ,\ G of L 1 (G)

on L2(G) since G is amenable. That is, C*(G) ~ C~(G) = B(L2(G))-closure of ,\G(L1(G)), the

reduced C*-Algebra of G.

Mn(C) with the operator norm (on C") is a C*-Algebra. Each FE Mn(Co(A)) can be written F =

(Fc,d)c,dED with each Fc,d E Co(A). Then Mn(Co(A)) is naturally identified with Co(A, Mn(C))

which is a C*-Algebra, so the norm on Mn(Co(A)) given by IIFII := supiiF(a)llmakes it a C*-Algebra. ae.A

PROPOSITION. M : Mn(Co(A)) -+ 8(1i), which, for F E Mn(Co(A)) and fl E 1i is given by

[M(F)l!Jc := LdeD Fc,dhd (for each c ED), is an isometric embedding.

PROOF: If M(F)fl = 0 for all h. in ?i, then each component ofF must be zero, so F = 0, and so M

is one-to-one. Involution ofF is the usual transposition and then conjugation of each component.

Using the definition of inner product on a Hilbert direct sum, one shows by rearranging summations

that< M(F)*fl,! >=< M(F*)ll,! >for any twoelementsfland!of?i. So M preserves involution.

Since it is also a homomorphism, then, by 2.1, M is an (isometric) C*-isomorphism onto its range.

So Mn(Co(A)) is isomorphic to a C*-subalgebra of 8(1i) via M. L 1 (G) is now shown to be

embedded in the range of M in the following proposition, and this embedding leads to a natural

embedding of C*(G) in Mn(Co(A)) as well.

PROPOSITION. For f E L 1(G), w..\yw-1 E M(Mn(Co(A)))

PROOF: Let f E L 1(G). First we compute the effect of ,\Y on an element h E L 2(G). As before,

/d E L 1(A), hd E L 2 (A) where now we are using the alternate notation for G, so /d(a) := f(d, a),

and hd(a) := h(d, a) (a.e. on A). Note the following (long) calculations for the G-product of g- 1 h

for g, h in G:

(c, a)- 1 = (c- 1 , a(c, c- 1 )- 1(c.a- 1 )) = (c- 1 , (c.a(c-1 , c)- 1 )(c.a-1 )), so

( c, a)- 1(d, b) = (c- 1, a(c, c-1 )-1(c.a-1 )(d, b)) = (c- 1d, a(c- 1, d)(d- 1 .(a(c, c- 1 )- 1 (c.a- 1 ))b) and

d- 1.(a(c,c- 1)- 1(c.a- 1)) = [d- 1.a(c,c-1)]- 1(d-1c.a- 1) and the 2-cocycle identity then implies

[d- 1 .a(c,c-1)]- 1 = a(c,c- 1)- 1a(c- 1,d)-1 (by noting that a(ln,x) = lA Vx ED). In short, one

has

43

For I and h from above, one then has, for each d E D and almost all b E A,

(>.7 h)d(b) ==<I. h)(d, b) == L l(y)h(y- 1(d, b))dy

(defn ofHaar on G)= I: f l(c,a)h((c,a)- 1(d,b))da cEDjA

(by above calculation)= I: 1 l(c, a)h(c- 1d, a(c, c-1d)- 1(d-1c.a-1 )b)da cED A

(let a'= c-1d.a,which does not affect Haar measure on A by above lemma)

= I: { lc(c- 1d.a)hc-td(a(c, c- 1d)a- 1b)da ceDJA

(let a'= aa(c, c- 1d)) = L f lc(c- 1d.a(c, c-1d)- 1a)hc-ld(a- 1b)da cED }A .

= L { 9c,d(a)hc-td(a- 1b)da ceDJA

=: L(9e,d * hc-ld)(b) cED

where 9c,d(a) := lc(c- 1d.(a(c, c- 1d)- 1a)) defines (a.e. on A) an element of L1 (A) . So

('l!>.}h)d = ('1!(1 * h))d := P(l * h)d = P L 9c,d *he-ld

(properties of P) = L Uc,dP(hc-ld) cED

(let c' = c- 1d) = L Udc-I,dP(hc) · cED

cED

From 3.III we have each Udc-l,d E Co(A) and P(hc) E L2 (A) and so 'l! >..7 h E 11,. Thus with

F = (Udc-t,d)dc-t,deD,! E 1l such that h = w- 1(!), we have w>..r'l!- 1/i = 'll>..7h = M(F)£ so

w>..r'l!-1 is in the range of M.

In view of the above, since we have the injection L1(G) -+ M (Mn(C0 (A))) ~ 8(1l) by I~

'l!>..f'l!-1, and since M : M0 (C0 (A)) -+ 8(1l) is an embedding, then the map :F : L1(G) -+

M0 (C 0 (A)) that pulls back the image of L1 (G) by :F(I) := M-1('l!>..f'l!- 1 ) can be extended

. to an isomorphism of C*(G) into M0 (C0 (A)) since M is actually a C*-isomorphism. Thus we

have C*(G) connected with M0 (C0 (A)) as desired. In the following, the map :F and its range in

M0 (C 0 (A)) is made explicit.

The D-action on the abelian subgroup A induces an action on the dual A: for dE D, X E A, let

d.x be the map defined on A by

d.x(a) := x(d- 1 .a) Va EA.

44

Then d.x: A---+ Tis easily shown homomorphic by using the action of Don A, and so the properties

required for a group action are easily verified. One property of this action which will be used later

is d.x(a-1) = x(d-1.a-1) = x((d-1.a)-1) = x(d-1.a) = d.x(a). With this D-action on A, the image

of L1(G) in Mn(Co(A)) has a simple description.

PROPOSITION. For f E L1(G), :F(f) (E Mn(Co(A)) by above) has (c, d)-entry (for c, dE D) given

by

Vxe.A.

PROOF: From the above proof [:F(/)]e,d = 9ed-1,d with Ued-l,d E L1(A) given by

Ued-l,d(a) = /ed-l(d.(a(cd-1,d)-1a)) Va EA. Now for each X E A,

9ed-l,e(X) := L Ued-l,e(a)x(a)da

= L /e•-•(d.(o(ctl 1, d)- 1a))x(a)da

(Let a' = a(cd- 1, d)- 1a) = L !ed-1 (d.a)x(a(cd-1, d)a)da

ex a hom)= x(a(cd_,. d)) L 'e·-· (d.a)x(a)da

(Let a'= d- 1.a) = x(a(cd- 1 ,d)) L !ed-1(a)x(d- 1 .a)da

= x(a(cd- 1 ,d)) L ,ed-l(a)(d.x)(a)da

=: x(a(cd- 1, d))ied-l(d.x)

Let p be the right-regular unitary representation of the group G on L 2( G) given, for each g E G,

by p(g)f(x) := f(xg) (Vf E L2(G), x E G). For each dE D, define U(d) := 'lip("Y(d))w- 1 • Then

U (d) : 1-l ---+ L 2( G) ---+ L 2 (G) ---+ 1-l is a unitary transform of 1-l since w and p(g) (for any g E G) are

unitary transformations.

PROPOSITION. For each dE D, the map

f3(d)F := M- 1(U(d)M(F)U(d)*) (VF E Mn(Co{A)))

defines an injective homomorphism D---+ AUT( M 0 (C0 (A)) ).

PROOF: First, the range of M (=M(Mn(Co(A)))) in 8(1-£) will be shown invariant under conju

gation by U(d) for any d E D. Fix d E D. For l! = (he)eeD E 1-l, let h E L 2{G) be such that

'li(h) = [P(he)]eeD = (he)eeD· Note that if he E L 1(A) n L 2(A), then P(he) =he. Then forb ED,

45

and x E A, note the following four items.

(1) For a E A,[P("Y(d))*h]b(a) := P("Y(d- 1))h(b,a)

= h((b,a)(d-1 ,a(d,d-1)-1 ) (by inverse in G)

= h(bd- 1 , a(b, d-1)(d.a)a(d,d-1)- 1)

= hbd-t(a(b,d- 1 )a(d, d-1)-1(d.a)) (a.e. on A)

(2) The action of U(d)* on fl.:

[wp("Y(d))*h],(x) = P((P("Y(d- 1))h)t,)(x)

= ([P("Y(d- 1))h]b}(X)

:= i [P("Y(d- 1))h]b)(a)x(a)da

(by (1)) = i hbd-t(a(b,d- 1)a(d,d-1)-

1(d.a))x(a)da

(Let a= d- 1.a') = i hbd-' (Ot(b,d- 1)0t(d, d- 1)-

1a)x(d-1.a)da

(Let a= a(b,d- 1)-

1a(d,d-1 )a' and recall xis a hom)

= i hbd-' (a)x(d- 1 .(01(b, d- 1 )-

101(d, d- 1 )))x( d- 1.a)da

= (d.x)(a(b, d- 1 )-

1a(d, d- 1 ))hbd-• (d.x)

= (d.x)(a(b, d-1 )-1a(d, d-1 ))&d-• (d.x)

= [w p( "Y( d))* q,- 1 hh(x)

(3) Now for [U(d)h]b(x), note for a.a. a E A:

(P("Y(d))h)b(a) := P("Y(d))h(b, a)

= h((b, a)(d, 1))

= h(bd,a(b,d)(d- 1 .a))

= hbd(a(b,d)(d- 1 .a))

46

= ['lfp("'f(d))h]b(X)

= ([P("Y(d))h]b)(X)

= L hbd(a(b, d)(d- 1 .a))x(a)da (by 3)

(Let a= d.a') = L hbd(o:(b, d)a)x(d.a)da

(Let a= a(b, d)-1a') = L h6d(a)x(d.( a(b, d)-'a))da

= x(d.a(b, d))hbd(d- 1.x)

= (d- 1.x)(a(b, d))fh]bd(d- 1.x)

Putting the above together we calculate conjugation by U(d). Let F = (Fe,d]c,dED E Mn(Co(A)).

Then

[U(d)M(F)U(d)*l!]b(x) = (d- 1.x)(a(b, d))[M(F)U(d)* l!]bd(d- 1.x) (by 4)

(defn of M) = (d- 1.x)(a(b, d))(L(Fbd,c)[U(d)*lt]c(d- 1.x) cED

cED

cED

cED

= L(d- 1 .x)(a(b, d)- 1a(c, d))Fbd,cd(d- 1 .x)fh.1c(X) cED

So for appropriate G E Mn(C0 (A)) defined in terms ofF and the factors in the above expression,

one sees conjugation by U(d) leaves the range of M in 8(1i) invariant. Then the action of f3(d) is

seen to be, for F = (Fb,-ch,ceD E Mn(Co(A)),

since M is a C*-isomorphism.

To show f3 is a homomorphism, pick y, z E D, x E A, calculate f3(yz) and f3(y)f3(z):

[f3(yz)F]b,c(X) = (yz)- 1 .x(a(b, (yz))- 1a(c, (yz))) Fb(yz),c(yz)((yz)- 1 .x)

[f3(y)f3(z )F]b,c(X) = (y- 1.x(a(b, y)- 1a(c, y)))((yz)- 1 .x(a(by, z)- 1a(cy, z))) Fbyz,cyz((yz)- 1 .x)

47

and f3 is a homomorphism if these two components are equal. Note that it suffices to show equality

of the scalar factors in these expressions since the matrix positions are the same for each b, c. Using

the definition of D-action on A, it suffices to show:

y.(a(b, y)- 1a(c, y))(yz).(a(by, z)- 1a(cy, z))

= "Y(Y)( a(b, y)- 1a(c, y))"Y(Y)- 1"Y(Y)(z .(a( by, z)- 1a(cy, z)))"Y(Y)- 1

= "Y(Y)(a(b, y)- 1(z.(a(by, z)- 1a(cy, z)[z- 1 .a(c, y)]))"Y(Y)- 1

= "Y(Y)"Y(z)(z- 1 .a(b, y)- 1 )"Y(z)-1"Y(z)a(by, z)- 1a(cy, z)[z- 1 .a(c, y)]"Y(z )-1"Y(Y)-1

= "Y(Y)"Y(z)[a(by, z)(z- 1 .a(b, y)]- 1[a(cy, z)z- 1 .a(c, y)]"Y(z)- 1"Y(Y)-1

= "Y(Y)"Y(z)[a(b, yz)a(y, z)]- 1 (a(c, yz)a(y, z)]"Y(z)- 1"Y(Y)- 1 (apply the 2-cocycle identity)

= "Y(Y)"Y(z)a(y,z)- 1a(b,yz)- 1a(c,yz)a(y,z)"Y(z)- 1"Y(Y)- 1

= "'f(yz)a(b, yz)- 1a(c, yz)"'f(yz)-1"Y(Y)- 1 (by defn of a)

= (yz).(a(b, yz)- 1a(c, yz))"Y(yz)- 1"Y(Y)- 1 , so the arguments to x are equal in the two expressions,

as required. To show f3 is monomorphic, note that for any non-identity dE D, f3(d) will move the

non-zero entry of the standard basis element e6,c to the position (bd, cd) which will be the same as

the position (bd', cd') for {J(d') iff bd = bd'; that is, iff d = d'. So f3 : D ~AUT( Mn(Co(A)) ) is an

injective homomorphism.

PROPOSITION. Using the above monomorphism f3 : D ~AUT( M0 (C 0 (A)) ), the fixed point alge-~ D ~ ~

bra M0 (C0 (A)) :={FE M0 (C0 (A)) 1: Vd ED, {3(d)F = F} is a C*-subalgebra of M0 (Co(A)) and ~ D

the :F extension defined above is a C*-isomorphism of C*(G) with Mn(Co(A)) .

PROOF: First note that the representations p and Aa commute; that is, for IE L1(G) and g E G,

p(g)AY p(g)* = Al. This is because p(g) is unitary for each g e G, and so, for hE L2 (G) and x E G,

(p(g)Ayh)(x) := (Ayh)(xg) := (I* h)(xg) := fa l(u)h(u- 1xg)du = fa l(u)(p(g)h)(u- 1x)du =

1 * (p(g)h)(x) = Al p(g)h(x).

Recall :F(I) := M- 1(wAyw- 1) for IE L1(G). Then for any dE D we have

{3(d):F(I) := M- 1(U(d)M(:F(I))U(d)*)

= M- 1(wp("'f(d))w- 1 M(:F(I))wp("'f(d))*w- 1)

= M- 1(wp("'f(d))w- 1 M(M- 1(wAyw- 1))w P("Y(d))*w-1)

= M- 1('1tp("'f(d))Ay p("'f(d))*w- 1)

= M- 1(wAyw- 1 ) (by above commutativity)

= :F(I)

~ D so :F(I) E Mn(Co(A)) whenever IE L1(G).

48

To show .1"(L1(G)) is dense in M0 (C0 (A))D and, therefore, that the extended .1" maps C*(G) 0 1!!'0

Mn(Co(A))D, we approximate a given, fixed F E M0 (C0 (A))D within a given f > 0 by an image

point of .1". Recall that the C*-norm for F is given by IIFII := supaeA IIF(a)ll· Choose 6 > 0 such

that IIG- Fll < (whenever G e Mn(Co(A)) is such that IIGb,e- Fb,elloo (on A) < 6 (Vb, c e D).

Now for each bED, choose/bE L1(A) such that llfb- Fb,lolloo < 6. Then f E L1(G) by above, A D

where f(b, a) := /b(a) (b E D, a E A). Then .1"(/) e Mn(Co(A)) , and for each b, c E D, note

{3( c ).1"(/) = .1"(/) and /3( c- 1 ).1"(/) = .1"(/), so for x E A,

=I (c.x( a(b, c- 1 )-

1a(c, c- 1 ))x(a(bc- 1, cc- 1 )Jbc-lcc-l (c.x)

- (c.x(a(b, c- 1 )-

1a(c, c-1 ))x(a(bc- 1, cc- 1 )Fbc-l,cc-1 (c.x) I

49

=I (c.x(a(b, c- 1)-

1a(c,c- 1)) II (!be-l - Fbc-l,lo)(c.x) I (since x(a(bc- 1

' cc- 1)) = 1)

:5llfbc-t - Fbe-l,lo)lloo < b (by assumption),

so IIF{/) - Fll < f. Thus any element of M0 {C0 {A))D can be approximated arbitrarily close

in .1"(L1{G)), so .1"(L1(G)) is dense in this fixed point algebra, and the extension of .1" is a C*

isomorphism of C*(G) and M0 (C0 (A))D.

The above proof gives the explicit characterization of C*{G) in Mn(Co{A)) as those elements

consisting of n arbitrary C 0 (A) elements {Fd,lo 1: de D} in the ''first" column (or lD-column),

and the other columns determined by the formula for the {3-invariance,

Fb,e(X) = [f3( c-1 )F] b,e (x)

= c.x(a(b,c- 1)-

1a(c,c- 1))Fbc-l,lo(c.x) Vx e A

which will be used in the sequel to describe C*(G) for each planar crystal group.

II The identification C*(G) ~ r o(E).

The second aim of the chapter will now be addressed. C* (G) will now be tied to the space of

sections r(E) on a certain C*-Bundle E . First, some preliminary notation and definitions must

be given.

Let A/ D denote the space of D-orbits of A given by the D-action on A with the quotient topology.

II.l The tilde bundle E.

DEFINITIONS. Recalling the bundle definitions from 2.111, we further define the following terms.

(1) A D-C*-Bundle E = (p, E, X) is a group D and C*-Bundle E together with aD-action onE

and X such that the d-action for each dE D forms a C*-Bundle map (over the homeomorphic

d-action on X).

(2) For a D-C*-Bundle E = (p, E, X), a section s is D-equivariant (or just equivariant, if D is

understood) if d.s(z) = s(d.z) (Vz EX).

(3) Let r~(E) := {s E fo(E) 1: sis D-equivariant}.

Recall A, A, D, and G from section I above, and recall B(L2(D)) ~ Mn(C) (where n =Card(D)).

Let E be the trivial C*-Bundle over A:

E = (p, E, A), where E := Mn(C) X A

and p(N, a) := a for each element of E. Note that

r o(E) = {s: A-+ Mn(C) X A 1: sis continuous, vanishes at oo, p(s(a)) =a}

~ {s: A-+ Mn(C) 1: sis continuous, vanishes at oo}

So we can now identify elements of the cross-sectional C*-Algebra r o(E) with elements of the

C*-Algebra M0 (C0 (A)) . With a certain D-action on E , this identification will tie together

where E is a C*-bundle which will be defined below. The bijection r o(E) +-+ Co(A, Mn(C)} is

accomplished by projection onto the first factor. Given s E r o(E) define s' E Co( A, Mn(C)) by

s'(a) := proj1(s(a)). Conversely, givens' E C0 (A, Mn(C)), define s(a) := (s'(a), a). This bijection

is a C*-isomorphism.

FordED, define the d-action onE as follows. Let A E M0 (C), a E A, and so (A, a) E E. Then

d.( A, a):= (Ad, a, d.a)

where Ad,a E M0 (C) is defined by

(for each pair b, c E D)

which is easily seen to beaD-action on E by the D-action on A. For convenience, denote Ad,a by

d.A when a is understood.

50

CLAIM. The above D-action makes the C*-bundle E into a D-C*-bundle.

PROOF: Obviously the following commutes (set-wise) for each dE D:

d E -----+ E

A -----+ A d

It will first be shown that each fibre map d : Mn(C) --+ Mn(C) is an isometric isomorphism for a

fixed a. (Actually, d: Ea --+ Ed.a-) Since the action of don A E Mn(C) simply permutes entries of

A then twists each entry by an element ofT (each of whose elements have modulus 1) depending

only on d, a, and matrix position, then d does not change the norm of A (as an operator on C")

so the action is clearly an isometry, and also injective since only 0 permutes to 0 in Mn(C) , and

surjective since a basis is permuted to another basis of Mn(C) . So it is an isometric transformation

of Mn(C) (since n is finite).

Now that the action of d has been shown to be a fibre-wise C*-map, we must establish continuity

of d as a bundle map. Simply recall E = Mn(C) x A has the product topology and so the continuity

of the component functions of d : (A, a) 1-+ (d.A,d.a) establishes continuity of don the product

space.

Thus E is a D-C*-Bundle.

Construction of E. With D-C*-Bundle E above, a new bundle E is constructed such that fo(E) ~ r~(E), the

D-equivariant sections of E . r o(E) will then be shown C*-isomorphic to C*(G) . With this

in mind and following [3] ( ch. 2), a bundle F is first constructed as "the part of E seen by

equivariant sections".

For each a E A, let Da := {dE D 1: d.a = a} be the isotropy subgroup of D for a, sometimes

called the stability subgroup.

Define

Fa:= {bE Ea 1: b is fixed by Da}

={(A, a) 1: A E Mn(C) and d.A = A(Vd E Da)}

as those elements in a's fibre which are fixed by a's stability subgroup. Notice that for a fixed a E A

and any equivariant section s E r~(E) and d E Da, one has d.s(a) = s(d.a) = s(a), and so Do.

actually fixes each such fibre element s(a) E Ea for all equivariant sections s E r!'(E). Thus, Fa

contains the point-evaluation of r~ (E) at a.

51

CLAIM. Fa is a C*-subalgebra ofEa (for any fixed a E A).

PROOF: Simply check. To show it is a linear subspace, note 04 is fixed by Da so Oa E Fa. For a E C

and b, c E Fa and dE Da we have (since dis a C*-map on fibres of E ) d.(ab +c) = d.(ab) + d.c

= a( d.b) + d.c = ab + c since b and c are fixed by Da by definition. So d fixes linear combinations

from Fa. One shows in a similar fashion that Fa is a *-subalgebra. For multiplicative closure pick

b, c E Fa and dE Da. Then d.(bc) = (d.b)(d.c) =be. For involution note d.(b•) = (d.b)• = b•, again

because d is a C*-mapping.

It remains to show Fa is closed in Ea. If (bi) is a sequence of elements from Fa convergent to,

say, b E Ea and d E Da, then d.bi = bi -+ b and d.bi -+ d.b by continuity of d's action and so d.b = b

since E is a bundle, and, thus, bE Fa as required.

Define the set

CLAIM. D.F = F

F := U Fa. a eA.

PROOF: First, note the following relation for isotropy subgroups which is valid for any group action

H on a set X: for hE H, x EX

Hh.z = {k E H 1: k.(h.x) = h.x}

= {k E H 1: h- 1 .k.h.x = x}

= {hkh- 1 E H 1: k.x = x}

= hH z h - 1 (that is, just conjugate).

Now, for a fixed a E A and bE Fa and dEDit is sufficient to show Dd.a(d.b) = d.b. For any c E Dd.a

the above relation gives c = dgad- 1 for some ga E Da. Then c.(d.b) = (dgad- 1 )d.b = dga.b = d.b

since ga E Da and bE Fa. Thus d.b E F d. a· The reverse inclusion is trivial (using the trivial action

lv).

Give F the subspace topology from E and induced action of D. Then, restricting the projection

of E to 1r : F -+ A makes F = ( 1r, F, A) a bundle. Finally, define the "quotient bundle"

E := F/D

with the quotient topology; two points b, c E F are identified iff d.b = c for some d E D. 1r IS

D-equivariant (since each d-action is a bundle map on all E ) so there is an induced projection

1f: E-+ A/ D. Let

E := (i,E,A/D).

52

Elements of E will be denoted [b] (where b e F) and elements of A/ D will be denoted by [a]. Letting

~ A~ b fix d d · h 10 Quotient D F /D h b"" · b t a E e e an usmg t e set maps Fa ~ D.Fa ---+ . a , one as a IJectiOn e ween

E[a] and D.Fa/ D. The injective map f{)a : Fa !:! E[a] defined by the above bijections is also onto

and so we use tpa (for each a) to carry the C*-structures from F to the fibres of E. Operations in

these quotient fibres are the usual coset operations.

CLAIM. The C*-algebra operations on the fibres E[a] are well-defined.

PROOF: Fix [a] E A/ D. Addition will be shown. Scalar multiplication is similar. For two elements

[b], [c] E E[a] take any two representatives b, b' E [b] and c, c' E [c] such that 1r(b) = 1r( c) and

1r(b') = 1r(c') (so addition will make sense). We need to show b + c = d.(b' + c') for some dE D.

Without loss of generality one may assume 1r(b) = a. But b = d.b' for some dE D since they are

D-equivalent, and so c = d.c' since d carries fibres to fibres in F and c, c' are also equivalent by

assumption. Thus b + c = d.b' + d.c' = d.(b' + c') since dis a C*-map of Fa~ Fd.a·

The well-definedness of the norm for any [b] E E[a] is obvious since, by definition, ll[b]ll := llbll for

any bE [b], and any dE Dis an isometry of Fa and F d. a.

Involution and product in E[a] are similarly shown well-defined by using the C*-maps defined by

each dE D: if b' = d.b and if c' = d.c then d.(b*) = ( d.b )* = (b')* shows the involution defined by

[b]* := [b*] works and d.(bc) = ( d.b )( d.c) = b' c' shows the multiplication defined by [b][c] := [be] also

works.

THEOREM (FOLLOWING (3], 2.5). With the above notation, E := (7r, E, A/ D) is a C*-Bundle over

A/D. Further, the cross-section C*-algebra r o(E) on the quotient is isomorphic to the D-equivariant

sections ofE (that is, r o(E) ~ r!'(E)).

The proof of the above theorem is rather long. One first verifies that E is a C*-bundle. In order

to show this, two lemmas are required.

LEMMA 1. For a point a E A and a point b e Fa, define the equivariant function s' on the orbit

D.a of a : s'(d.a) := d.s'(a) and s'(a) =b. Then there is an equivariant section s on all E which

extends s'.

PROOF: Since the original E is a C*-bundle, 2.111 shows that E has enough sections. Let r be a

section which paSses through the given b: r(a) = b = s'(a). Let #Da be the size of the isotropy

subgroup Da. Since A is locally compact, one can use Urysohn's lemma to pick a continuous function

of compact support f: A~ [0, 1] such that f(a) = 1 and f(d.a) = 0 for all dE D \Da (a finite set).

53

Let fr denote pointwise product of these two functions. Define

s(x) := #~· 2: d.(fr)(d- 1 .x) a dED

for each x E A.. s(x) defines an element of E.t for each x . s is obviously bounded and contin

uous since r and f are, and the sum is finite. Further, let c E D and x E A. Then c.s( x) =

c.#k LdeD d.(fr)(d- 1 .x) = #1. LdeD cd.(fr)(d-1.x). Let d' = cd, then d = c- 1d' and d- 1 =

(d')- 1c. Substituting, c.s(x) = #baLdeD d.(/r)(d- 1 .c.z) = s(c.z) by definition. Therefore, s is

D-equivariant. Finally, /(d-1 .a) is one iff dE Da and it is zero iff dE D \ Da. Thus, s(a) simply

adds b up #Da-times, then divides out this same number, and so s(a) =bas required.

LEMMA 2. For a net ( ba) in F such that

(1) aa := 11"(ba) ___.a in A., and

(2) [ba]---+ [b] in F/D(= E) (sob E Fa wlog)

we have ba ---+ b in F .

PROOF: Define s'(a) := b, and for each d E D define s'(d.a) := d.s'(a). Note that this is well

defined since if d.a = c.a then c- 1d E Da and so c- 1d.s'(a) = s'(a) since s'(a) E F, and finally

d.s'(a) = c.s'(a). Use Lemma 1 above to get an equivariant sections onE which agrees with s' on

D.a. For each e > 0 define an open neighborhood Wf of b = s(a) = s(11"(b)) as follows:

Wf :={wE F 1: llw- s(11"(w))ll < E},

So the image Wf/ D in E is open. So, given e > 0, [ba] is eventually in every neighborhood of

(b), and especially in Wf/2/ D. But Wf/2 ~ F and if w E Wf/2 then equivariance of s and 11" gives

d.w- s(11"(d.w)) = d.w- s(d.11"(w)) = d.w- d.s(11"(w)) = d.(w- s(11"(w))). Since the action of dis

norm-preserving on F, we then have lld.w- s(11"(d.w))ll = lld.(w- s(11"(w)))ll = ll(w- s(11"(w)))ll <!

by definition of Wf/2 • Thus d.w E Wf/2 shows Wf/2 is a D-invariant set. So once any representative

ba of (ba] is in Wf, then all its representatives are in Wf. In particular for our given net, ba E Wf/2

for large enough a. Therefore llba - s(aa)ll = llba - s(11"(ba))ll < ! for large enough a. Also

s(aa)---+ s(a) =band so lls(aa)- s(a)ll <!for large enough a. Thus we finally have llba- bll =

llba- s(a)ll ~ llba- s(aa)ll + lls(aa)- s(a)ll < ! +! = f for sufficiently large a. That is, ba ---+b.

PROOF THAT E IS A C*-BUNDLE:

Continuity of norm' and operations must be verified as well as the "zero-limit" property.

( 1) llilli!l: The norm on F is continuous. The quotient map F -+ F / D ( = E) is open and the

norm on E is directly induced from the norm on F. Therefore any open set in R pulls back

54

to a set in F / D whose preimage in F (via the quotient map) is open. Thus the set is open in

F / D = E. Hence the norm is continuous on E. (2) scalar product: Let Ai -+A in C and [bi]-+ [b] in E both be nets over the same index set I.

Then Ai[bi]-+ A[b] needs to be shown for continuity. Again, note that scalar multiplication is

defined by A[b] := [Ab]. Now, since [bi] -+ [b], the projected net in A/ D, namely [iii] := 1f[bi] -+

i[b] =: [a], also converges in A/ D . Note that if N is a base of neighborhoods for a point z in

A , then N / D is a base for [z] in A/ D . Thus, [iii] -+ [a] implies that there is a convergent

net of representatives (iii) with each iii e [iii], and~ -+a (some a E [a]), since D is finite. So

[bi] -+ [b] and iii -+ a and lemma two then imply that bi -+ b in F. So, by continuity of scalar

multiplication in F, we have Aibi -+ Ab and so in Ewe have Ai[bi] = [Aibi]-+ [Ab] =: A[b].

{3) addition: Let [bi], [ci] be two nets over the same index set I such that [bi] and (ci] are over the

same base point[~] (so that addition makes sense). Suppose further that these nets converge

to [b], [c] respectively. Pick a convergent net (iii) of representatives for ([~]) as in proof of

scalar multiplication and a net of representatives (bi), (ci) for the nets [bi], [ci] respectively

with each bi, Ci over the point iii. Then lemma two gives bi -+ band Ci -+ c in F . Finally,

[bi] + [ci] := [bi + ci] -+ [b + c] =: (b] + (c] again by continuity of addition in F .

( 4) multiplication and involution are similar.

(5) Finally, the "zero-limit" property for a C*-bundle is verified. Let ([bi]) be a net in E such that

ll[bi]ll -+ 0 and [a;] := 1f([bi]) -+(a] e A/ D. It must be shown that [bi] -+ O(a]· Again, choose

a net (ai) in A with ai E [ai] such that ai-+ a, and a net (bi) in F such that bi E Fai. Then

llbill =: ll[biJII-+ 0 and so bi -+ Oa by the zero-limit property on F . Thus [bi] -+ [Oa] = O[a]·

PROOF THAT r!'(E) ~ ro(E):

Forse r~(E), define ~(s)[a] := [s(ii)] for [a] e A/D. Since sis equivariant, s(a) e F for each

a EA. So well-definedness of ~(s) is easily shown: if a= d.z for some de D, then s(a) = s(d.x) =

d.s(x) by equivariance, and so by definition [s(a)] = [s(x)]. Since the norm onE is induced from F

and the quotient map A -+ A/ D is continuous, then ~(s) is continuous and vanishes at infinity as

s does. In fact, lls(a)IIF = ll[s(a)]ll by definition, and so llslloo = ll[s]lloo: that is,~ is an isometry.

With pointwise operations on r~(E) and ro(E) (which makes them C*-algebras), ~is easily seen to

be linear. Therefore, r~ (E) is isometrically isomorphic to its image in r o(E). We want the image

to be all r o(E).

To finish the proof we will use the "Bundle Stone-Weirstrass" theorem. Since r~ (E) and its ~

image are full algebras of operator fields on A/ D ([4], p.233), we will be done if we show the image

55

separates r o(E) in the following sense: if [a], [x] e A/ D and if [b] e E[a] and [c] e E[~], then there is

an image point c)( s) that passes through both [b] and [c). For this, lemma 1 can be extended to take

any finite D-invariant subset of A and yield a invariant section passing through each representative

of [b] and [c). Thus c) is onto, and so r~(E) ~ r o(E).

11.2 C*(G) ~ ro(E).

Recall, from section I above, the injective homomorphism fJ: D--+- AUT(Mn(Co(A))) with respect

to which C*(G) ~ M0 (Co(A))D. We also have r~(E) ~ r o(E) by section II. Thus, it is sufficient to

show r~(E) is isomorphic to Mn(Co(A))D. This last isomorphism is accomplished in the following

two claims.

Since E (constructed in section II) is the trivial bundle over A , there is a natural identification

of Mn(Co(A)) and r o(E) given by

for each F E M0 (C0 (A)), where (F,id..4)(a) .- (F(a),a). This is a C*-identification since the

bijection c)o is obviously an isometry.

Define aD-action on r o(E) pointwise, using the D-action forE: fordED and s E r o(E), then,

for all a E A, define (d.s)(a) :=d. s(a). Using the above identification c) 0 , lets= (S, id_..d for some

S e Mn(Co(A)). The D-action on fo(E) then becomes (d.s)(a) = d.(S,id.A)(a) = d.(S(a),a) =

(S(a)d,a,d.a). From the definition of D acting onE, one has, for b,c e D, (S(a)d,a)b,c

:= a(a(b,d)- 1a(c,d))(S(a))bd,cd· But notice that (f3(d)S)b,c(d.a) = a(a(b,d)- 1a(c,d))(S(a))bd,cd,

and so, solving, one has a(a(b,d)a(c,d)- 1)f3(d)Sb,c(d.a) = (S(a))bd,cd· Substituting and simplifying,

one has the expression (S(a)d,ah,c = (f3(d)S)b,c(d.a).

CLAIM. <~>o takes fixed-point elements (M 0 (C 0 (A))D) to equivariant elements (T~(E)).

A D PROOF: If S E Mn(Co(A)) , then, by section I, one has f3(d)S = S for any dE D. So, by the above

expression, (S(a)d,ah,c = (f3(d)S)b,c(d.a) = (S)b,c(d.a) = (S(d.a))b,c for each pair b,c ED. That is,

d.(S, id_..t)(a) = (S(a)d,a, d.a) = (S(d.a), d.a) = (S, id..4)(d.a), and so (S, id..4) = <~>o(S) is equivariant.

CLAIM. The restriction of<l> 0 to M0 (C 0 (A))D maps onto r!'(E).

PROOF: Let s E f~(E). So s(d.a) = d.s(a) for all d E D and a E A. When unrestricted, <1>0

is surjective onto r o(E), so let S e Mn(Co(A)) be such that s = c)0 (S) = (S, id.A)· It will be

shown that Sis actually a fixed-point element. Fix de D. Since sis equivariant, (S(d.a),d.a) =

(S,id.A)(d.a) = d.(S(a),a) := (S(a)d,a,d.a). Thus, S(d.a) = S(a)d,a. So, using the definition of

S(a)d,a, one has, for each b,c ED, (S(d.a))b,c = (S(a)d,ah,c = a(a(b,d)- 1a(c,d))(S(a))bd,cd· But

56

then, substituting the expression derived above the previous claim for ( S( a) )bd,cd and simplifying,

one has (S)b,c(d.a) = (S(d.a))b,c = (j3(d)S)b,c(d.a) for each b, c e D, and so S(d.a) = (j3(d)S)(d.a) is

valid for any dE D and any a e A.. Using d- 1 .a in this expression, one has S(a) = S(d.(d- 1 .a)) = (j3(d)S)(d.(d- 1.a)) = (j3(d)S)(a) for any de D and a eA.. Thus, Sis a fixed-point element, as

required.

We now have all the machinery necessary to calculate C* (G) and the irreducible representations A A D •

of C*(G) . For each a E A, .Aa := {F(a) 1: F E M0 (C0 (A)) (:!! C*(G))} IS a C*-subalgebra of

M0 (C) which corresponds to the fibre Fa,. The next chapter uses r 0 (E) (:!! C*(G)), the section

C*-Algebra of E , to find all representations of r 0 (E) by finding those of each fibre and extending.

57

Chapter 6 The C*-Algebras of the Planar Crystal Groups

and Their Irreducible *-Representations

This chapter applies the theory developed in chapter five to the planar crystal groups listed in

chapter four. For a two-dimensional crystal group G, c• (G) is constructed and then an alternative

description is determined. The irreducible representations of c• (G) are then determined and

described. The full listing for all seventeen crystal groups can be found in appendix A. Throughout

this chapter, let G be a fixed two-dimensional crystal group, A ~ G the lattice subgroup, and

D = G/A the point group of G. Recall from 4.1 that these three groups form the exact sequence

In this case A~ Z2 • Let n be the (finite) size of the point group D.

The first step which must be carried out before applying the formula for c• (G) is to find an

explicit identification between the two formats used to describe G. On the one hand, 5.1 views G

as {(d, (~)) 1: dE D, i,j E Z} with a certain product. On the other hand, 4.11 uses a (fixed)

cross-section '"( for D to express G as {(Md, ud)(l, G)) 1: d E D, (Md, ud) = '"f(d), i,j E Z} by

A-coset decomposition. Using the particular cross-section ; from 4.1I.2 (for each group), we will

map each element ;(d)= (Md,ud) to (d,O,O) to define a (fixed) cross-section of Din the format of

chapter five. Then the map (Md, v) 1-+ (d, M;J 1(v- ud)) is the desired isomorphism and has inverse

(d, G)) I-+ (Md,Ud Md G)). ! Description of A.

In order to apply the formula from chapter five which describes c• (G) one needs the dual space

A of the lattice A. A~ V turns out to be T 2 in the planar case. For let (z, w) be an element of

T 2 (so z, w are elements of C with unit modulus). Define X(z,w) by

(i) .. X(z,w) j := z'w (V G) e Z2

).

Then X(z,w) is a homomorphism of Z2 ~ T, and since Z2 is discrete, X(z,w) is continuous. Hence

- - (z) X(z,w) E Z2 • Conversely, if IE Z2 then I acts on each component of Z2 homomorphically: I 0 =

I (~)'and I G) =ICY (for any i, j E Z). Let z =I ( ~) and w =I ( ~). Then I= X(z,w)·

Therefore A ~ Z2 ~ T 2 as claimed.

58

II D-action on A and A.

The action of the point group D on the lattice A ( = Z2) and its dual A ( = V = T 2) can be

explicitly calculated as follows. Note that the cross-section "Y and the lattice A are preserved by

our identification of G (with ''itself" via the above isomorphism) and so the D-action on A may

be calculated directly by using the description of G found in 4.11.2. Now, for a fixed d E D,

recall the orthogonal transformations identified with d and d- 1 , namely Md, Md-l E 9(R2 ). Let

Md = [au a12 ] and let Md-l = [6611

6612 ]. Recall that the D-action on A is defined ford E D

a21 a22 21 22

and a E A by d.a := "'f(d)a"Y(d)- 1 . In the notation of chapter four this evaluates to the action of Md

on Z2: d.( 1' G ) ) = ( 1, M d G ) ) (identifying A with Z2). Therefore, the action has the explicit

formulation

In the dual A, the D-action is defined for d E D and x E A by the action on A: ( d.x)( a) .

x(d-1.a) (Va E A). Now, given dE D and (z, w) E T 2 , we have, for each G) E Z2 ,

d.X(•,w) G) := X(•,w) ( d-'. G)) = X(•,w) ( Md-•· G))

= (zbuwb:~1)i (zbt:~wb22 )i

= X(•'""''"·''""'"'') (;)

which gives a formula for the action of don A . Note that an element X(z,w) of A may be written

simply as (z, w) without any ambiguity, and this naming convention will be followed henceforth.

Appendix A lists the D-action on A for each planar crystal group.

The 2-cocycle.

The remaining piece of information required to apply the formula for C* (G) is the 2-cocycle

a : D x D ~ A. This is found by using the cross-section "Y and the definition of a: a(b, c) = "Y(bc)- 1"Y(b)"Y(c) for each pair b, c E D. Since the cross-section is preserved by our identification

and A is preserved, we may use chapter four's description of G to calculate the 2-cocycle a directly.

Appendix A lists the 2-cocycle for each planar crystal group.

Description of C*( G).

By 5.1, one has the following formula for C*(G) . For each dE D let Fd(z, w) = Fd,1D(z, w) be

59

an arbitrary continuous function over T 2 • Then the matrix whose entries are defined by

is an element of C* (G) . Further, every element of C* (G) is obtained in this way for some set of

n functions, Fd(z, w) E C(T2) (for each dE D). A general element of C*(G) is listed in Appendix

A for each planar crystal group.

III Determination and Description of A/ D.

In order to construct the tilde bundle E (5.11), one needs the base space. In the planar crystal

group case, the base is the quotient space A/ D = T 2 /D. It has been shown that the group C*

Algebra C*(G) is isomorphic to the section C*-algebra fo(E) . Again, this section C*-algebra

is desired because the irreducible representations on it are accessible, and these representations

determine those of C* (G) . Further, one can give an alternate description for C* (G) using the

quotient space.

Recall that the action of each dE D is a homeomorphism of T 2 • The quotient space identifies

each D-orbit by definition. Thus, an appropriate way to find the quotient space is to determine a

particularly nice ''fundamental domain" of the orbit space and then determine how its boundary

points are identified by the D-action. A fundamental domain is an (open) subset I< s; T 2 satisfying

c.I< n d.I< = 0 whenever c, d E D are distinct elements, and UdeDd.I< = T 2. That is, a fundamental

domain is a subset of T 2 whose D-translates are disjoint and whose closure translates over all T 2 •

In the case of two-dimensional crystal groups it often happens that the stability sets Sd := {(z, w) E

T 2 1: d.(z, w) = (z, w)} (for each d E D) delineate an obvious choice for a fundamental domain.

Note that each point of such a fundamental domain must have a trivial isotropy subgroup. The

frontier of a fundamental domain I< will always map to the frontier of any D-translate of I<, and

all these boundaries are disjoint from any translate of I<. Thus the quotient space is determined by

finding the identification produced by translating the boundary of I<. In the planar crystal group

case, some of the D-actions on A/ D are identical for distinct groups and so these quotient spaces

will be listed together in Appendix A.

For some purposes, such as proving a set is a fundamental domain, it is easier to work in R 2 /(2Z)2

rather than in the (homeomorphic) space T 2 • R 2 /(2Z)2 is represented diagramatically by [-1, 1] x

[-1, 1] with opposite edges identified, as usual. An element(} E [-1, 1] maps to the element z = e1riB

ofT C C. So (z, w) E T 2 and (0,() E [-1, 1] x [-1, 1] can be identified (with due regard for the

edge identification in [-1, 1] x [-1, 1]). Direct calculation shows the action of D on T 2 induces a

60

"linear" action on [-1, 1] x [-1, 1]:

d.(8,() = (8,()Md-l,

where the right-hand side of the equation is right matrix multiplication with a row vector (and then,

of course, factoring out by (2Z)2•

IV Alternate Description of C* (G).

The identification of C* (G) with r o(E) leads to an alternate description of C* (G) . Recall

the isomorphism Fi' ~ E[!) from 5.11 for any [x] e A/ D and then any X e [x]. If X has trivial

isotropy subgroup D1-, then F1- ~ M0 (C) by the definition of the bundle F . Thus the fibres of

E are trivially M0 (C) over each such quotient point [x]. This is the case, for example, if x is an

element of a fundamental domain for T 2 . In general, Eisa C*-bundle so it has enough sections in

ro(E) and so, recalling the map 4> from chapter five, we have F! ~ E[x] = {4>(s)([x]) 1: s E r!'(E)}

~ {s(x) 1: s E r!'(E)} ~ {F(x) 1: FE M0 (Co(A))}D which is just point evaluation of the equivariant

function matrices describing C* (G) . Let L be a cross-section of the D-action on T 2 that is contained

in the closure of a (given) fundamental domain]{. Then the only points in L which have non-trivial

isotropy subgroup are contained in the frontier of K (in L). By appropriate identification, C*(G) is

isomorphic to a subalgebra B C M0 (C(L)) where FEB iff F(x) E Fx for every x E L, and F(x)

satisfies the following limit condition as x approaches a point of K \ L: for a E J{ \ L, there exist

b E K u L and dE D such that d.b = a. As x approaches a from within L,

lim1--aF(x) = d.F(b),

where one recalls the notation for the D-action on such a matrix from 5.11. This provides the most

easily visualized description of C"'(G) . For (z0 , w0 ) E T 2 , let A(zo,wo) denote the algebra, obtained

by point evaluation, {.E(z0 , w0 ) 1: .E E M0 (Co(A))D}. Appendix A lists each such boundary point

(zo, wo) which has non-trivial isotropy subgroup in a given cross-section for the D-action, and also

lists the associated finite dimensional C*-subalgebra A(zo,wo) C M0 (C).

V Irreducible Representations of C*(G).

From 5.111, C*(G) ~ r o(E). Further, from 3.111, one has that the irreducible representations

of r o(E) are in one-to-one correspondence with the set of irreducible representations over all the

C*-fibres E{a)· As noted above in the alternative description for C*(G) , each fibre of E can be

explicitly written down as a C*-subalbgebra of M0 (C) , namely Aa. Also, for a= (zo, wo) E T 2,

A(zo,wo) is all Mn(C) if and only if D(zo,wo) = {lD }. The only irreducible representation (up to

61

equivalence) of Mn (C) is the trivial one, so the representation over any point with trivial isotropy

subgroup lifts to the following representation of C* (G) :

viewing C*(G) as M0 (C0 (.A))D. If A(zo,wo) is not M0 (C) , then, from 3.1, the following procedure

is sufficient to determine (a representative of) each of the irreducible representations of A(zo,wo) .

There are only finitely many irreducible representations since A(zo,wo) is finite dimensional, so call

them 7r(1zo,wo)' ••• ,7rk(•o.•o> Each of these lifts to an irreducible represent.ation on all C*(G): (zo,Wo) •

That is, apply the irreducible representation to F evaluated at (z0 , w0 ).

METHOD FOR FINDING THE IRREDUCIBLE REPRESENTATIONS OF A ) Given a C* subalgebra (z 0 ,Wo · -

(1) Determine the form of the commutant A(zo,wo) of A(z0 ,w0 ) •

(2)

(3)

(4)

Determine the form of projections in A'( ) . Zo,Wo

Determine the form of minimal projections in A(zo,wo) .

Find a set P1, ... , PR in A(zo,wo) of orthogonal minimal projections that sum to the identity

I.

(5) Find a change of basis U (Unitary) that Block-diagonalizes A(zo,wo) with respect to the set

from (4). Each block is a simple C*-algebra.

(6) Pick a maximal subset Pip ... , Pis of inequivalent projections from ( 4).

Then the representations defined, for each A E A(zo,wo)' by 1rj (A) = Alp,; for 1 ~ j ~ S are

inequivalent, irreducible and all there is for A(zo,wo) . They have a particularly nice form:

Finally, one gets a mutually inequivalent list of all the irreducible *-representations of C*(G) as

where L is any cross-section of the D-orbits in T 2 •

62

REFERENCES

(1] W. Arveson, An Invitation to C*-algebras, Springer-Verlag, N.Y., 1976.

[2] M. J. Dupre and R. M. Gillette, Banach Bundles, Banach Modules and automorphisms of C*

algebras, Pitman Publishing Inc., Mass., 1983.

[3] B. D. Evans, C*-bundles and compact transformation groups, Mem. Amer. Math. Soc., vol 39,

no 269 (1982).

(4) J. M.G. Fell, The Structure of Algebras of Operator Fields, Acta. Math. 106 (1961) 233-280.

(5) J. M. G. Fell and R. S. Doran, Representations of *-Algebras, Locally Compact Groups, and

Banach *-Algebraic Bundles, vol. I, Academic Press, San Diego, 1988.

(6] F. P. Greenleaf, Invariant Means on Topological Groups, American Book Company, 1969.

(7] I. Hargittai and M. Hargittai, Symmetry Through the Eyes of a Chemist, VCH Publishers, New

York, 1986.

[8] H. Hiller, Crystallography and Cohomology of Groups, ______ , Dec. 1986.

[9) R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol.I,

Academic Press, N.Y., 1983.

[10] B. O'Neil, Elementary Differential Geometry, Academic Press, N.Y., 1966.

[11] W. Rudin, Real and Complex Analysis, McGraw-Hill, N.Y., 1974.

[12] W. Rudin, Fourier Analysis on Groups, Interscience Publishers, N.Y., 1960.

[13] R. L. E. Schwarzenberger, The 17 plane symmetry groups, Math. Gaz., 58 (1974) 123-131.

[14] K. F. Taylor, C*-Algebras of Crystal Groups, Operator Theory: Advances and Applications, 41

(1989), Birkhauser Verlag Basel, 511-518.

[15) D. R. Farkas, Crystallographic Groups and their Mathematics, R. Mtn. J. Math., 11 (1981),

511-551.

63

Appendix A

Group: p1

As usual, all information for p 1 is trivial.

Group: p2

Let (z, w) E T 2, and (8, () E (-1, 1] x (-1, 1] =: 12•

1. D-Action on T 2 :

D tr

d.(z, w)

d.(8, ()

(z,w)

(-8,-()

2. Stability sets for each dE D:

For tr: {(±1, ±1) E T 2 }.

3. Fundamental domain K forD-action:

K = {(8,() E 12 1: 0 < 8 < 1, -1 < ( < 1}

4. Cross-section L for D-action:

s1 = {(O,() e 12 I= o ~ c < 1}

52 = { ( 8' 1) E 12 I: 0 ~ 8 ~ 1}

Sa= {(1,() E 12 1: 0 ~ ( < 1}

L = K u S1 u S2 u Sa

5. Quotient space T 2 / D:

Sphere.

Stability SPts and Fundamental Domain

A-1

Group: p3

Let (z,w) E T 2 , and (0,() E [-1,1] x (-1,1] =: 12 •


D u u2

(zw,z) (w,zw) d.(z, w)

d.(O, () (-0- (,0) ((, -0- ()

2. Stability sets for each d E D:

For u: {(j,j),(-j,-j),(O,O) E 12}

For u 2 : same as for u


K = {(0,() 1: -1 =50 =5 -j, -!O- 1 < ( < -!O}

u{(O,() I= -j =5o< o, o < (<-to}

u{(O, () 1: 0 < 0 =5 j, -!O < ( < 9}

U{(O,() 1: j $0$1, -!O < ( < -!0+ 1}


St = {(0,0) 1: 0 =59 =5 j}

52= {(9,9) 1: -i =59< 0}

L = KUSt US2


Sphere.

A-2

t (0,1) 1" -

(0 I 1) :....--------··········-·--··-·····-------

Stability Sets and Fundamental Domain

Group: p4

Let (z, w) E T2 1. . • and (9,() E [-D-Ac~on on T2: 1, 1) x [-1, 1) =: P.

2. Stabilit y sets for each d eD· Foro-· {( · . ±1, ±1) E T2}

For o-2. 8 • . arne as for o-

For o-3. 8 . arne as for o-

3. Fundamental d . omain K r: 10r D t" K = {(9 -ac Jon:

'() 1: 0 < 0 < 1 4. Cross-sect" ' 0 < ( < 1}

Ion L forD . -action·

St- {(0 . - , O) 1: 0 5: 0 < 1}

52- {(0 -- , 1) 1:0 < 0 < L- _1}

- K USt US2

5 Qu . . otlent space T2 / D:

Sphere.

·--·--·--··· ( 0 1 ~

(-l,O)t----4~· (0,-ll 1 i __. ~

Stabilitu s ... t " "' sand F undamenta 1 D . om am

A-3

Group: p6

Let (z, w) E T 2 , and (0, () E [-1, 1] x (-1, 1] =: 12•


D tr

d.(z,w)

d.(O, ()

(zw,z)

(0- (,0)

(w,zw) (z, w)

( -(' 0 - () ( -0' -()


For tr: {(0, 0) E 12}

For o-2: {( -~, ~), (~, -~), (0, 0) E 12 }

For o-3 : {(±1, ±1) E T 2}.

For tr5 : same as for tr

For o-3 : same as for fT


K = {(0,() 1:-1 ~ 0 < -i, 20 + 2 < ( < !O+ 1}

u{(O, () 1: 0 < 0 ~ i, -0 < ( < !O}

u{(O, () 1: i ~ 0 ~ 1, 20-2 < ( < !O}


s1 ={(O,-O) l=o~o~ il s2 = {(o, 20- 2) I= i < o ~ 1}

L = KuS1 us2


Sphere.

(zw, z) (w,zw)

(-0 + (, -0)((, -0 + ()

• ~ (0 -1) ~ ·-···-···---·-·······!.··-··· .. -··----·-·--··-..


A-4

Group: em

Let (z, w) E T 2 , and (8, () E (-1, 1] x [-1, 1] =: ! 2•

1. D-Action on T 2:

D Pl

d.(z, w)

d.(O,()

(z,zw)

(8,8- ()

2. Stability sets for each dE D:

For P1: {(w2 ,w) E T 2 }


K = {(8,() 1: -1 < 8 50, -!8 < ( < -!8 + 1}

u{(8,() 1:0 58< 1, -!8 < (51}

u{(8,() 1:0 58< 1, -1 5 ( < -!8- 1}


5t = {(8, -!8) 1: -1 58 51}

52 = {(8, -!8 + 1) 1: -1 5 8 5 0}

53= {(-1,() 1: --! <' < il L = K u 5t U 52 U 53

5. Quotient space T 2 f D:

Mobiiis Band.

~II:' '• ~l : ~1; ~ (0,-1) I .. ,,~4,.~~: ·-·-·---··-··--·······~.·"it'· ·············-··········-···-·-··-····-

• Stabilit'1 Sets and Fundamental Domain

A-5

Group: pm, pg

Let (z,w) e T 2, and (8,() e [-1,1] X [-1,1] =: 12•


D Pt

d.(z, w)

d.(8, ()

(z,w)

(8, -()


For Pt: {(z, ±1) E T 2 }


K = {(8,() 1: -1 $. 8 $. 1, 0 < ( < 1}


St = {(8, 1) 1: -1 < 8 $. 1}

s2 = {( 8, o) 1: -1 < 8 $. 1}

L = KUSt US2


Cylindar

(0 ~-1) ! ;: ·-------------------__,_: Stability Sets and Fundamental Domain

A-6

Group: cmm2

Let (z, w) E T 2 , and (8, () E (-1, 1) x (-1, 1) =: 12 •

1. D-Action on T 2: D tr Pl P2

(z, w) (z, zw) (z, zw) d.(z, w)

d.(8,() (-8, -() (8, 8- () (-8, -8 + ()


For u: {(±1, :t:l) E T 2}.

For P1: {(w2 , w) E T 2}

For P2: {(1, w) E T 2}


K = {(8,() 1:0 < 8 < 1, !8- 1 < ( < ~}


51= {(O,() I= -1 ~ ( ~ o}

52 = {( 8, !O) 1: 0 < 8 ~ 1}

53 = {(1,() 1: -! ~' ~ 0}

54 = {(8, !8- 1) 1: 0 < 8 < 1}

L = K u S1 u 52 u 53


Disc

1

(-1,0) l

i fkl • P.~~-~------------·

I -1 Stability s~ts and Fundamental Domain

A-7

Group: pmm2 'pmg2, pgg2

. , and ( 9, () E [- . Let (z, w) E T2 1. D-Act•on on T•· 1, 1] x [-1, 1] =: [2 D . . .

2. Stabilit y sets for each d eD·

ForO"" {(± · . 1,±1) E T2}

For p . {( · 1· z, ±1) E T2}

For P2: {(±1, w) E T2}

oma1n K £ 3. Fundamental d . K or D-actio .

= {(8,() 1: 0 < 8 n.

. Cross-se t" ' < 1} 4 < 1, 0 < ~

cIon L forD . -action·

s, = {( 8, o) I: o ::; 8 < 1} .

82 = {(0 I' -,,)1:0<(<1}

Sa- {(9 -- ,1)1:0<9~1}

84 = {(1,() 1: 0 < ( < 1}

L=KUS 1 u 82 us us 5 Q .

3

4 . uotient space T2 / D:

Disc

A-8

Stability Set ~ sand Fundamental D . ~ om am

A-9

Group: p3ml

Let (z,w) E T 2 , and (8,() E (-1,1] x (-~,1] =: 12 •

1. D-Action on T 2:

D tr Pl

(w,zw) (zw,z) (z,zw) d.(z,w)

d.(8,() (-(, 8- () (-8 + (, -8)(8, 8- ()


For tr: {( -j, j), (j, -j), (0,0) E 12 }

For tr2: same as for tT

For P1: {(w2 , w) E T 2 }

For P2: {(z, z2) E T 2}

For p3: {(z,z) E T 2}


K = {( 8, () 1: -1 ~ 8 < - j, 28 + 2 < ( < !8 + 1}

U{(8, () 1: 0 < 8 ~ ft, -8 < ( < !8}

U{(8,() 1: ft ~ 8 ~ 1, 28-2 < ( < !8}


s1 = {( o, -8) 1: o ~ o ~ i} s2 = {(o, !B) I= o < o ~ 1}

83 = {(8, !B + 1) 1: -1 < 8 ~ -j}

84 = {( 8, 28 - 2) 1: i < 8 ~ 1}

8s = {(8,28+ 2) 1:-1 < 8 < -j}

L = K u S1 u 82 u S3 u 84 u Ss


Disc

P2

(zw,w) (w,z)

(-8 +(,() (-(,-8)

.. li f;


Group: p31m

Let (z,w) E T 2 , and (6,() E [-1,1] x [-1,1] =: 12 •


d.( z, w) ( ziifi, zw2) (z2 w3, zw) ( z, zw) (z2 w3, zw )

d.(6, () (6- 3(, 6- 2() ( -26 + 3(, -6 + () (6, 6- () ( -26 + 3(, -6 + 2()


For tr: {(0, j), (0, -j), (0, 0) E 12 }

For o-2 : same as for tr

For Pt= {(w2 , w) E T 2}

For P2: {(z, z) E T 2}

For p3: {(z, 1) E T 2}


K = {(0,() 1: 0 < 6 ~ 1, !O- j < ( < 0}

u{(6,() 1: -1 ~ 6 < 0, !O < ( < 0}


St = {(6,0) 1:-1 < 6 ~ 1}

52 = {(0, () 1: -j ~ ' < 0}

L = KUSt US2


Disc

t (0,1) 1' ·-----·-·-·-··--· r-·---·-······--··--······

~ 1

A-10

(-1,0) (0 1 0) l Cl,Ol.l

~ ~~2~~"" ~·~~a f, ""i ill i

1 r ! i (0,-l) j [ e. ~. ···-·····-·········-···-·······-/,·~·· ········-····-········--··--·······..:

2. Stability Sets and Fundamental Domain

_"! ..

A-ll

Group: p4mm, p4mg

Let {z, w) E T 2 , and (8, () E [-1, 1] x [-1, 1) =: 12•


D (1 (12

d.(z, w) (w,z) {z,w)

d.(O,() (-(,8) (-8,-()


For t1, t12 , and t13 : {{±1, ±1) E 12}

For Pt: {{z,±1) E T 2}

For P2: {{z,z) E T 2}

For p3: {(±1, w) E T 2}

For p4: {(z, z) E T 2}


K = {(8,() 1: 0 < 8 < 1, 0 < ( < 0}


5t = {(0, 0) 1: 0 ~ 0 ~ 1}

s2 ={(I,() I= o < ( ~ 1}

s3 = {(0,8) I= o <8 < 1}

L = K u 5t u 52 u 53


Disc

(13 Pt P2 P3 P4

(w,z) (z,w) (w,z) (z,w) (w,z)

((, -8) (8,-() ((,8) (.-8, () (-(, -0)

~ 0 (0,1) J.~~------~~~----------4

(0,-1)

Stability Sets and ~ndamenta 1 Domain /3 4

Group: p6mm

Let (z, w) E T 2 , and (9, () E (-1, 1] x (-1, 1] =: 12•

1. D-Action on T 2:

D (T u2 a-3 d.(z, w) (zw, z) (w,zw) (z,w) d.(9,() (9- (, 9) (-(, 9- () (-9, -()

D Pl P2 Pa d.(z,w) (z, zw) (w,z) (zw,w) d.(9,() (9, 9- () ('' 9) (-9 + (,()


For u and u5 : {(0, 0) E 12 }

For u 2 and u 4: {( -j, j), (j, -j), (0, 0) E 12 }

For u3 : {{±1, ±1) E 12}

For P1: {(w2 , w) E T 2 }

For P2: {(z, z) E T 2}

For p3: {(z,z2) E T 2

}

For p4: {(1, w) E T 2}

For Ps: {(z, z) E T 2}

For P6: {(z, 1) E T 2 }

3. Fundamental domain I< for D-action:

K = {(9,() 1: 0 < 9 ~ i, -9 < ( < 0}

U{(9,() 1: ~ ~ 9 < 1, 29-2 < ( < 0}

4. Cross-section L for n:action:

A-12

(T4 u5

(zw,z) (w,zw) ( -9 + (, -9) ((, -9 + ()

P4 Ps P6 (z,zw) (w,z) (zw, w)

( -9,-9 + () ( -(, -9) (9- (, -()

s1 = {(9,o) I= o ~ 9 ~ 1}

s2 = {(9, -9) I= o < o ~ il Sa = {( 9, 29 - 2) I: i < 9 < 1}

L = K u S1 u S2 uSa

(-110)

"~~~!--------~~~~~~~~ 9

11 5. Quotient space T 2 / D:

Disc

Cocycle Tables.

Only the non-trivial cocycle tables are listed. Each entry is of the form a( c, d) = i, j representing

the cocycle value a( c, d) = (1, ( ~) ). Note that c runs down the left-hand side.

GROUP pg:

Pt 0,0 1, 0

GROUP pmg:

a(c, d) 1n (T p, P2

1D 0,0 0,0 0,0 0, 0 (T 0, 0 0,0 1. 0 -1,0 Pt 0, 0 0,0 1, 0 -1.0 P2 0,0 0,0 0,0 0,0

GROUP pgg:

a(c,d) 1n (T Pt P2

1n 0, 0 0,0 0,0 0,0 (T 0, 0 0,0 1, 0 -1,0 Pt 0, 0 0,-1 1, 0 -1, 1 P2 0,0 0,-1 0, 0 0, 1

GROUP p4mg:

a(c, d) 1n (T (T2 (T3 Pt P2 P3 P4

1D 0,0 0,0 0, 0 0,0 0, 0 0, 0 0, 0 0,0 (T 0, 0 1, -1 -1,0 0, 1 1, 0 0, 0 -1, 1 0, -1

(T'2 0,0 1, 0 0,0 -1,0 0,-1 0, 0 0, 1 0, 0 a-3 0,0 1, 0 -1,0 0, 0 0, 0 0, 0 0, 1 0, -1 Pt 0,0 0,0 -1,0 0, 1 1, 0 0, 0 0, 0 0, -1 P2 0,0 0,0 -1, 1 0, 0 0, 0 0,0 0,0 1, -1 P3 0,0 1, 0 -1,0 -1, 1 1, -1 0,0 0, 1 0,-1 P4 0,0 0,-1 -1, 1 0, 1 1, 0 0,0 -1,0 1, -1

A-13

GROUP p2

General element Fin C*(p2) :

Torus Point: (1, 1).

Typical element in A{l,l):

A set of orthogonal minimal projections in the commutant with sum I:

Change of basis used to block diagonalize A{l,l) with respect to these projections:

Lifted Representations for C*(p2) at (1, 1):

Torus Point: ( -1, 1).

Typical element in A(-1,1):


Change of basis used to block diagonalize A( -l,l) with respect to these projections:

A-16

Lifted Representations for C"'{p2) at ( -1, 1):

Torus Point: ( -1, -1).

Typical element in A(-1,-1):

ll(_1,1)(F) = [F1 0 {-1, 1) + Fl7{-1, 1)]

F( _ 1 _ 1) = [ F1 0 ( -1, -1) Fl7( -1, -1) l - ' F17 ( -1, -1) Ft 0 ( -1, -1)


Change of basis used to block diagonalize Ac _1,- 1) with respect to these projections:

Lifted Representations for C"'(p2) at ( -1, -1):

Torus Point: (1, -1).

Typical element in A(l,- 1):


A-17

A-18

Change of basis used to block diagonalize A(t,-l) with respect to these projections:

Lifted Representations for C*(p2) at (1, -1):

GROUP p3



Typical element in .A(1,1):


[

1 1 1 ] [ 1 1 1 ·; l 1 1 ·; 1

] [ 1 3 3 3 3 -6 - 2z 3-, -6 + 2 1 3-, 3 1 1 1 1 1· 1 1 1 1· 1 1 1· 3 3 3 ' -6 + 21/3" 3 -6- 21/3" ' -6- 2z/3; 1 1 1 1 1 '/3.1 1 1 '/3.1 1 1 1 '/3.,\. 3 3 3 -6- 21 ., -6 + 21 ., 3 -6 + 21 J

Change of basis used to block diagonalize ..4.(1,1) with respect to these projections:


A-19

1 1 '/3 1 ] -6- 2l . 2

1 1 '/3 I --::- + ':'jl 2 . ti -

1 3

I1[1,1)(F) = ( F1 0 (l, 1)- !Fu,(l, 1) + !( -3); F17 ,(1, 1)- !Fu(1, 1)- !( -3); Fu(l, 1)]

I1~1,l)(F) = ( F1 0 (l, 1)- !F17,(1, 1)- !( -3); Fu,(1, 1)- !Fu(1, 1) + !( -3); Fu(l, 1)]

• .1 . .1 . Torus Pomt: (e- 311'

1, e- 311'1 ).

Typical element in .A( -!tri -!"i): e ,e


1 1 '/3.1 1 1 '/3.1] [ 1 --- -1 , -- + -z , -6 2 6 2 3

i -!- !i/3~ ' -!- !i/3~ _l + 1ij3~ 1 _1 + 1ij3~

6 2 3 6 2

1 1 '/3.l -- + -z 2 6 2 1 3

1 1 '/3.l -6- 2z 2

1 1 '/3.l] -6- 2l 2

1 1 '/3.l -6+2z 2 •

1 3

Change of basis used to block diagonalize A -~•i -~•i with respect to these projections: (e 3 ,e 3 )

Lifted Representations for C*(p3) at (e-t"'i, e-t"'i):

ll2 . . (F) =F (e-!"'i e-!"'i)- ~F :r(e-i"'i e-!"'i) + ~(-3)~ F :r(e-!"'i e-!1ri) (e-i•• ,e-i"'') - lD ' 2 t7 ' 2 t7 '

_ ~F(7( e-!1ri, e-!1ri) _ ~( _ 3) ~ F(7( e-!1ri, e- .;1ri)

ll3 . . (F) =F (e-!"'i e-ilri)- ~F :r(e-!"'i e-!1ri)- ~(-3)~ F :r(e-!ri e-!ri) (e-i••,e-i••)- 1D ' 2 t7 ' 2 t7 '

- ~F(7(e-t"'i, e-!1ri) + ~( -3)~ F(7(e-t"'i, e-t"'i)

Torus Point: (e-j1ri, e-i"'i).

Typical element in A(e-!•i,e-!•i):

A set of orthogonal minimal projections in the commutant with sum 1:

Change of basis used to block diagonalize A _2,..i _2.,; with respect to these projections: ( e 3 ,e 3 )

A-20

ll~e-i .. ; ,e-i .. ;) (F) =Fto ( e-fw-i, e-fw-i)- ~ Fe12(e-iw-i, e-fw-i) + ~( -3)! Fe12( e- fw-i, e- f""i)

- ~Fe1(e- fw-i, e-fw-i)- ~( -3) 1 Fe1(e- fw-i, e- f""i)

ll~e-i .. i,e-i .. ;/F) =Ft0 (e-iw-i, e-iw-i)- ~Fe12(e-iw-s, e-i""')- 4( -3)1 Fe12(e-i""i, e-i""i)

- 4FC1(e-iw-i, e-iw-i) + ~( -3)1 Fe1(e-jw-i, e-iri)

A-21

GROUP p4





[ I

1. 1

1 .] [ 1 1 . 1

1] [ 1 1 1 !] [ -\ 4 -41 -4 -1 - 4' -4

~\' I 4 4 1 • 1 1 . ~* -\i 1 1 . 1 1 4' 4 -41 4 4' 4 4

1 1 . 1 1 . ' 1 1 . 1 1 . ' 1 1 1 1 ' 1 -4 4' 4 -41 -4 -41 4 41 4 4 4 4 4 1 . 1 1 . 1 1 . 1 1 . 1 1 1 1 1 1

-41 -4 4' 4 41 -4 -41 4 4 4 4 4 -4

1 -4

1 4

1 -4

1 4

Change of basis used to block diagonalize A(1,1) with respect to these projections:

[ I

1 1

}:] 2 2 2 1 . 1 . 1 2' -2! 2

1 1 1 -2 -2 2

1 . 1 . 1 -21 2' 2


Il~1 , 1 )(F) = (F1 0 (1, 1)- FC13(1, 1)i- FC1:~(1, 1) + Fe1(1, 1)i]

IIf1,1)(F) = ( F1 0 (1, 1) + FC13(1, 1) + FC1:~(1, 1) + Fe1(1, 1)]

II(1,1)(F) = ( F1 0 (1, 1)- FC13(1, 1) + FC1:~(1, 1)- FC1(1, 1)]


Typical element in .A(-1,-1):

A-22

1

-ttl 4 1

-4 1 1 . 4 -4

1 1 -4 4


[1 1 1 1] [ 1 1 1 1] [ 1 4 4 4 4 4 -4 4 -4 4 1 1 1 1 1 1 1 1 1· 4 4 4 4 -4 4 -4 4 41 1111' 1 1 1 1' 1 4 4 4 4 4 -4 4 -4 -4 1 1 1 1 _1 1 _1 1 -1i 4 4 4 4 4 4 4 4 4

Change of basis used to block diagonalize .A(-1,- 1) with respect to these projections:

[I 1 1

1 l 2 2 2 1 1 . 1 •

-1 l' ~tl 1 1 2 -2

1 1 • 1 . -2 -21 2'

Lifted Representations for C*(p4) at ( -1, -1):

nt -l,-l)(E.) = [ F1 0 ( -1, -1) + Fq3( -1, -1) + Fql( -1, -1) + Fq( -1, -1)]

llf-1,-1)(F) = [ F10 ( -1, -1)- Fq3( -1, -1) + Fql( -1, -1)- Fq( -1, -1)]

nr-l,-l)(F) = [F1 0 (-1, -1) + Fq3(-1, -1)i- Fql(-1, -1)- Fq(-1, -1)i]

nt-l,-l)(F) = [ F1 0 ( -1, -1)- Fq3 ( -1, -1)i- Fql( -1, -1) + Fq( -1, -1)i]


Typical element in Ac- 1,1):


0 1

[

! 0

-! ~ 0 -!

~! ~!] [~ ! ~ !] 1 0 '1 010. 2 2 2

0 ! 0 ! 0 ! Change of basis used to block diagonalize Ac _1,1) with respect to these projections:

[ 12" 0 12~

:t~~ 2

12J. 0 2 l

0 121 2

-!2~ 0

A-23

1 "] ~y

1 . . 4z 1 4

A-24

Lifted Representations for C*(p4) at ( -1, 1):

GROUP p6

General element F in C* (p6) :

F1 0 (z, w) FD(z, w) Ff72(z, w) Ff73(z, w) FD•(z, w) FDs(z, w)

FDs(zw, z) F1 0 (zw,z) FD(zw,z) Ff72(zw, z) F(73(zl1J, z) FD•(zw, z)


Typical element in A(1,1):

FD•(w, zw) FDs(w, zw) F1 0 (w,zw) FD(w,zw) FD,(w,zw) Ff73(w, zw)

F(73(z, w) FD•(z, w) FDs(z, w) F1 0 (z, w) FD(z, m) Ff72(z, w)

FD,(zw, z) Ff73(zw, z) FD·(zw, z) FDs(zw, z) F10 (zw, z) FD(zw, z)

FD(w,zw) Ff72(w, zw) F(73(w, zw) FD•(w,zw) FDs(w, zw) F1 0 (w,zw)

F1 0 (1, 1) FDs{1, 1) F(7·{1,1) F(73(1, 1) F(72(1, 1) F(7(1,1) F(7(1, 1) F10 (1,1) FDs{1, 1) F(7·(1, 1) F(73(1, 1) F(72(1, 1)

F(1, 1) = F(72(1, 1) F(7(1, 1) F1 0 {1, 1) FDs{1, 1) F(7·(1, 1) F(73(1,1) F(73(1, 1) F(72(1, 1) F(7(1, 1) F1 0 (1, 1) FDs{1, 1) F(7·(1,1) F(7·(1, 1) F(73 ( 1, 1) F(7,(1,1) F(7(1,1) F1 0 (1,1) FDs(1, 1) FDs(1, 1) FD·(l, 1) F(73(1, 1) F(72(1, 1) F(7(1,1) F1 0 (1,1)

The following is a set of orthogonal minimal projections in the commutant with sum I:

1 . 1 1 1 1 1 6 6 6 6 6 6 1 l 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 6 6 6 6 6 6

1 1 1 1 1 1 6 -6 6 -6 6 -6 1 1 1 1 1 1

-6 6 -6 6 -6 6 1 1 1 -l 1 1 6 -6 6 6 -6 1 1 1 1 1 1

-6 6 -6 6 -6 6 1 1 1 1 1 1 6 -6 6 -6 6 -6

1 1 1 1 1 1 -6 6 -6 6 -6 6

l n + ti/31 ..1. ..1.. '31 1 ..1. 1 '3 .1 1 1 "/3.1 6 -12+ 12 1 -6 -12 - 121 2 -- -z 2 12 4

112 - ii/31 1 112 + ii/31 1 1 '31 1 1 1 '3l 6 -12 + 121 -6 -12- 121 2

1 1 '3 .1 1 1 '/3.1 1 1 1 '/3.1 1 1 '3 .1 1 -12- 121 2 -- -z 2 6 12 + 4% 2 -12 + 121 2 -6 12 4 -l - 112 - 112 i31 1 1 '/3.1 l 1 + 1 '/3.1 1 1 '3 .1 -- -z 2 12 4' 2 - 12 + 12 z 2 12 4 6

1 1 '3 .1 1 _..L- ..Li31 1 1 '/3.1 1 1 + 1 '/3l -12 + 121 2 -6 -- -z 2 6 12 4' 2 12 12 12 4 1 + 1 '/3.1 1 1 '31 1 1 1 '3 .1 1 1 '/3.1 1

12 4' 2 -12 + 121 -6 -12- 121 2 -- -z 2 6 12 4

1 1 1 '/3.1 1 1 '3 .1 1 _..L + ..Li31 1 1 "/3.1 6 -- + -z 2 -12- 121 2 6 -12- 4z 2 12 4 12 12 1 1 '/3j, 1 1 1 '/3j, 1 1 '3 j, 1 1 1 '3 j, -12- 41 2 6 -- + -z 2 -12- 121 2 6 -12 + TIZ 2 12 4 1 1 '3 j, 1 1 '/3j, 1 1 1 '/3j, 1 1 '3 j, 1

-12 + 12 1 2 --- -z 2 6 -- + -z 2 -12- 121 2 6 12 4 12 4 l 1 1 '3 .1 -l2- ti/31 l 1 1 "/3j, 1 1 '3 j, - 12 + 12z 2 -- + -z 2 -12- Til 2 6 6 12 4

1 1 '3 j, l 1 ..1. '3j, _112 - ti/31 l 1 1 '/3j, -12-12 1 2 -n+ 12 1 2 -- + -z 2 6 6 12 4 1 1 '/3j, _..L- ..l.i31 l 1 ..1. .3j, 1 1 '/3j, 1 -- + -z 2 -n+ 12 1 2 6 12 4 12 12 6 -12- 41 2

A-25

A-26

..!... + lij3! 12 4

1 1 "3 1 -n + 121 ~ 1 -s-

-1~ - l~i3! 1 1 "/31 T2- 4' 2

1 6

Lifted Representations for C* (p6) :

4 1 1 1 1 1

IT(l,1)(F) =[-2 F~:a(1, 1)-2

( -3)~ F~:a(1, 1) + 2

( -3)' F~(1, 1) + F~3(1, 1)

1 ~ 1 ~ 1 1 1 - 2(-3) 2 F~~(1, 1) + 2(-3) 2 F~·(1, 1)- 2F~·(1, 1) + Ft 0 (1, 1)- 2F~(1, 1)- 2Fas(1, 1)]


Typical element in .A(-l,l):

F(-1,1)=

F1 0 ( -1, 1) Fo( -1, 1) F02( -1, 1) F0 r.( -1, 1) F0 •( -1, 1) F0 r.( -1, 1)

For.( -1, -1) Fl0 (-1, -1) Fo( -1, -1) F02( -1, -1) F03( -1, -1) Fo•( -1, -1)

Fo•(1, -1) F0 s(1,-1) F10 (1,-1) Fo(1, -1) F0 2(1, -1) F0 3(1, -1)

F03( -1, 1) Fo•(-1,1) F0 r.( -1, 1) F1 0 (-1, 1) Fo(-1, 1) F02( -1, 1)

F02( -1, -1) F03( -1, -1) F0 •( -1, -1) F0 r.( -1, -1) F10 ( -1, -1) Fo( -1, -1)


1 0 0 1 0 0 2 2 0 1 0 0 1 0 2 2 0 0 1 0 0 1

2 2 1 0 0 1 0 0 ' 2 2 0 1 0 0 1 0 2 2 0 0 1 0 0 1

2 2

1 0 0 1 0 0 2 -2 0 i 0 0 1 0 -2 0 0 1 0 0 1

2 -2 1 0 0 1 0 0 -2 2

0 1 0 0 1 0 -2 2 0 0 1 0 0 1

-2 2

Change of basis used to block diagonalize .A(- 1,1) with respect to these projections:

0 121 2 ~ 0 0 12! 2 0

0 0 !2; 0 0 12! 2

12! 2 0 0 12!

2 0 0

0 121 2 ~ 0 0 -t2! 0

0 0 12! 2 0 0 -t2!

12! 2 0 0 -i2; 0 0

Lifted Representations for C*(p6) at ( -1, 1):

Fo(1, -1) F0 :or{1, -1) F0 3(1, -1) F0 •(1,-1) F0 r.(1, -1) F1 0 (1, -1)

[

F10 (1, -1) + F0 3(1, -1) F0 2( -1, 1) + F0 r.( -1, 1) Fo( -1, -1) + F0 •( -1, -1) ] II(-1,1)(F) = Fo•(1,-1) + Fo(1,-1) F1 0 (-1, 1) + F0 3(-1, 1) F0 r.(-1,-1) + Fo:or(-1, -1)

For.(1, -1) + F0 :or(1, -1) Fo( -1, 1) + Fo .. ( -1, 1) F1 0 ( -1, -1) + F03( -1, -1)

J. . ~ . TorusPoint: (-e3~'~"~,e-3n).

Typical element in A( ~ ,.., _ .:~. •i : -e"l" ,e "3 )

A-27

F( -el1ri, e-f"'i) =

Flo Fq,. Fq• Fq:s Fq~ Fq Fq F1 0

Fq,. Fq. Fq:s Fq~

Fq~ Fq Flo Fq,. Fq• Fq:s ( -ei"'i, e-i"'i) Fq:s Fq~ Fq Flo Fq'- Fq• Fq. Fq:s Fq~ Fq Flo Fq'-Fq,. Fq. Fqs Fq, Fq Flo


1 l 0 -l + !il31 0 -l- !il31 0 i 0 -i + !il31 0

0 3 -i- !il31 0

1 -i - !il31 0 i 0 -i + !il31 0 3 0 -i - !il3; 0 i 0 1 1 '13 1 -6 + 2 1 .,

1 1 '13 1 0 1 1 '13 1 0 1 -6 + 2 1 ., -6 - 2 1 ., 3 0

0 -i + !il3i 0 -i - !il3i 0 1 3

1 1 1 I .1 1 1 .13 1

0 3 0 -6 - 2i 3~ 0 -6 + 2z 'l

0 1 0 1 1 '13.1 0 1 1 'l3.l 3 -6 - 2z ~ -6 + 21 2

-l + !il31 0 l 0 -l- !il31 0 1 1 I .l 1 0 1 1 .13.1 0 -6 + 2i 3~ 0 3 -6 - 2z 2

-i - !il31 0 -i + !il31 0 i 0

0 -i- !il31 0 -i + !il31 0 i Change of basis used to block diagonalize A _t,.; _2.,;) with respect to these projections:

(-e'! ,e 3"

13! 0 0 13! 13! 0 3 3 3

0 13.1 13! 0 0 131 3 2 3 3

31(-t- !il31) 0 0 13.l 3 2 .l( 1 1·1 .l) 32 - 6 + 2 1 32 0

0 1 3 0

0 31( -l- !il31) 13.l 3 2 0 0 .l( 1 1'/3.1) 32 - 6 + 2 1 2

31(-i + !il31) 0 0 131 3

.l( 1 1'13.1) 32 - 6 - 2z 2 0

0 31(-i + !il31) 131 3 0 0 .l( 1 1 '/3.l) 32 - 6 - 2 z 2

Lifted Representations for C*(p6) at ( -el"'i, e-i11'i):

[F + 31i-lp _rl.w.p 1o 2 <7 4 2 <7 2

F + J~i-lp - 3~i±1p o 2 o" 2 o3

A-28

0 1 0 1 0 3 3 1 0 1 0 1 3 3 3 0 1 0 1 0 3 3 1 0 1 0 1 3 3 3 0 1 0 1 0 3 3 1 0 1 0 1 3 3 3

GROUPcm

General element Fin C*(cm) :

[Ft 0 (z,w) Fp 1 (z,zw) l Fp1 (z, w) Ft 0 (z, zw)

Torus Point: (w2 , w) for all wET have non-trivial and equal isotropy subgroup.

Typical element in A(w,,w):


Change of basis used to block diagonalize A(wl,w) with respect to these projections:

Lifted Representations for C*(cm) at (w2 , w):

A-29

GROUPpm

General element Fin c•(pm) :

[F1 0 (z,w) Fp 1 (z,w)] FPl (z, w) Flo(z, w)

Torus Point: (z, 1) for all z E T have non-trivial and equal isotropy subgroup.

Typical element in A(z,l):


[ t t] ' [ }! -t!] . Change of basis used to block diagonalize A(z,l) with respect to these projections:

[ 121 121 ] t2' .:!2'

Lifted Representations for C*(pm) at (z, 1)

ll(z,l)(F) = [F1 0 (z, 1) + Fp 1 (z, 1)]

llfz,l)(F) = [F1 0 (z, 1)- Fp 1 (z, 1)]

Torus Point: (z, -1) for all z E T have non-trivial and equal isotropy subgroup.

Typical element in A(z,-1):


[ t t] ' [it -/] . Change of basis used to block diagonalize A(z,-l) with respect to these projections:

[ 121 121 ] !2; -

2

!21

Lifted Representations for C*(pm) at (z, -1)

ll(z,-l)(F) = [ F1 0 (z, -1) + Fp1 (z, -1)]

llfz ,-l)(.E) = [ F1 0 (z, -1) - Fp 1 (z, -1)]

A-30

GROUP pg

General element Fin C*(pg) :

Torus Point: (z, 1) for all z E T have non-trivial and equal isotropy subgroup.



Change of basis used to block diagonalize A(z,l) with respect to these projections:

Lifted Representations for C*(pg) at (z, 1):

Torus Point: (z, -1) for all z E T have non-trivial and equal isotropy subgroup.



-~z~ l 1 . 2

Change of basis used to block diagonalize A(z,-l) with respect to these projections:

A-31

Lifted Representations for C*(pg) at (z, -1):

rrtz,-l)(F) = [F1 0 (z,-l) + z;Fp1 (z,-1)]

IIfz,-l)(F) = [ F1 0 (z, -1)- z; Fp 1 (z, -1)]

A-32

GROUP cmm2

General element F in C*(cmm2) :



Fp 1 (z, zw) Fp2 (z, zw) Ft 0 (z,zw) Ft7(z, zw)


Change of basis used to block diagonalize A(l,l) with respect to these projections:

Lifted Representations for C*(cmm2) at (1, 1):

Df1,1)(F) = ( Ft0 (1, 1)- Ft7(1, 1)- Fp 1 (1, 1) + Fp2 (1, 1))

n~l,l)(F) = [F1o(l, 1) + Ft7(1, 1) + FP1 (1, 1) + Fp'J(1, 1)]

n{ 1,1) (.E.) = [ F1 D ( 1, 1) - Ft7 ( 1 , 1) + Fp. ( 1, 1) - Fp2 ( 1 , 1) ]

Torus Point: (w2 , w) for all w E T such that Im(w) < 0 and Re(w) ~ 0 OR Im(w) > 0 and

Re( w) 2: 0 all have non-trivial and equal isotropy subgroup.

Typical element in A(w2,w):

A-33


2 0 0 1

[

1

-i ~ 0 -i

0 _1 0 1 0 1 2 2 .2 -! 0 l [! 0 i 0]

i 0 'i 0 i 0. 0 ! 0 i 0 !

Change of basis used to block diagonalize .A(w~,w) with respect to these projections:

Lifted Representations for C*(cmm2) at (w2 , w):

n2 ~ (F)= [F1 0 (w2, w) + Fp 1 (w

2, w) F(f(w2

, w) + Fp~(w2 , w) ] (w ,w)- F ... (w2 , w) + Fp .. (w2 , w) F (w2 -w) + F (-w2 -w)

., • 1o ' Pl '


Typical element in .A(- 1,1):


[

i -i -! ! 0 0 0 0

0 0 l [1 1 0 0] 2 2 0 0 1 1 0 0 1 _1 ' ~ ~ 1 1 . 2 2 2 2

_1 1 0 0 1 1 2 2 2 2

Change of basis used to block diagonalize .A( _ 1,1) with respect to these projections:

[ 12, 0 124 2 2 2 2 -r 0 121 2

121 0 2

-!21 0

Lifted Representations for C*(cmm2) at ( -1, 1):

A-34


Typical element in .A(l,-1):


Change of basis used to block diagonalize .A(l,- 1) with respect to these projections:

[Jt 1 1

ll 2 2 1 1

-2 -2 1 1 2 -2

1 1 -2 2

Lifted Representations for C*(cmm2) at (1, -1):

!Ill

Torus Point: (1, w) for all w E T such that Im(w) < 0 all have non-trivial and equal isotropy

subgroup.

Typical element in .A(l ,w r

A-35

A-36


0 ! -! 0 0 ! ! 0

[ ! 0 0 -!] [! 0 0 !] 0 _1 ! 0 , 0 1 ! 0 .

2 2 2 2 -! 0 0 ! ! 0 0 !

Change of basis used to block diagonalize A(1,w) with respect to these projections:

[ 12! 0 12~

2 2 0 12~ 0 2 0 -!2~ 0 1 1 0 121 -22" 2 .,

Lifted Representations for C*(cmm2) at (1, w):

II 2 (F)_ [F1 0 (1,w) + Fp2 (1,w) F(1(1,w) + Fp1 (1,w) l (1,w) - - Fu(1, w) + Fp

1 (1, w) F1 0 (l, w) + Fp2 (1, w)

GROUP pmm2

General element Fin C*(pmm2) .




1 1] [1 1 1 1] [ 1 4 -4 4 4 4 4 4

-* * * * * * -! 1 1 '1 111' 1 4 -4 4 4 4 4 -4 1 1 1 1 1 1 1 -4 4 4 4 4 4 4

1 1 l [ 1 -4 -4 4

-* -* -* 1 1 ' 1 4 4 4 1 1 1 4 4 -4

Change of basis used to block diagonalize .A(1,1) with respect to these projections:

Lifted Representations for C*(pmm2) at (1, 1):

llf1,1)(F) = ( Fto (1, 1)- F(7(1, 1) + Fp 1 (1, 1) - Fp2 (1, 1))

ll~1 , 1 )(F) = [ Ft 0 (1, 1) + F(7(1, 1) + Fp 1 (1, 1) + Fp1 (1, 1)]

nt1,1)(F) = [Ft 0 (1, 1)- F(7(1, 1)- Fp1 (1, 1) + Fp1 (1, 1)]

Torus Point: (1, w) for all wET such that Im(w) > 0 have non-trivial and equal isotropy subgroup.

Typical element in .A(l,w):

A-37


[ ~ } ! gl [ ~ ! ~! ~!] 0 l l 0 ' 0 _l 1 0 .

2 2 2 2 t 0 0 t -! 0 0 !

Change of basis used to block diagonalize A(t,w) with respect to these projections:

Lifted Representations for C*(pmm2) at (1, w):


Typical element in A(l,- 1):


Change of basis used to block diagonalize .A(l,- 1) with respect to these projections:

Lifted Representations for C*(pmm2) at (1, -1):

A-38

!Ill·

ll(1,-t)(F) = [ F1 0 (1, -1) + Fe1(l, -1) + Fp1 (1, -1) + Fp2 (1, -1)]

Torus Point: (z, -1) for all z E T such that Im(z) > 0 have non-trivial and equal isotropy subgroup.



2 0 0 .!.

[

1

-t ~ 0 -t

~! ~!] , [~ i ! i]. 0 .!. 0 -2

1 0 .!. 2 2

Change of basis used to block diagonalize .A(z,- 1) with respect to these projections:

[ 12! 0

~i2; 121 2 .,

0 1 J. -222

Lifted Representations for C*(pmm2) at (z, -1):


Typical element in .A( -1, _ 1):

121 2 .,

0 12; 2

0

A-39


[

1 1 1 1] [ 1 1 1 1] [1 1 1 1] [ 1 4 -4 4 -4 4 4 -4 -4 4 4 4 4 4 -! i -! t i i -! -! i i i i -! 1 1 1 1, 1 1 1 1,1111, 1 4 -4 4 -4 -4 -4 4 4 4 4 4 4 -4 1 1 1 1 1 1 1 1 1111 1

-4 4 -4 4 -4 -4 4 4 4 4 4 4 4

Change of basis used to block diagonalize .A(-1,-1) with respect to these projections:

[}: 1 1

~t l 2 2 1 1 2 2 1 1

-2 2 1 1 -2 2

Lifted Representations for C*(pmm2) at ( -1, -1):

IIl-1,-1lF) = [ F1 0 ( -1, -1) + F<T( -1, -1)- Fp1 ( -1, -1)- Fp2 ( -1, -1)]

II( -1,-1)(F) = [ F1 0 ( -1, -1) + F<T( -1, -1) + Fp 1 ( -1, -1) + Fp2 ( -1, -1)]

I1(_ 1,- 1)(F) = [ F1 0 ( -1, -1)- F<T( -1, -1)- Fp1 ( -1, -1) + Fp2 ( -1, -1)]

-\] 1 . -4

1 4

Torus Point: (-1,w)forallw E Tsuchthatlm(w) > Ohavenon-trivialandequalisotropysubgroup.

Typical element in .A(-1,w):


[l 0 0 !] [ ! 0 0 -!] 0 t t 0 0 t -t 0 0 1 1 0 ' 0 _1 1 0 .

2 2 2 2

t 0 0 t -t 0 0 t Change of basis used to block diagonalize .A(-l,w) with respect to these projections:

0 0 12.1. 2 2

12.1. 2 2

12.1. 2 2 -!2~ 0 0

A-40

Lifted Representations for c• (pmm2) at ( -1, w ):

rrt (F)- [.FtD(-1,w)+Fp2 (-1,w) F~(-1,w)+Fp1 (-1,w)] (-t,w)-- F~(-1,w)+Fp1 (-1,w) FtD(-1,w)+Fp2 (-1,w)

II2 (F)_ [FtD(-1,w)- Fp2 (-1,w) F~(-1,w)- Fp1 (-l,w)] (-t,w)- - F~(-1,w)- Fp

1(-1,w) F1D(-1,w)- Fp2 (-1,w)


Typical element in A(- 1,1):

Fp,( -1, 1)] Fp 1 ( -1, 1) F~(-1, 1)

FtD(-1,1)


[ i -i t -i l [ t t -i -i l [ t -i -i t l [ * 1 1 1 1 1 1 1 1 1 1 1 1 1

-4 4 -4 4 4 4 -4 -4 -4 4 4 -4 4 1 1 1 1' 1 1 1 1' 1 1 1 1 '1 4 -4 4 -4 -4 -4 4 4 -4 4 4 -4 4

1 1 1 1 1 1 1 1 1 1 1 1 1 -4 4 -4 4 -4 -4 4 4 4 -4 -4 4 4

Change of basis used to block diagonalize A(- 1,1) with respect to these projections:

Lifted Representations for C*(pmm2) at (-1, 1):

II(_ 1,1)(F) = (F1D(-1, 1)- F~(-1, 1) + Fp 1 (-1, 1)- Fp,(-1, 1)]

IIl-1, 1)(F) = [ F1D( -1, 1) + F~( -1, 1)- Fp 1 ( -1, 1) - Fp,( -1, 1)]

II~ _1,1)(F) = ( F1D ( -1, 1)- F~( -1, 1) - Fp 1 ( -1, 1) + Fp,( -1, 1)]

II{-t,1)(F) = [ FtD( -1, 1) + F~( -1, 1) + Fp 1 ( -1, 1) + Fp,( -1, 1)]

!Ill·

Torus Point: (z, 1) for all z E T such that Im(z) > 0 have non-trivial and equal isotropy subgroup.


Fp,(z, 1)] Fp 1 (z, 1) F~(z, 1)

F1 0 (z, 1)

A-41


2 0 0 1

[

1

-! ~ 0 -!

-! 0 l [! 0 ! 0] 0 -t 0 t 0 t 1 0 '1 010. 2 2 2 0 t 0 ! 0 t

Change of basis used to block diagonalize .A(z,l) with respect to these projections:

Lifted Representations for c•(pmm2) at (z, 1):

A-42

GROUP pmg2

General element Fin C*(pmg2) :


Typical element in .A(l,l):

zFp 1 (z, w) Fp2 (z, w) Ft 0 (z, w) zF(7(z, tii)


Change of basis used to block diagonalize .A(l,l) with respect to these projections:

Lifted Representations for C*(pmg2) at (1, 1):

ITl1,1)(F) = ( Ft 0 (1, 1)- F(7(1, 1) + Fp1 (1, 1)- Fp2 (1, 1)]

n~l,l)(F) = (FtD(1, 1)- F(7(1, 1)- FPt (1, 1) + Fp'J(1, 1) 1

IT(1,1)(F) = ( FtD(1, 1) + F(7(1, 1) + Fp 1 (1, 1) + Fp2 (1, 1)]

I Ill

Torus Point: (1, w) for wET such that Im(w) > 0 have non-trivial and equal isotropy subgroup.

Typical element in A(l,w):

A-43


[ t ! ~t ~il [g ! ! gl 0 _1 1 0 , 0 1 -21 0 .

2 2 2 -~ 0 0 ~ ~ 0 0 ~

Change of basis used to block diagonalize .A(l,w) with respect to these projections:

Lifted Representations for C*(pmg2) at (1, w):

II1 (F)_ [ F10 (1, w)- Fp,(l, w) F11(l, w)- Fp 1 (1, w) l (l,w)- - F11 (1,w)- Fp

1(1,w) F10 (1,w)- Fp,(1,w)

II 2 (F)_ [Fl 0 (1,w) + Fp,(1,w) F11(1,w) + Fp 1 (1,w) l {l,w) - - F11 (1, w) + Fp

1 (1, w) F1 0 (l, w) + Fp,(1, w)


Typical element in .A{l,-l):


Change of basis used to block diagonalize .A(l,-l) with respect to these projections:

Lifted Representations for C*(pmg2) at (1, -1):

A-44

ll·

Torus Point: {z, -1) for z E T such that Im{z) > 0 have non-trivial and equal isotropy subgroup.


zFp1 (z, -1) Fp'J(z, -1) F10 (z, -1) zFq(z, -1)


!z;

0 l [ 1

0 1 .1 2 2 -2Z'l

0 !:z; 0 1 0 2 0 ' 1-.1

2 1 0 1 2 -2Z'l 2 0 ! 0 1 .1 0 2 -2Z'l

-jz! l Change of basis used to block diagonalize A(z,- 1) with respect to these projections:

!2! 0

12_1_1 -2 .,. z" 0

Lifted Representations for C*(pmg2) at (z, -1):

Torus Point: (-1,-1).

Typical element in A(- 1,- 1):

-Fp1 (-1, -1) Fp'J( -1, -1) F1 0 ( -1, -1) -Fq(-1, -1)

A-45


[

! -! 1 1

-2 2 0 0 0 0

0

0 1 2 -!

3!]~[~ t ~ ~-2211]· ! 0 0 ! Change of basis used to block diagonalize A(- 1,- 1) with respect to these projections:

Projection 2 is equiv to 1

Lifted Representations for C*(pmg2) at ( -1, -1):

Torus Point: (-1,w) for wET such that Im(w) > 0 have non-trivial and equal isotropy subgroup.

Typical element in A(-1,w):

-Fp1 (-1, w) Fp,(-1, w) F10(-1, w) -Fq(-1,w)


[

l 0 2 0 1

0 _:! ! 0 2

0 0 0 -!] 1 1 0 2 2 l l 0 . 2 2 0 0 ! 2

Change of basis used to block diagonalize A(-l,w) with respect to these projections:

Lifted Representations for C*(pmg2) at ( -1, w):

A-46



-Fp1 ( -1, 1) Fp,( -1, 1) F1 0 (-1, 1) -F41 ( -1, 1)


[t t ~ 0 0 ! 0 0 !

0 0 l 0 0 ! -! .

1 1 -2 2



Lifted Representations for C*(pmg2) at ( -1, 1):

Torus Point: (z, 1) for z E T such that lm(z) > 0 have non-trivial and equal isotropy subgroup.

Typical element in A(z,1):

zFp 1 (z, 1) Fp,(z, 1) F1 0 (z, 1) zF41 (z, 1)


[ ,t 0 lz~

0 l [ 1

0 -~z~ 2 2 1 0 !:z~ 0 1 0 2

2 0 ' 1-~ 2 0 1 0 1 2z2 2 -2z, 2

0 !z~ 0 ! 0 1 ~ 0 2 2 -2z2

1-~

0 l -r.

A-47

A-48


Lifted Representations for C*(pmg2) at (z, 1):

GROUP pgg2

General element F in C* (pgg2) :


Typical element in A(t,l):

zFp1 (z, iii) Fp,(z, iii) Ft0 (z, iii) zF~(z, w)


Change of basis used to block diagonalize A(l,l) with respect to these projections:

Lifted Representations for C*(pgg2) at (1, 1):

II~t,l)(F) = [ Ft 0 (1, 1) + F~(1, 1)- Fp 1 (1, 1)- Fp,(1, 1) 1

II(1,1)(F) = ( Ft 0 (1, 1)- F~(1, 1)- Fp 1 {1, 1) + Fp,{1, 1) 1

II{t,l)(F) = [Ft 0 {1, 1) + F~{1, 1) + Fp 1 {1, 1) + Fp,{1, 1)]

!Ill·

Torus Point: {1, w) for all wET such that lm{w) > 0 have non-trivial and equal isotropy subgroup.

Typical element in A(t,w):

A-49


[it 0 0

lw! l [ l 0 0 -!rl· 2 2 1 1w; 0 0 1 -~w! 2 2 2

1-; 1 0 , 0 1-1 1 2w 2 -2w 2

0 0 ~ -~w1 0 0

Change of basis used to block diagonalize A(1,w) with respect to these projections:

[ 12! 0 121

!:t, 2

121 0 2

~21w! 0

0 12,_.1 -2 w~

Lifted Representations for c•(pgg2) at (1, w):

II1 (F)= [F1 0 (l, w) + wi Fp~(1, w) (1,w) - F17 (1, w) + w1 Fp

1 (1, w)

F 11 (1,w) + Fp 1 (1,w)wi ] F10 (l, w) + Fp~(I, w)w!

II2 (E.)_ [F1 0 (1, w)- w! Fp~(l, w) (1,w) - F17 (1,w)- w1Fp

1(1,w)

Fa(l, w)- Fp 1 (1, w)w! ] F1 0 (1, w)- Fp~(1, w)wi


Typical element in A(1,-1):


[

! ! 1 1 2 2 0 0 0 0

0 l [ ~ -! 0 0 l 0 0 -~ ! 0 0 1 _.!. ' 0 0 .!. 1 . 2 2 2 2

-! ! 0 0 ! ~

0

Change of basis used to block diagonalize A( 1,- 1) with respect to these projections:


!21 !21 !21 -!21

0 0 0 0

Lifted Representations for c•(pgg2) at (1, -1):

II1 (F)_ [Ft 0 (1,-1)- Fa(1,-1) Fp1 (1,-1)- Fp~(1,-1)] (1,-l)-- Fp1 (1,-1)+Fp~(1,-1) F1 0 (1,-l)+Fa(l,-1)

A-50

Torus Point: (z, -1) for all z E T such that Im(z) > 0 have non-trivial and equal isotropy subgroup.


zFp1 (z, -1) Fp2 (z, -1) F1 0 (z, -1) zF17(z, -1)


[ -iz! 0 -!z!

0 l [ ! 0 1z!

-!z! ~ 2

1 0 1 0 2 2 0 1 0 ' !:z! 0 1

2 2 2 -!z! 0 1 0 1z! 0 2 2

!l! ]· Change of basis used to block diagonalize A(z,- 1) with respect to these projections:

[ 12! 0 12!

0 l 2 2

0 12! 0 12! 2

!:~.; 1 1_1

0 !2izi -2~~z~ 1 1 1 0 -22~z~

Lifted Representations for C*(pgg2) at (z,-1):


Typical element in A(-1,-1):

-Fp 1 (-1,-1) Fp,( -1, -1) F1 0 ( -1, -1) -F11 ( -1, -1)


[ -\ 1 1 . I] [ I 1 1 . I] [ I 1 1 .

-4 -4z -4z 4 4 41 --z - -4 41 1 1• 1 . 1 ! 1 . -!i -4~ ! -~i 4 41 41 4 4 41 4

1 . 1 . 1 1 I 1 • 1 . 1 1 I 1 • 1 . 1 41 -41 4 4 -4 1 -41 4 -4 -4 1 41 4 1 . -~i l l !• 1 . _l ~ -*i 1• 1 41 4 4 41 4' 4 41 4

I·] [ I 41 4 1 • 1

-41 4 1 I } '

4 41 1 1 . 4 -4z

1 1. 4 -4l 1 1. 4 -4l 1 . 1 4z 4 1. 1 -4l -4

A-51

ti l li 4

1 . -4

1 4

Change of basis used to block diagonalize .A(-1,- 1) with respect to these projections:

1 l 2 1 2 1 • 2'

1 • -21

Lifted Representations for C*(pgg2) at ( -1, -1):

Til-1,-1)(F) = (FtD( -1, -1) +Fa( -1, -1) + Fp 1 ( -1, -1)i- Fp:~( -1, -1)i]

IT~- 1 ,-l)(F) = ( FtD ( -1, -1)- Fq( -1, -1) + Fp 1 ( -1, -1)i + Fp:~( -1, -1)i]

ll{-t,-l)(F) = (FtD( -1, -1) + F<T( -1, -1)- Fp 1 ( -1, -1)i + Fp:~( -1, -1)i]

Torus Point: ( -1, w) for all w E T such that Im( w) > 0 have non-trivial and equal isotropy subgroup.

Typical element in .A( -l,w):

-Fp1 ( -1, w) Fp:~(-1,w) F1D( -1, w) -Fa(-1,w)

wFp:~( -1, w) ] -wFp

1 ( -1, w)

-Fq(-1, w) FtD(-l,w)


[Jw~ 0 0 -~ow!] [ ~ 0 0

tf]· 1 lw! 1 1 l 2 2 2 -2w:~

1-.1 1 0 , 0 1-.1 1 2w~ 2 -2w~ 2

0 0 1 1-1 0 0 2 2w

Change of basis used to block diagonalize .A( _1,w) with respect to these projections:

0 121

0 l 2 121 0 12.1 2 2 ~

!21w1 0 1 .1-.1 -22o~w~

0 12.1-.1 2 ~w~

Lifted Representations for C*(pgg2) at ( -1, w):

nt (£.) = [FtD(-1,w)- w1Fp~(-l,w) Fq(-1,w)- Fp 1 (-1,w)w1 l (-

1,w) Fq( -1, w) + w1 FP1 ( -1, w) F1D( -1, w) + Fp~( -1, w)w-!

n2 (F)= [ F1D( -1, w) + w1 Fp~( -1, w) F<T( -1, w) + Fp 1 ( -1, w)w-! ] (-

1,w)- Fq(-1,w)- w1Fp

1(-1,w) F1D(-1,w)- Fp~(-1,w)w1

A-52



[

F1 0 (-1, 1) F17 (-1, 1) F(-1, 1)= Fa(-1,1) F10 (-1,1)

Fp1 (-1,1) Fp,(-1,1) Fp,( -1, 1) Fp1 ( -1, 1)

-Fp 1 (-1, 1) Fp,( -1, 1) F10 (-1, 1) -F17(-1, 1)


0 l [1 1 0 0] 2 2

0 0 1 l. 0 0 1 _1 ' ~ ~ 1 1 . 2 2 2 2

1 1 0 0 1 1 -2 2 2 2

0


12~ 2 0

-!2~ 0

0 12~ 2

0 12~ 2


Lifted Representations for C*(pgg2) at ( -1, 1):

rrt (F)= [ Ft 0 (-1,1)+F11 (-1,1) Fp1 (-1,1)-Fp,(-1,1)] (-1,1)- -Fp

1(-1, 1)- Fp,(-1, 1) Ft

0(-1, 1)- Fq(-1, 1)



[

Ft 0 (z, 1) F17 (z, 1) F(z, 1) = F17 (z, 1) F10 (z, 1) - Fp 1 (z, 1) Fp,(z, 1)

Fp,(z, 1) Fp 1 (z, 1)

zFp1 (z, 1) Fp,(z, 1) F10 (z, 1) zF17 (z, 1)


[ l

0 1 J.

_;z1] [ ~ 0 l.z~

2 -2z2 2

0 1 0 1 0 2 2 1-.l. 0 1 0 ' 1-.l. 0 1

-~z, 2 2z2 2 1 J. 0 l. 0 lz~ 0 -2Z2 2 2

I~" l 2z, 0 . 1 2

A-53

A-54

Change of basis used to block diagonalize .A(.-,l) with respect to these projections:

Lifted Representations for c•(pgg2) at (z, 1):

nt (F)= [Ft0 (z, 1)- z~Fp1 (z, 1) FtT(z, 1)- Fp,(z, 1)z; ] (z,t) - FtT(z, 1)- Fp,(z, 1)z~ Ft

0(z, 1)- z~ Fp 1 (z, 1)

n2 (F)= [Ft0 (z, 1) + z;Fp 1 (z, 1) FtT(z, 1) + Fp,(z, 1)z; ] (z,t) - FtT(z, 1) + Fp,(z, 1)z! Ft0 (z, 1) + z! Fp

1 (z, 1)

GROUP p3ml

General element F in C* (p3ml) :

F1 0 (z, w) Fe1(z, w) Fe12(z, w) Fp1 (z, w) Fp2 (z, w) Fp3 (z, w)

Fe12(w,zw) F10 (w,zw) Fe1(w,zw) Fp2(w,zw) Fp3 (w,zw) Fp 1 (w,zw)

Fe1(zw, z) Fe12(zw, z) F1 0(zw, z) Fp3 (zw,z) Fp 1 (zw,z) Fp2(zw,z)

Fp1 (z, zw) Fp2(z, zw) Fp3 (z, zw) F1 0 (z, zw) Fe1(z, zw) Fe12(z, zw)

Fp2(zw, w) Fp3 (zw, w) Fp 1 (zw, w) Fe12(zw, w) F10(zw, w) Fe1(zw, w)

Fp3 (w, z) Fp 1 (w, z) Fp2 (w, z) Fe1(w, z) Fe12(w, z) F1 0 (w,z)

Torus Point: (w2 ,w) for all wET such that Im(w) > 0 and Re(w) >-!all have non-trivial and

equal isotropy subgroup.

Typical element in A(w2,w):

F(w2, w) =

F1 0 (w2 , w) Fe1(w2, w) Fe1'J(w2, w) FP• (w2' w) Fp2 (w2 ,w) Fp3 (w2 , w)

Fe12(w,w) F1 0 (w, w) Fe1(w,w) Fp2 (w, w) Fp3 (w, w) Fp1 (w,w)

Fe1(w, w2) Fe12(w, w2) Ft 0 (w,w2 )

Fp3 (w, w2 )

Fp1 (w, w2)

Fp2 (w, w2)

Fp1(w2 ,w)

Fp'J(w 2 , w) Fp 3 (w2 , w) F1 0 (w2

, w) Fe1( w2 , w) Fe12(w2, w)

FP'J(w, w) Fp3 (w, w) Fp 1 (w,w) FC1'J(w,w) F10 (w,w) Fe1(w, w)


1 0 0 1 0 0 2 -2 0 1 0 0 1 0 2 -2 0 0 1 0 0 1

2 -2 1 0 0 1 0 0 -2 2

0 1 0 0 1 0 -2 2 0 0 1 0 0 1

-2 2

1 0 0 1 0 0 2 2 0 1 0 0 1 0 2 2 0 0 1 0 0 1

2 2 1 0 0 1 0 0 2 2 0 1 0 0 1 0 2 2 0 0 1 0 0 1 2 2

Fp 3 (w, w2)

Fp 1 (w,w2)

Fp'J(w, w2)

FC1(w, w2)

Fe1'J(w,w2)

Ft 0 (w, w2)

Change of basis used to block diagonalize A(w'l,w) with respect to these projections:

0 0 121 2

0 121 2 0

121 2 0 0 121

2 0 0

0 121 2 0 0 0 121

2

0 0 1 .1 0 121 0 -22'J 2 1 .1 0 0 121 0 0 -22'J 2

0 1 ..\. 0 0 0 12~ -22'J 2

Lifted Representations for C*(p3ml) at (w2 , w):

[

F1o(w, w)- Fp3(w, w) FC1'J(W, w2)- FP1 (w, w2) FC1(w2' w)- Fp2(w 2

' w) ] ITtw2,w)(F) = Fe1(w, w)- Fp1 (w, w) F1 0 (w, w2)- Fp2(w, w2) Fe1'J(w2, w)- Fp3(w 2

, w) Fe1'J(w, w)- Fp2 (w, w) Fe1(w, w2)- Fp3(w, w2 ) F1 0 (w2

, w)- Fp 1 (w2, w)

A-55

Torus Point: (z, z2) for all z E T such that Re(z) < -j all have non-trivial and equal isotropy

subgroup.

Typical element in A(z,z2):

F10 (z, z2 )

Fe1(z, z2)

Fe12(z, z2)

Fp 1 (z, z2)

Fp,(z,z2 )

Fp 3 (z,z2 )

FC1,('z2, z) F1 0 (z2 , z) Fe1(z2 , z) Fp,(z2

, z) Fp3 (z2

, z) FPl (z2' z)

Fe1(z, z) Fe12(z, z) F1 0 (z, z) Fp3 (z, z) Fp 1 (z,z) Fp,(z, z)

Fp1 (z, z) Fp,(z, z) Fp3 (z, z) F1 0 (z, z) Fu(z, z) Fu,(z,z)

Fp,(z, z2)

Fp3 (z,z 2)

Fp 1 (z, z2)

Fu,(z, z2)

Ft 0 (z, z2 )

Fu(z,z 2)


1 0 0 0 1 0 2 -2 0 1 0 0 0 1

2 -2 0 0 1 1 0 0 2 -2 0 0 1 1 0 0 -2 2

1 0 0 0 1 0 -2 2 0 1 0 0 0 1

-2 2

1 0 0 0 1 0 2 2 0 ! 0 0 0 1

2

0 0 1 1 0 0 2 2 0 0 1 1 0 0 2 2 1 0 0 0 1 0 2 2 0 1 0 0 0 1

2 2

Fp3 (z2 , z) Fp 1 (z

2, z)

Fp,(z2 ,z) Fu(z2

, z) Fu2 (z2 , z) Ft 0 (z2 , z)

Change of basis used to block diagonalize A(z,z2) with respect to these projections:

0 0 12~ 2 0 121 2 ., 0

12~ 2 0 0 12~

2 0 0

0 12! 2 0 0 0 12!

2

0 -!2! 0 0 0 12! 2

0 0 1 .1. 0 121 0 -222 2 1 .1. 0 0 121 0 0 -22:a 2

Lifted Representations for C*(p3ml) at (z, z2):

A-56

Torus Point: (z,z) for all z E T such that Im(z) > 0 and Re(z) >-tall have non-trivial and equal

isotropy subgroup.

Typical element in .A(z,zr

F(z,z) =

F1 0 (z, z) F(l(z,z) F(12(z, z) Fp 1 (z, z) Fp2 (z, z) Fp3 (z, z)

F(12(z, z2)

Ft 0 (z, z2)

F(l(z, z2)

Fp2 (z, z2 )

Fp3 (z, z2 )

Fp1 (z, z2 )

F(1(z2, z)

F(12(z2 ,z} FtD(:z2,-z) Fp3 (z2 , z) Fp1 (-z2, z) Fp2(-z2, z)

FPt (z, z2) Fp2 (z, z2)

Fp3 (z, z2)

F1 0 (z, z2 )

F(l(z, z2 )

F(12(z, z2)

Fp2(-z2 ,z) Fp3 (z2,z) Fp1 (-z2 ,z} F(12(z2 , z) F1 0 (z2

, z) F(1(z2, z)


1 0 0 0 0 1 2 -2 0 1 0 1 0 0 2 -2 0 0 1 0 1 0 2 -2 0 1 0 1 0 0 -2 2 0 0 1 0 1 0 -2 2 1 0 0 0 0 1

-2 2

1 0 0 0 0 1 2 2 0 1 0 1 0 0 2 2 0 0 ! 0 1 0 2 2 0 1 0 1 0 0 2 2 0 0 1 0 1 0 2 2 1 0 0 0 0 1 2 2

Fp3 (z, z) Fp1 (z, z) Fp2 (z, z) F(l(z, z) F(12(z, z) F1 0 (z, z)

Change of basis used to block diagonalize .A(z,T) with respect to these projections:

121 2 ., 0 0 0 121 2 ., 0

0 12-! 2

0 0 0 121 2 .,

0 0 12-! 12-! 0 0 2 2

0 1 1 0 0 0 121 -22~ 2

0 0 1 .1 !21 0 0 -222 2 1 .1 0 0 0 12~ 0 -222 2

Lifted Representations for C*(p3ml) at (z, z):

A-57

A-58

Torus Point: ( -eiri, e-jri).

Typical element in .A( -ei ri ,e-f .. i):

F1o Fq, Fq Fpl Fp, FP3 Fq F1o Fq, Fp, FP3 Fpl

F( -eiri' e-jri) = Fq, Fq F1o FP3 FPl Fp, ( -eiri, e-jri) FPl Fp, Fps F1o F(l, F(l Fp, FP3 FPl F(l F1o Fq, FP3 FPl Fp, F(l, F(l Fto

A set of orthogonal minimal projections in the commutant with sum I: 1 1 1 1 1 1 3 -6 -6 6 -3 6

1 1 1 1 1 1 -6 3 -6 6 6 -3

1 1 1 1 1 1 -6 -6 3 -3 6 6

1 1 1 1 1 1 6 6 -3 3 -6 -6

1 1 1 1 1 1 -3 6 6 -6 3 -6

1 1 1 1 1 1 6 -3 6 -6 -6 3 1

i 1 1 1 1

6 6 -6 -6 -6 1 1 1 1 1 1 6 6 6 -6 -6 -6 1 1 1 1 1 1 6 6 6 -6 -6 -6

1 1 1 1 1 1 -6 -6 -6 6 6 6

1 1 1 1 1 1 -6 -6 -6 6 6 6

1 1 1 1 1 1 -6 -6 -6 6 6 6

1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 6 6 6 6 6 6

1 1 1 1 1 1 3 -6 -6 -6 3 -6 1 1 1 1 1 1

-6 3 -6 -6 -6 3 1 1 1 1 1 1

-6 -6 3 3 -6 -6 1 1 1 1 1 1

-6 -6 3 3 -6 -6 1 1 1 1 1 1 3 -6 -6 -6 3 -6

_1 1 _1 _1 _1 1 6 3 6 6 6 3

Change of basis used to block diagonalize .A ~ .. , _1 .. ,) with respect to these projections: (-e ,e

0 131 3

161 6

161 6 0 131

3 1 -i31 161 161 1 -i31 2 6 6 2

1 -i31 16J. 16J. 1 -i31 -2 6 l 6 l -2 1 13J. 1 J. 16J. 1 -i31 2 6 l -66, 6 l -2 0 -k31 -t61 161

6 0 131 3

1 131 -i61 161 1 -i31 -2 6 6 2


Lifted Representations for C*(p3ml) at ( -et"i, e-jri):

ll1 (F)= (-ei•i,e-j•i) -

[Flo- !FO'~ + Fp~- !Fp1 - !FO'- !Fp3 !F0'~3! - !F0'3! + iFp1 31 - !Fp3 3!

na . . (F) =F (-et"i e-jri) + F (-et"i e-jri) + F (-et"i e-jri) ( -e1•• ,e-1••) - 10 ' 0':1 ' 0' '

+ Fp1 ( -e*"i, e-jri) + Fp~( -e*"i, e-iri) + Fp3 ( -ei"i, e-i"i)

. 2 . ~ . Torus Pomt: ( e- 3 "', -e 3"

1 ).

Typical element in .A(e-1•i,-e1ri):

Flo FO'~ FO' FP1 Fp~ FP3 FO' F1 0 FO'~ Fp~ FP3 FPt

F(e-i"i, -e'i"i) = FO'~ FO' Flo FP3 FPt Fp~ ( e-ill'i, -e'i"i) FPt Fp~ FP3 Flo FO'~ FO' FP:I FP3 FPt FO' Flo FO':~

FP3 FP1 Fp~ FO'~ FO' Flo


1 1 1 1 1 1 1 1 1 1 1 1 3 -6 -6 6 -3 6 3 -6 -6 -6 3 -6

-t 1 1 1 1 1 1 1 1 1 1 1 3 -6 6 6 -3 -6 3 -6 -6 -6 3

1 1 1 1 1 1 1 1 1 1 1 1 -6 -6 3 -3 6 6 -6 -6 3 3 -6 -6

1 1 1 1 1 1 1 1 1 1 1 1 6 6 -! ! -6 -6 -6 -6 ! ! -6 -6 1 1 1 1 1 1 1 1 1 1 1 1

-3 6 6 -6 3 -6 3 -6 -6 -6 3 -6 1 1 1 1 1 1 1 1 1 1 1 1 6 -3 6 -6 -6 ! -6 3 -6 -6 -6 3

1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 6 -6 -6 -6 l l l l l 1 l ! 1 _l _1 -i 6 6 6 6 6 6 6 6 6 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 6 -6 -6 -6 1 1 1 1 ! 1 ' -i _1 -t l 1 1 6 6 6 6 6 6 6 6 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6 6 -6 -6 -6 6 6 6 1 l l l 1 l _l _l _l l l l 6 6 6 6 6 6 6 6 6 6 6 6

Change of basis used to block diagonalize .A _1 .. i ! ri with respect to these projections: (e ,-e )

0 131 3 0 13~ 3 'J 161

6 161 6

1 1 ~ 1 1 ~ 161 161 2 -63:~ 2 -63:~ 6 6 1 1 ~ 1 1 ~ 161 161 -2 -63:~ -2 -63:~ 6 6

1 131 -! 1 ~ 16~ -t61 2 6 -63:~ 6 'J

0 13~' -3 'J 0 13~ 3 'J 16~ 6 'J -i61

-! 131 l -i31 16~ -t61 6 2 6 'J

A-59

Projection 2 is equivalent to 1

Lifted Representations for C*(p3m1) at (e-fri, -e!"'i):



F10 (1,1) F17 (1, 1)

F( ) F17~(1, 1)

-1,1 = FPt(1,1) Fp~(1, 1) Fp3 (1, 1)

F17~(1, 1) F1 0 (1,1) F17 (1,1) Fp~(1, 1) Fp3 (1, 1) Fp1 (1, 1)

F 17 (1, 1) F17 ~(1, 1) F1 0 (1, 1) Fp3 (1,1) Fp1 (1,1) Fp~ (1, 1)

Fp 1 (1, 1) Fp~(1, 1) Fp3 (1,1) F1 0 (1,1) F17 (1,1) F17~(1,1)

Fp~(1, 1) Fp3 (1, 1) Fp1 {1, 1) F17 ~(1,1) F1 0 (l,1) F17 (1,1)

A set of orthogonal minimal projections in the commutant with sum I: 1 1 6 6 1 1 6 6 1 1 6 6

1 1 -6 -6

1 1 -6 -6

-i -l

1 1 6 -6 1 1 6 -6 1 1 6 -6

1 1 -6 6

1 1 -6 6

1 1 -6 6

1 1 -6 -6

1 1 -6 -6

1 1 -6 -6

1 1 6 6 1 1 6 6 1 1 6 6

1 1 6 6 1 1 6 6 1 1 6 6 1 1 6 6 1 1 6 6 1 1 6 6

1 1 6 6 1 1 6 6 1 1 6 6 1 1 6 6 1 1 6 6 1 1 6 6

1 1 6 6 1 1 6 6 1 1 6 6 1 1 6 6 1 1 6 6 1 1 6 6

Fp3 {1, 1) Fp1 (1, 1) Fp~(1, 1) F17 (1,1) F17 ~(1, 1) F1 0 (1, 1)

1 1 1 1 1 1 1 1 1 1 1 1 3 -6 -6 -3 6 6 3 -6 -6 3 -6 -6

1 1 1 1 1 1 1 1 1 1 1 1 -6 3 -6 6 -3 6 -6 3 -6 -6 3 -6 -t -t i t t -i -t -t i -t -t i

1 1 1 1 _.l 1 1 1 1 1 1 1 -3 6 6 3 6 -6 3 -6 -6 3 -6 -6 t -i t -t i -t -t i -t -t i -t .l .l _.l _l _.l l _.l _.l .l _l _.l 1 6 6 3 6 6 3 6 6 3 6 6 3

Change of basis used to block diagonalize A( 1 ,1) with respect to these projections:

0 1 2

1 -2 0

-! 1 2

0 1 2

1 -2 0 1 2

-!

A-60


Lifted Representations for C*(p3m1) at (1, 1):

ll~1 ,1)(F) = [Ft 0 (1, 1) + F(72(1, 1) + F(7(1, 1) + Fp1 (1, 1) + Fp,(1, 1) + Fp3 (1, 1)]

ll(t,l)(F) =

tF173; - tF17,3i - tFp,3i + tFp3 3i l (1 1) F1 0 - !F17 :z - !F17 - Fp1 + !Fp:z + !Fp 3 '

A-61

A-62

GROUP p31m

General element Fin C*(p31m) :

F1 0 (z, w) Fa2(z~,zw2) Fa(z2w3 ,zw) FPl (z, zw) F (z2w3 zw2 ) Fp3 (zw3 ,w) P2 ' Fa(z, w) F1 0 (z~, zw2

) Fa:~(z2w3 ,zw) Fp2(z, zw) F (z2w3 zw2 ) Fp 1 (z~,w) P3 '

Fa:~(z, w) Fa(z~,zw2) F1 0 ('Z2w3, zw) Fp3 (z, zw) F (z2w3 zw2) Fp2 (z~,w) Pt '

Fp1 (z, w) Fp2 (z"iiP, zw2) F (-2 3- ) F1 0 (z, zw) Fa2(z2w3, zw2) Fa(z~,w) p3 Z W , ZW

Fp2 (z, w) Fp3 ( z"iiP, zw2) F (-2 3- ) Fa(z, zw) F1o(z2w3' zw2) Fa2(z~,w) Pt Z W , ZW

Fp3 (z, w) Fp1 ( z"iiP, zw2) F (-2 3- ) Fa2(z,zw) Fa(z2w3

, zw2) F1 0 (zw3

, w) p2 Z W , ZW



F1 0 (1,1) Fa2(1, 1) Fa(1, 1) Fp1 (1,1) Fp2(1, 1) Fp3 (1,1) Fa(1, 1) F1 0 (1, 1) Fa2(1,1) Fp2(1, 1) Fp3 (1, 1) Fp1 (1,1)

F(1, 1) = Fa:~(1, 1) Fa(1,1) F10 (1,1) Fp3 (1, 1) Fp 1 (1, 1) Fp2(1, 1) FP1 (1, 1) FP2 (1, 1) Fp3 (1, 1) F1 0 (1,1) Fa2(1, 1) Fa(1,1) Fp2(1, 1) Fp3 (1,1) Fp1 (1,1) Fa(1,1) F1 0 (1, 1) Fa:~ (1, 1) Fp3 (1, 1) Fp1 (1, 1) Fp2 (1, 1) Fa2(1, 1) Fa(1,1) F1 0 (1, 1)


1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 6 -6 -6 -6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 6 -6 -6 -6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 6 -6 -6 -6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6 6 -6 -6 -6 6 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6 6 -6 -6 -6 6 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6 6 -6 -6 -6 6 6 6

1 1 1 1 1 1 1 1 1 1 1 1 3 -6 -6 6 -3 6 3 -6 -6 -6 3 -6 1 1 1 1 1 1 1 1 1 1 1 1

-6 3 -6 6 6 -3 -6 3 -6 -6 -6 3 1 1 1 1 1 1 1 1 1 1 1 1

-6 -6 3 -3 6 6 -6 -6 3 3 -6 -6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 -3 3 -6 -6 -6 -6 3 3 -6 -6

1 1 1 1 1 1 1 1 1 1 1 1 -3 6 6 -6 3 -6 3 -6 -6 -6 3 -6

1 1 1 1 1 1 1 1 1 1 1 1 6 -3 6 -6 -6 3 -6 3 -6 -6 -6 3


161 6

161 6

1a! 3 0 0 1a! 3 161 161 1 .L 1 1 1 .L 6 6 -632 2 2 -632 16.\. 16.\. -i-31 _l -! -i-31 6 2 6 2 2 16.\. -!61 1a.L 1 -! 1 .L 6 2 6 2 2 -632 16.\. 6 2 -i-61 -sa! 0 0 1a.L 3 2

161 -!61 131 1 1 -!31 6 6 -2 2


Lifted Representations for C*(p31m) at (1, 1 ):

A-63

lll1,1)(F) = [ F1 0 (1, 1) + F(1~(1, 1) + F(1(1, 1)- Fp1 {1, 1)- Fp~(1, 1)- Fp3 (1, 1)]

II3 (F)_ [Flo- ~Fq~- ~Fq + ~Fp1 - Fp~ + ~Fp3 ~F(1~3-i - ~F(13; + ~Fp1 3;- ~Fp3 3; l (1 1) (l,l) - - ~Fq31 - ~F(1~3; - ~Fp3 3; + ~Fp1 3; F10 - ~Fq~ + Fp~ - ~Fp1 - ~Fq - ~Fp3 '

Torus Point: (z, 1) for all z E T such that z #;. 1 all have non-trivial and equal isotropy subgroup.


F1 0 (z, 1) Fq2(z, z) Fq(z2 , z) Fp1 (z, z) Fp,(z2,z) Fp3 (z, 1) F(1(z,1) F10 (z, z) Fq2(z2 ,z) Fp2(z, z) Fp3(z2,z) Fp 1 (z, 1)

F(z, 1) = Fq2(Z, 1) Fq(z, z) F10 (z2 ,z) Fp3(z, z) Fp1 (z2, z) Fp,(z, 1) Fp1 (z, 1) Fp,(z, z) Fp3 (z2 , z) F10 (z, z) F(1~(z2 , z) F(1(z,1) Fp,(z, 1) Fp3 (z, z) Fp. (:z2, z) Fq(z,z) F10 (z2, z) Fq2(z,1) Fp3 (z, 1) Fp 1 (z, z) Fp,(z2 , z) Fq2(z, z) F(1(z2 , z) F10 (z, 1)

A set of orthogonal minimal projections in the commutant with sum I: 1 0 0 0 0 1 1 0 0 0 0 1 2 -2 2 2 0 1 0 -~ 0 0 0 1 0 1 0 0 2 2 2 0 0 1 0 1 0 0 0 1 0 1 0 2 -2 2 2 0 1 0 1 0 0

, 0 1 0 1 0 0 -2 2 2 2

0 0 1 0 1 0 0 0 1 0 1 0 -2 2 2 2 1 0 0 0 0 1 1 0 0 0 0 1 -2 2 2 2

Change of basis used to block diagonalize A(z,1) with respect to these projections:

0 121 2 ., 0 121 2 ., 0 0

0 0 121 2 0 121 2 ., 0

121 2 0 0 0 0 121

2

0 0 1 1

0 121 0 -22~ 2

-~2~ 0 0 0 0 1 2J. 2 ~

0 -~21 0 121 2 0 0

Lifted Re12resentations for C*(p31m) at (z, 1):

[

F1 0 (z2, z)- Fp 1 (z2

, z) Fq~(z, 1)- Fp2(z, 1) Fq(z, z)- Fp3 (z, z) ] n[z,1)(F) = Fq(z2 , z) - Fp~(z2 , z) F1 0(z, 1)- Fp3(z, 1) Fq2( z, z) - Fp 1 (z, z)

Fq2(z2 , z)- Fp3(z2 , z) Fq(z, 1)- Fp1 (z, 1) F1 0 (z, z)- Fp,(z, z)

[

F10 (z, 1) + Fp3 (z, 1) F(1~(z, z) + Fp 1 (z, z) F(1(z2, z) + Fp,(z2

, z) ] nlz,1)(F) = Fq(Z, 1) + FPl (z, 1) F1o(z, z) + Fp,(z, z) Fq':l(z2

, z) + FP3 (z2' z)

Fq2(z, 1) + Fp,(z, 1) Fq(z, z) + Fp 3 (z, z) F1 0 ('z2 ,z) + Fp 1 (z2

, z)

Torus Point: (1, e-j1ri).

Typical element F(1,e-1""i) in A(l,e-i•i):


1 -i- tifa1 -i + tifa1 0 0 0 3 -i + tiJa1 i -!- tiJa1 0 0 0

-!- tiJa' -i + tiJa' 1 0 0 0 3 0 0 0 1 -i + tiJa' -i- ji/3; ! 0 0 0 -i- ji/3; 1 -i + ji/3; 3 0 0 0 -i + !i/3; -i- !i/3; 1

3

1 1 1 0 0 0 3 3 3 1 1 1 0 0 0 3 3 3 1 1 i 0 0 0 3 3 0 0 0 1 1 1

3 3 3 0 0 0 1 1 1

3 3 3 0 0 0 1 1 1

3 3 3 1 -i + !ifa' 1 1 "ja 1 0 0 0 3 -6- 2 1 .,

1 1 ·;a 1 1 -i + jiJa' 0 0 0 -6- 2 1 ., 3 -i + !i/3; -i- !i/3; 1 0 0 0 3

0 0 0 1 -i- !iJa' 1 1 "/3 1 3 -6 + 2 1 .,

0 0 0 -! + !i/3! 1 1 1 '/31 3 -6- 2 1 .,

0 0 0 -!- !iJa' 1 1 '/3 1 1 -6 + 21 ., 3

Change of basis used to block diagonalize A( -i•• with respect to these projections: 1,e )

!a! 3 0 0 13i

3

a1( -! + !i/31) 0 0 !a! 3

a!(-t- !i/31) 0 0 !a! 3

0 131 3

131 3 0

0 31( -!- !i/3!) 131 3 0

0 a1(-t + !i/31) !a! 3 0

Lifted Representations for C*(p31m) at (1, e-i"'i):

TI 1 (F)-(1,e-i•') - -

[ F1 0 - !Fql- ~( -3)1 Fql- !Fq + ~( -3)1 Fq

FPl- !FPl- !(-a)1FPl- !FP3 + !(-a)!FP3

13l 3 :l 0

a1(-t- !iJa1) 0 .1( 1 1"/3.1) al -6 + 2' l 0

0 13.1 3 l 0 31(-t + !i/31) 0 .1( 1 1 '/3.1) 3l -6- 21 ~

A-64

A-65

GROUP J24mm

General element Fin C*(p4mm):

F1D(z, w) F(7:s(w, z) F(7:a(z, w) F(7(w,z) Fp1 (z, w) Fpl(w, z) Fp3 (z, w) Fp,.(w,z) F(7(z, w) F1D(w, z) F(7:s(z, w) F(7l(w, z) Fpl(z, w) Fp3 (w, z) Fp,.(z, w) Fp 1 (w,z) Ft7l(z, w) F(7(w, z) F1D(z,w) F(7:s(w, z) Fp3 (z, w) Fp .. (w, z) Fp 1 (z, w) Fpl(w,z)· F(7:s(z, w) F(7l(w, z) F(7(z, w) F1D(w,z) Fp .. (z, w) Fp1 (w, z) Fpl(z, w) Fp3 (w, z) Fp 1 (z, w) Fpl(w,z) Fp3 (z, w) Fp .. (w,z) F1D(z, w) Ft7:s(w, z) F(7l(z, w) F(7(w, z) Fpl(z, w) Fp3 (w, z) Fp .. (z, lli) Fp1 (w, z) F(7(z, w) F1D(w,z) F(7:s(z, w) Ft7'l(w, z) Fp3 (z, w) Fp .. (w,z) Fp1 (z,w) Fpl(w,z) F(7'J(z, w) Ft7(w, z) F1D(z,w) F(7:s(w, z) Fp4 (z, w) Fp1 (w,z) Fpl(z,w) Fp3 (w,z) F(7:s(z, w) Ft7l(w, z) F(7(z, w) F1D(w, z)


Typical element F(1, 1) in A(1,1):

F1D(1,1) Ft7:s(1, 1) Ft7'l(1, 1) F(7(1, 1) Fp1 (1,1) FP'l(1, 1) Fp3 (1, 1) Fp .. (1, 1) F(7(1, 1) F1D(1, 1) Ft7:s(1, 1) Ft7'l(1, 1) FPl(1, 1) Fp3 (1, 1) Fp4 (1, 1) FPl (1, 1) Ft7'l(1,1) F(7(1, 1) F1D(1, 1) Ft7:s(1, 1) Fp3 (1, 1) Fp .. (1,1) Fp

1 (1, 1) Fp'J( 1, 1)

Ft7:s(1, 1) Ft7l(1, 1) Ft7{1, 1) F1D{1, 1) Fp4 {1, 1) FPl (1, 1) Fpl(1, 1) Fp3 {1, 1) Fp1 (1, 1) Fpl(1, 1) Fp3 (1,1) Fp4 (1,1) F1D(1, 1) Ft7:s(1, 1) Ft7'l{1,1) F(7(1, 1) Fpl(1, 1) Fp3 (1,1) Fp4 (1,1) Fp1 (1,1) Ft7(1,1) F1D(1, 1) Ft7:s(1, 1) Ft7'l(1,1) Fp3 (1, 1) Fp4 (1,1) Fp1 (1,1) Fpl(1, 1) Ft7l(1, 1) Ft7(1,1) F1D(1, 1) Ft7:s(1, 1) Fp4 (1,1) Fp1 (1,1) Fpl(1, 1) Fp3 (1, 1) Ft7:s(1, 1) Ft7'l(1,1) Ft7(1,1) F1D(1, 1)


1 l 1 1 1 1 1 1 8 s 8 8 -8 -8 -8 -8 1 1 1 1 1 1 1 1 8 8 8 8 -8 -8 -8 -8 1 1 1 1 1 1 1 1 8 8 8 8 -8 -8 -8 -8 1 1 1 1 1 1 1 1 8 8 8 8 -8 -8 -8 -8

1 1 1 1 1 1 1 1 -8 -8 -8 -8 8 8 8 8

1 1 1 1 1 1 1 1 -8 -8 -8 -8 8 8 8 8

1 1 1 1 1 1 1 1 -8 -8 -8 -8 8 8 8 8

1 1 1 1 1 1 1 1 -8 -8 -8 -8 8 8 8 8

1 1 1 1 1 1 1 1 8 -8 8 -8 8 -8 8 -8

1 1 1 1 1 1 1 1 -8 8 -8 8 -8 8 -8 8

1 1 1 1 1 1 1 1 8 -8 8 -8 8 -8 8 -8

1 1 1 1 1 1 1 1 -8 8 -8 8 -8 8 -8 8 l _1 l -i l -t 1 1 s s s s s -8 1 1 1 1 1 1 1 1 -8 8 -8 8 -8 8 -8 8

1 _l l _l 1 _l 1 -t 8 8 s 8 8 8 8 1 1 1 1 -i 1 1 1 -8 8 -8 8 8 -8 8

1 1 1 1 1 1 l 1 8 8 8 8 8 8 8 8 1 1 1 l 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 l 1 1 1 1 1 8 8 8 8 8 s 8 8 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 l 1 1 l 1 l l 1 8 8 8 8 8 s 8 8

A-66

1 0 -i 0 1 0 1 0 4 4 -4 0 1 0 1 0 1 0 1

4 -4 4 -4 1 0 1 0 1 0 1 0 -4 4 -4 4

0 1 0 1 0 1 0 1 -4 4 -4 4

1 0 1 0 1 0 1 0 4 -4 4 -4 0 1 0 1 0 1 0 1

4 -4 4 -4 1 0 1 0 1 0 1 0 -4 4 -4 4

0 1 0 1 0 1 0 1 -4 4 -4 4 1 0 _! 0 1 0 1 0 4 4 -4 4 0 ! 0 1 0 1 0 1

4 -4 -4 4 1 0 1 0 1 0 1 0 -4 4 4 -4

0 1 0 1 0 1 0 1 -4 4 4 -4

1 0 1 0 1 0 1 0 -4 4 4 -4 0 1 0 1 0 1 0 1

-4 4 4 -4 1 0 1 0 1 0 1 0 4 -4 -4 4 0 1 0 1 0 1 0 1

4 -4 -4 4 1 1 1 1 -i 1 1 1 8 -8 8 -8 8 -8 8

1 1 1 1 1 1 1 1 -8 8 -8 8 8 -8 8 -8

1 1 1 1 1 1 1 1 8 -8 8 -8 -8 8 -8 8

1 1 1 1 1 1 1 1 -8 8 -8 8 8 -8 8 -8

1 1 1 1 1 1 1 1 -8 8 -8 8 8 -8 8 -8

1 1 1 1 1 1 1 1 8 -8 8 -8 -8 8 -8 8 1 1 1 1 1 1 1 1

-8 8 -8 8 8 -8 8 -8 1 1 1 1 1 1 1 1 8 -8 8 -8 -8 8 -8 8

Change of basis used to block diagonalize A(l,1) with respect to these projections:

12.l. 12.l. 12! 0 1 0 1 12! 4 2 4 2 4 2 2 4 12! -t2! 12! 1 0 1 0 -i2! 4 4 2 2 12.l. 12.l. 12! 0 1 0 1 12! 4 2 4 l 4 -2 -2 4 12! -i2! 12! 1 0 1 0 -i2; 4 4 -2 -2

-i2; 12! 12! 0 1 0 1 1 .l. 4 4 2 -2 -422

1 1 1 1 121 1 0 1 0 121 -42~ -42~ 4 ~ 2 -2 4 ~

-i21 121 121 0 1 0 1 1 1

4 4 -2 2 -42~

-i21 1 1 121 1 0 1 0 121 -42~ 4 ~ -2 2 4 ~


Lifted ReQresentations for C*(p4mm):

II(t,l)(F) = [F1 0 (1, 1) + F0 3(1, 1) + F0 2(1, 1) + F0 (1, 1) + Fp 1 (1, 1) + Fp2 (1, 1) + Fp3(1, 1) + Fp4 (1, 1)]

II4 (F)= [F1 0 (1, 1)- Fo2(1, 1)- Fp 1 (1, 1) + Fp3(1, 1) Fo(1, 1)- Fo3(1, 1) + Fp 2 (1, 1)- Fp4 (1, 1) l <1·1) - Fo3(1, 1)- Fo(1, 1) + Fp2 (1, 1)- Fp

4 (1, 1) Ft 0 (1, 1)- F0 2(1, 1) + Fp 1 (1, 1)- Fp3(1, 1)

A-67

Torus Point: (z, z) for all z E T such that Im(z) > 0 have non-trivial and equal isotropy subgroup.

Typical element in .A(z,z):

F10 (z, z) F(73(z, z) F(72('Z, z) F(T(z, z) F,1 (z, z) F,2(z, z) F,3(z, z) F,. {z, z) FD(z,z) F10 (z, z) F(73 ('!, Y) F(T'l(Z, '!) F,2(z,Y) F,3{z,z) F,.(Y,z) F,. (z, z) FD2(z, z) F(T(z, z) F10 (z,z) F(T3(z, z) F,3(z, z) F,4 (z,z) Fp 1 (z, z) Fp'J(z,z)

F(z, z) = FD3(z, z) F(72('Z, z) F(T(z, z) F10 (z, z) Fp4 (z, z) Fp 1 (z, z) Fp2 (z, z) Fp3(z, z) Fp1 (z,z) Fp2 (z, z) Fp3 (z, z) Fp4 (z, z) F10 (z, z) FD3(z,z) F(T'J(z, z) Fo(z, z) Fp2 (z, z) Fp3 (z, z) Fp.(z, z) Fp1 (z, z) F(T(z, z) F10 (z, z) F(T3(z, z) Fo'J(z, z) Fp3(z, z) Fp4 (z, z) Fp1 (z, z) Fp2 (z, z) F(72(z, z) F0 (z,z) F10 (z, z) F0 3(z, z) Fp 4 (z,z) Fp 1 (z, z) Fp2 (z, z) Fp3(z, z) Fo3(Z, z) FD'J(z, z) F0 (z, z) F10 (z, z)


1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 2 2 2 -2 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 2 2 2 -2 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1

2 2 2 -2 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 2 2 2 -2 0 0 0 1 1 0 0 0 ' 0 0 0 1 1 0 0 0 2 2 -2 2 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 2 2 -2 2 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 2 2 -2 2 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1

2 2 -2 2

Change of basis used to block diagonalize .A(z,z) with respect to these projections:

0 0 0 121 2

12i 2 0 0 0

121 2 0 0 0 0 12i

2 0 0

0 12i 2 0 0 0 0 121

2 0

0 0 121 0 0 0 0 121 2 2 ~

0 0 121 0 0 0 0 1 1

2 -22~

0 0 0 121 2 -~21 0 0 0

121 2 0 0 0 0 -~21 0 0

0 12l 0 0 0 0 1 .1 0 2 -22'J

Lifted Representations for C*(p4mm) at (z, z):

A-68


Typical element in A(- 1,-1):

FlD Ff73 Ff7l Ff7 FPt Fp, FP3 FP• Ff7 F1D Ff73 Ff7l Fp, FP3 FP• FPt Ff7l Ff7 F1D Ff73 FP3 Fp. FPt Fp,

F(-1, -1) = Ff73 Ff7, Ff7 FtD Fp. Fp. Fpl FP3 ( -1, -1) FP• Fp, FP3 Fp. FtD Ff73 Ff7l Ff7 Fp, FP3 FP4 FPt Ff7 FlD Ff73 Ff7, FP3 Fp. FPt Fp, Ff7, Ff7 F1D Ff73 Fp. FP• Fp, FP3 Ff73 Ff7, Ff7 F1D


1 1 1 1 1 1 1 1 s s s s s s s s 1 1 1 1 1 1 1 1 s 8 8 8 s 8 8 8 1 1 1 1 1 1 1 1 8 s s 8 s s s 8 1 1 1 1 1 1 1 1 s 8 s s s 8 s s 1 1 1 1 1 1 1 1 8 s s s 8 s 8 8 1 1 1 1 1 1 1 1 8 s 8 8 s 8 s 8 1 1 1 1 1 1 1 1 s s s 8 s s s s 1 1 1 1 1 1 1 1 s s s s s s s s

1 0 1 0 0 1 0 1 4 -4 -4 4 0 1 0 1 1 0 1 0 4 -4 4 -4

1 0 1 0 0 1 0 1 -4 4 4 -4 0 1 0 1 1 0 1 0 -4 4 -4 4 0 1 0 1 1 0 1 0 4 -4 4 -4

1 0 1 0 0 1 0 1 -4 4 4 -4 0 -i 0 1 1 0 1 0 4 -4 4 1 0 1 0 0 1 0 1 4 -4 -4 4

1 0 -i 0 0 ! 0 1 4 4 -4 0 1 0 1 1 0 1 0 4 -4 -4 4

1 0 1 0 0 1 0 1 -4 4 -4 4 0 1 0 ! 1 0 1 0 -4 4 4 -4 0 1 0 1 1 0 1 0 -4 4 4 -4 1 0 _! 0 0 1 0 1 4 4 4 -4 0 1 0 1 1 0 ! 0 4 -4 -4 4

1 0 ! 0 0 -i 0 ! -4 4 4

1 1 1 1 1 1 1 1 8 -8 8 -8 -8 8 -8 8

1 1 1 1 1 1 1 1 -8 8 -8 8 8 -8 8 -8

1 1 1 1 1 1 1 1 8 -8 8 -8 -8 8 -8 8

1 1 1 1 1 1 ! 1 -8 8 -8 8 8 -8 8 -8

1 1 1 1 1 1 ! 1 -8 8 -8 8 8 -8 8 -8 ! _! l -i -i l -i ! 8 8 8 8 8

1 1 1 1 1 1 1 1 -8 8 -8 8 8 -8 8 -8 l -i ! -i _l ! _l l 8 8 8 8 8 8

A-69

1 1 i 1 1 1 1 1 8 8 8 -8 -8 -8 -8 1 1 1 1 1 1 1 1 8 8 8 8 -8 -8 -8 -8 1 1 1 1 1 1 1 1 8 8 8 8 -8 -8 -8 -8 1 1 1 1 1 1 1 1 8 8 8 8 -8 -8 -8 -8

1 1 1 1 1 1 1 1 -w -w -8 -w w 8 8 8 1 1 1 1 1 1 1 1

-8 -w -8 -8 8 8 8 8 1 1 1 1 1 1 1 1 -w -w -8 -w w i i 8 1 1 1 1 1 1 1 1 -w -w -8 -8 8 8 8 8

1 1 1 1 1 1 1 1 w -8 w -w w -w i -w 1 1 1 1 1 1 1 1 -w i -w 8 -w 8 -8 8

1 1 1 1 1 1 1 1 8 -8 8 -8 i -8 8 -8

1 1 1 1 1 1 1 1 -8 8 -8 i -8 8 -w 8

1 1 1 1 1 1 1 1 8 -8 8 -8 8 -8 8 -8 1 1 1 1 1 1 1 1

-8 8 -8 8 -8 8 -8 8 1 1 1 1 1 1 1 1 8 -8 i -w i -w i -w

1 1 1 1 1 1 _1 1 -w 8 -w 8 -w i 8 i

Change of basis used to block diagonalize .A(- 1,- 1) with respect to these projections:

121 0 1 1 0 121 121 121 4 2 2 4 4 4

121 1 0 0 1 -i-2; 121 -i-21 4 2 2 4 121 0 1 1 0 121 !21 121 4 -2 -2 4 4 4

121 1 0 0 1 1 l 121 -i-21 4 -2 -2 -42:l 4 ~ 121 1 0 0 1 -i-2; -i-21 121 4 2 -2 4 12l 0 1 1 0 121 1 1 -t21 4 :l -2 2 4 -42~

121 1 0 0 1 -t21 -t21 121 4 -2 2 4

121 0 1 1 0 121 -i-21 -i-21 4 2 -2 4


Lifted ReQresentations for C*{p4mm) at ( -1, -1):

ll{_ 1,-1)(F) =F1 0 {-1, -1) + Fa3{-1, -1) + Fa:l{-1, -1) + Fa{-1, -1)

+ Fp1 ( -1, -1} + Fp:l( -1, -1} + Fp3( -1, -1) + FPt ( -1, -1)

n~-1,-l)(.E) =

ll(_1,- 1)(F) =F1 0 ( -1, -1)- Fa3( -1, -1) + Fa:l( -1, -1)- Fa( -1, -1)

- Fp1(-1, -1) + Fp:l(-1, -1)- Fp3{-1, -1) + Fp4 (-1, -1)

ll(-t,-t)(F) =Ft 0 ( -1, -1) + Fa3( -1, -1) + Fa:l( -1, -1) +Fa( -1, -1)

- Fp1 ( -1, -1)- Fp,( -1, -1)- Fp3( -1, -1)- FPt ( -1, -1)

A-70

ll~- 1 ,-t)(F) =Ft0 ( -1, -1)- F~3( -1, -1) + F~,( -1, -1)- F~( -1, -1)

+ FPl ( -1, -1)- Fp,( -1, -1) + Fp3( -1, -1)- Fp. ( -1, -1)

Torus Point: ( -1, w) for all wE T such that Im( w) > 0 have non-trivial and equal isotropy subgroup.

Typical element F( -1, w) in .A(-l,w):

Ft 0 (-1, w) F~( -1, w) F~,(-1, w) F~J(-1, w) Fp1 ( -1, w) Fp,( -1, w) Fp3 ( -1, w) Fp.( -1, w)

F~3(lii, -1) F1 0 (w, -1) F~(w,-1) F~,(w, -1) Fp,(w, -1) Fp3 (w, -1) Fp.(w, -1) Fp 1 (w, -1)

F~,(-1,w) F~3(-1, w) F10 (-1,w) F~(-1,w) Fp 3 (-1,w) Fp.(-1,w) Fp 1 (-1,w) Fp,(-1, w)

F~(w, -1) F~,(w,-1)

F~3(w,-1) F1 0 (w, -1) Fp.(w, -1) Fp1 (w,-1) Fp,(w, -1) Fp3 (w, -1)

Fp1 (-1,w) Fp2 (-1, w) Fp3 ( --1, w) Fp.(-1, w) F1 0 (-1, w) F~(-1,w) p~,(-1,w)

F~3(-1, w)

Fp2 (w, -1) Fp3 (w, -1) Fp4 (w, -1) Fp 1 (w, -1) F~3(W, -1) Ft 0 (w, -1) F~(w, -1) F~l(W, -1)


1 0 0 0 0 0 1 0 1 0 0 0 0 0 2 -2 2 0 1 0 0 0 0 0 1 0 1 0 0 0 0 2 -2 2 0 0 1 0 1 0 0 0 0 0 1 0 1 0 2 -2 2 2 0 0 0 1 0 1 0 0 0 0 0 1 0 1

2 -2 2 2 0 0 1 0 1 0 0 0

, 0 0 1 0 1 0 -2 2 2 2

0 0 0 1 0 1 0 0 0 0 0 1 0 1 -2 2 2 2 1 0 0 0 0 0 1 0 1 0 0 0 0 0 -2 2 2

0 1 0 0 0 0 0 1 0 1 0 0 0 0 -2 2 2

1 2 0 0 0 0 0 1 2 0

Fp3 ( -1, w) Fp4 (-1,w) Fp 1 ( -1, w) Fp,( -1, w) F~2( -1, w) F~3( -1, w) Ft 0 (-1,w) F~(-1, w)

0 1 2 0 0 0 0 0 1 2

Fp.(w, -1) Fp 1 (w, -1) Fp2 (w, -1) Fp3 (w, -1) F~(w, -1) F~2(w, -1) F~3(tv, -1) Ft 0 (w, -1)

Change of basis used to block diagonalize .A(-1,w) with respect to these projections:

0 0 0 12! 2 0 0 0 12!

2 12! 2 0 0 0 12!

2 0 0 0

0 12! 2 0 0 0 121 2 ~ 0 0

0 0 12! 2 0 0 0 12!

2 0

0 -!2; 0 0 0 12! 2 0 0

0 0 -!21 0 0 0 121 2 0

0 0 0 1 J. 0 0 0 121 -222 2 1 l. 0 0 0 121 0 0 0 -222 2

Lifted Re:Qresentations for c•(p4mm) at ( -1, w):

ll{-l,w)(F) =

[ F1 0 (W, -1)- F,, (W, -1) F ~3 ( -1, w) - F p, ( -1, w) F~2(w,-1)- Fp3(w,-1) F.( -1, w)- F,,( -1, w) ] F~(w, -1)- Fp,(w, -1) F10 (-1,w)- Fp3(-1,w) F~3(w,-1)- Fp4 (w,-1) F~,( -1, w)- Fp

1 ( -1, w)

F~,(w,-1)- Fp3(w,-1) F~(-1,w)- Fp.(-1,w) F10

(w, -1)- Fp 1 (w, -1) F~J( -1, w)- Fp,( -1, w) F~3(w,-I)- Fp.(w,-1) F~,( -1, w)- Fp 1 ( -1, w) F~(w, -1)- Fp2 (w, -1) Ft D ( -1' w) - Fp3 ( -1, w)

A-71

[

F1D(w, -1) + Fp1 (w, -1) F~3{-1, w) + Fp2 (-1, w) F~2(w, -1) + Fp3(w, -1) F~{-1, w) + Fp4 ( -1, w) ] F~(w, -1) + Fp2(w, -1) F1D{-1, w) + Fp3{-1, w) F~3(w, -1) + Fp 4 (w, -1) F~2{-1, w) + Fp 1 ( -1, w) F~2(w, -1) + Fp3(w, -1) F~( -1, w) + Fp.( -1, w) F1D(w, -1) + Fp1 (w, -1) F~3( -1, w) + Fp2 ( -1, w) F~3(w, -1) + Fp4 (w,-1) F~2(-1, w) + Fp1(-1, w) F~(w, -1) + Fp2(w, -1) F1D( -1, w) + Fp3 ( -1, w)


Typical element F( -1, 1) in .A(-1,1):

F1D(-1, 1) F~3(1, -1) F~2( -1, 1) F~(1,-1) Fp1 ( -1, 1) Fp2(1, -1) Fp3( -1, 1) Fp4 (1, -1) F~{-1,1) F1D{1, -1) F~3( -1, 1) F~2(l, -1) Fp2( -1, 1) Fp3(1,-1) Fp4 (-1, 1) Fp

1 (1, -1)

F~2{-1, 1) F~(1,-1) F1D{-1,1) F~3(1, -1) Fp3( -1, 1) Fp4 (1,-1) Fp1 (-1, 1) FP2 (1, -1) F~3(-1, 1) Fq2{1, -1) Fq( -1, 1) F1 0 (1, -1) Fp4 (-1,1) Fp1 {1,-1) Fp2( -1, 1) Fp3(1,-1) FP1 ( -1, 1) Fp2(1, -1) Fp3( -1, 1) Fp4 (1, -1) F1 0 ( -1, 1) Fq3(1, -1) F~2(-1, 1) F(7(1, -1) Fp2 (-1, 1) Fp3(1, -1) Fp4 {-1,1) Fp 1 {1, -1) F~{-1, 1) F1 0 {1, -1) F~J(-1, 1) F~2(1, -1) Fp3( -1, 1) Fp4 (1,-1) Fp1 ( -1, 1) Fp2 (1, -1) F~2(-1,1) F~(1,-1) F1 0 ( -1, 1) F~3(1, -1) Fp4 (-1,1) FPl {1, -1) Fp2( -1, 1) Fp3 (1, -1) F~3( -1, 1) F~2(1, -1) Fq(-1, 1) F1 0 (1, -1)


1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 4 4 -4 -4 4 -4 4 -4 0 1 0 1 0 _l 0 1 0 1 0 1 0 1 0 1

4 4 4 -4 4 -4 4 -4 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 4 4 -4 -4 -4 4 -4 4 0 ! 0 1 0 1 0 -~ 0 1 0 1 0 1 0 1

4 4 -4 -4 4 -4 4 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 -4 -4 4 4 4 -4 4 -4

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 -4 -4 4 4 4 -4 4 -:r

1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 -4 -4 4 4 -4 4 -4 4 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

-4 -4 4 4 -4 4 -4 4

1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 4 -4 -4 4 4 4 4 4 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

4 -4 -4 4 4 4 4 4 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 -4 4 4 -4 4 4 4 4

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 -4 4 4 -:r 4 4 4 4

1 0 1 0 l 0 1 0 ! 0 1 0 ! 0 1 0 -:r 4 4 -4 4 4 4 4 0 _! 0 1 0 1 0 1 0 1 0 1 0 ! 0 1

4 4 4 -4 4 4 4 4 1 0 _1 0 _1 0 1 0 1 0 1 0 ! 0 ! 0 4 4 4 4 4 4 4 4

0 1 0 1 0 1 0 l 0 1 0 1 0 1 0 1 4 -4 -4 4 4 4 4 4

Change of basis used to block diagonalize .A(- 1,1) with respect to these projections:

0 1 1 0 0 1 1 0 2 2 2 2 1 0 0 1 1 0 0 1 2 2 2 2 0 1 1 0 0 1 1 0 2 -2 -2 2 1 0 0 1 1 0 0 1 2 -2 -2 2 0 1 1 0 0 1 1 0 -2 2 -2 2

_1 0 0 1 _1 0 0 1 2 2 2 2

0 1 1 0 0 1 1 0 -2 -2 2 2 -~ 0 0 -~ 1 0 0 1

2 2

Lifted Representations for C*(p4mm) at ( -1, 1):

nt-l,l)(F) =

A-72

[ Ft0 {1, -1) + F172(1, -1)- Fp3 (1, -1)- Fp 1 (1, -1) F17 ( -1, 1) + F17:s( -1, 1)- Fp2( -1, 1) - Fp 4 ( -1, 1) l F17:s(1, -1) + F17 (1, -1)- Fp2(1, -1)- Fp4 (1, -1) Ft 0 (-1, 1) + F172( -1, 1)- Fp1 ( -1, 1)- Fp3 ( -1, 1)

IIl-t,t)(F) =

[ Ft0 ( -1, 1)- F172( -1, 1) + Fp 1 ( -1, 1)- Fp3 ( -1, 1) F17:s(l, -1)- F17 (1, -1) + Fp2(1, -1) - Fp 4 (1, -1) l F0'(-1, 1)- F17:s(-1, 1) + Fp2(-1, 1)- Fp

4(-1, 1) Ft

0(1,-1)- F172(1,-1) + Fp3 (1,-1)- Fp 1 (1,-1)

II~-t,1)(F) =

[ Ft 0 (1, -1) - F172(1, -1) - Fp3 (1, -1) + Fp 1 (1, -1) Fu( -1, 1) - F17 :s( -1, 1) - Fp2( -1, 1) + Fp .. ( -1, 1) l F17:s(l, -1)- F17 (1, -1)- Fp2(1, -1) + Fp

4(1, -1) Ft

0( -1, 1)- F172( -1, 1)- Fp 1 ( -1, 1) + Fp3 ( -1, 1)

II( -t,t)(F) =

[Ft 0 (-1, 1) + F172(-1, 1) + Fp 1 (-1, 1) + Fp3 (-1, 1) F17:s(1,-1) + F17 (1,-1) + Fp2 (1, -1) + Fp 4 (1, -1) l FIT( -1, 1) + F17:s( -1, 1) + Fp2( -1, 1) + Fp

4 ( -1, 1) Ft

0(1, -1) + F172(1, -1) + Fp 3 (1, -1) + Fp 1 (1, -1)



Ft0 (z, 1) F17:s(1,z) F172(z, 1) Fu(1, z) Fp 1 (z, 1) Fp2(1,z) Fp3 (z, 1) Fp. (1, z) F17 (z, 1) Ft 0 (1,z) F17:s(z, 1) F17 2(1, z) Fp2(z, 1) Fp3 (1, z) Fp4 (z, 1) Fp

1 (1, z)

F172(z, 1) F17 (1,z) Ft 0 (z, 1) F17:s(1, z) Fp 3 (z, 1) Fp4(1, z) Fp 1 (z, 1) Fp2(1, z)

F(z, 1) = F17:s(z, 1) F172(1,z) Fu(z, 1) Ft 0 (1, z) Fp 4 (z, 1) Fp 1 (1,z) Fp2 (z, 1) Fp 3 (1, z) Fp 1 (z,1) Fp2(1, z) Fp3 (z, 1) Fp4 (1, z) Ft 0 (z, 1) F17:s(1, z) F17 2(z, 1) Fu(1, z) Fp2(z, 1) Fp 3 (1,z) Fp4 (z, 1) Fp 1 (1, z) F17 (z,1) Ft 0 (1, z) F17 3(z, 1) F17:z(1, z) Fp3 (z, 1) Fp 4 (1,z) Fp1 (z, 1) Fp2(1, z) F172(z, 1) F17 (1,z) F10 (z, 1) F17:s(1, z) Fp4 (z, 1) Fp 1 (1,z) Fp2 (z, 1) Fp3 (1, z) F17 3 ( Z, 1) F172(1, z) FO'(z, 1) Ft 0 (1,z)


1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 2 2 2 -2 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 2 2 2 -2 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 2 2 2 -2 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1

2 2 2 -2 1 0 0 0 1 0 0 0

, 1 0 0 0 1 0 0 0 2 2 -2 2 0 l 0 0 0 l 0 0 0 _l 0 0 0 l 0 0 2 2 2 2 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 2 2 -2 2 0 0 0 l 0 0 0 l 0 0 0 _l 0 0 0 l

2 2 2 2

Change of basis used to block diagonalize .A(z,l) with respect to these projections:

!21 0

0

0

!21 0

0

0

0

!21 0

0

0

-!2; 0

0

Lifted Representations for C*{p4mm) at {z, 1):

ntz,l)(F) =

0

0 121 2

0

0

0

-!2' 0

0

0

0

!21 0

0

0

-!2;

12.1 2 l

0

0

0

-!2; 0

0

0

[

F10 {1, z) + Fp3(1, z) Ff73(z, 1} + Fp,.(z, 1) Ff7l{1, z) + Fp1 {1, z) Ff7(z, 1) + Fp2 (z, 1) ] Ff7(1,z) + Fp,.(1,z) F10 (z, 1) + Fp 1 (z, 1} Ff73{1,z} + Fp2(1,z) Ff72(z, 1} + Fp3(z, 1) Ff72(l, z} + Fp 1 (1, z) Ff7(z, 1} + Fp2 (z, 1) F10 {1, z) + Fp3 {1, z) Ff73 (z, 1) + Fp 4 (z, 1) Ff7,(1,z) + Fpl(1,z} Ff72(z, 1} + Fp3(z, 1) Ff7(1,z) + Fp,.(l,z) F1 0 (z, 1) + Fp 1 (z, 1)

TI~z,l)(F) =

A-73

GROUP p4mg

General element Fin C*(p4mg) :

F1 0 (z, w) F(7(z, w) F(7~(z, w) F(73(z, w) Fp1 (z, w) Fp~(z, w) Fp3 (z, w) Fp4 (z, w)

zF(73(w, z) F10 (iii, z)

zwF(7(w,z) zF(7,(iii, z) Fp,(iii,z)

zFp3(w, z) iiiFp4 (iii,z)

Fp1 (iii, z)


F(7,(z, w) zF(73(z, w) F1 0 (z,w) zF(7(z, iii) zFp3(z,iii)

zwFp4 (z, w) zFp1 (z,iii) zwFp~(z, iii)

Typical element F{1, 1) in A(1,1):

F1 0 (1,1) F173(1, 1) F17,(1, 1) F17 (1, 1) F10 (1, 1) F173(1, 1) F17~(1,1) F17 (1, 1) F1 0 (1,1) F173(1, 1) F17 ~(1, 1) F17 (1, 1) Fp1 {1,1) Fp,(1, 1) Fp3(1,1) Fp,(l, 1) Fp3(1,1) Fp4 (1, 1) FP3(1, 1) FP• (1, 1) Fp1 (1,1) Fp 4 (1, 1) Fp1 (1,1) Fp,(1, 1)

wF(7(w,z) zF(7,(w,z) F(73(w, z) F1 0 (w, z)

wFp4 (w, z)

zFp1 (z, w) Fp,(z, iii)

wFp3(z, w) zFp

4(z, w)

Fp 1 (z, w) Fp,(z, w)

wF(7~(z, w) wFp1 (w, z) Fp,(w,z)

zwFp3(w,z)

zwFp3(z, w) zFp4 (z, iii) F1 0 (z, w) zF(7(z, w)

iiiF(7,(z, iii) F(73(Z, iii)

Fp~(w, z) Fp3 (w, z) Fp4 (w, z) Fp 1 (w, z) F(73(w, z) F1 0 (w,z) F(7(w,z) F(7,(w, z)

wF(73(z, w) F10 {z, w)

zwF17 (z,w)

F17 (1,1) FPl (1, 1) Fp~(1, 1) Fp3(1, 1) Fp4 (1, 1) F17 ,(1, 1) Fp~(1, 1) Fp3(1, 1) Fp. (1, 1) FPl (1, 1) F17 3(1, 1) Fp3(1, 1) Fp4 (1, 1) Fp

1 (1, 1) Fp,(1, 1)

F1 0 (1,1) Fp4 (1,1) Fp1 (1,1) Fp~(1, 1) Fp3 (1, 1) Fp4 (1, 1) F1 0 {1,1) F(73(1,1) F17,{1, 1) F17 (1, 1) Fp1 (1,1) F17 (1,1) F1 0 (l,1) F173(1,1) F17,(1, 1) Fp,(1, 1) F17 ,(1, 1) F17 (1, 1) F1 0 (1, 1) F173(1, 1) Fp3(1, 1) F173(1,1) F17,{1, 1) F17 (1,1) F1 0 (1, 1)


1 ! i i 1 1 _! 1 8 8 -8 -8 8 -8 1 1 1 1 1 1 1 1 8 8 8 8 -8 -8 -8 -8 1 1 1 1 1 1 1 1 8 8 8 8 -8 -8 -8 -8 1 1 1 1 1 1 1 1 8 8 8 8 -8 -8 -8 -8

1 1 1 1 1 1 1 1 -8 -8 -8 -8 8 8 8 8 1 1 1 1 1 1 1 1 -8 -8 -8 -8 8 8 8 8

1 1 1 1 1 1 1 1 -8 -8 -8 -8 8 8 8 8

1 1 1 1 1 1 1 1 -8 -8 -8 -8 8 8 8 8

1 1 1 1 1 1 1 1 8 -8 8 -8 8 -8 8 -8

1 1 1 1 1 1 -1 1 -8 8 -8 8 -8 8 8 8 ! 1 1 1 1 1 ! 1 8 -8 8 -8 8 -8 8 -8

1 ! _! ! -i 1 _! 1 -8 8 8 8 8 8 8 1 -i 1 -i 1 -i 1 -i 8 8 8 8

1 ! _! ! 1 1 1 1 -8 8 8 8 -8 8 -8 8 1 _1 1 _1 1 -i 1 -i 8 8 8 8 8 8

1 1 1 1 1 1 1 1 -8 8 -8 8 -8 8 -8 8

1 1 1 1 ! 1 1 ! 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 ! 1 1 1 1 ! 1 1 8 8 8 8 8 8 8 8 1 1 1 ! 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 ! 1 1 1 8 8 8 8 8 8 8 8 1 1 1 ! ! ! 1 ! 8 8 8 8 g g g g 1 1 1 1 1 1 1 1 8 8 8 i 8 8 8 8 ! ! ! ! ! 1 ! 1 g g g g g g g g

A-74

zwFp.(w,z) wFp

1(w,z)

zwFp,(w,z) wFp3 (w,z) wF17 (w,z) F17~(w, z)

wF17 3(w,z) F1 0 (w, z)

A-75

1 0 -i 0 1 0 1 0 4 4 -4 0 1 0 1 0 1 0 1

4 -4 4 -4 1 0 1 0 1 0 1 0 -4 4 -4 4

0 1 0 1 0 1 0 1 -4 4 -4 4 1 0 1 0 1 0 1 0 4 -4 4 -4 0 1 0 1 0 1 0 1

4 -4 4 -4 1 0 1 0 1 0 1 0 -4 4 -4 4

0 1 0 1 0 1 0 1 -4 4 -4 4 1 0 1 0 1 0 1 0 4 -i -4 4 0 1 0 -i 0 1 0 1

4 -i 4 1 0 1 0 1 0 1 0 -4 4 4 -4

0 1 0 1 0 1 0 1 -4 4 4 -4

1 0 1 0 1 0 1 0 -4 4 4 -4 0 1 0 1 0 1 0 1

-4 4 4 -4 1 0 1 0 1 0 1 0 4 -4 -4 4 0 1 0 1 0 1 0 1

4 -4 -4 4 1 1 1 1 1 1 1 1 8 -8 8 -8 -8 8 -8 8

1 1 1 1 1 1 1 1 -8 8 -8 8 8 -8 8 -8

1 1 1 1 1 1 1 1 8 -~ ~ -~ -~ s -s 8

1 1 1 1 1 1 1 1 -8 8 -8 8 8 -8 8 -8

1 1 1 1 1 1 1 1 -~ ~ -~ ~ ~ -~ 8 -~

1 1 1 1 1 1 1 1 8 -8 8 -8 -8 8 -8 8

1 1 1 1 1 1 1 1 -8 8 -8 8 8 -8 8 -8

1 1 1 1 1 1 1 1 8 -8 8 -8 -8 8 -8 8


121 121 121 0 1 0 1 12J. 4 4 4 2 2 4 :l 121 1 J. 121 1 0 1 0 -i21 4 -42:l 4 2 2 121 121 121 0 1 0 1 121 4 4 4 -2 -2 4 121 -!21 12J. 1 0 1 0 -i21 4 4 l -2 -2

-!21 12J. 12J. 0 1 0 1 -t21 4 l 4 l 2 -2 1 1 1 1 121 1 0 1 0 121 -421 -421 4 2 -2 4 1

-t2i 121 121 0 1 0 1 -t21 4 4 -2 2 -!21 1 1 121 1 0 1 0 121 -421 4 -2 2 4


Lifted ReQresentations for C*(p4mg) :

llf1,1)(F) = [F1 0 (1, 1)- Fq3(1, 1) + Fql(1, 1)- F<7(1, 1) + Fp 1 (1, 1)- Fp2 (1, 1) + Fp3(1, 1)- Fp 4 (1, 1)]

nf1 ,1) (F) = { F1.o ( 1, 1) + Fq3 ( 1, 1) + Fql ( 1, 1) + F<T ( 1, 1) + Fp 1 ( 1, 1) + Fp:l ( 1, 1) + Fp 3 ( 1, 1) + Fp. ( 1, 1)]

TI\ 1

(F)= [ Ft 0 (1, 1)- F<72(1, 1)- Fp 1 (1, 1) + Fp3 (1, 1) F<T(l, 1)- Fq3(1, 1) + Fp 2 (1, 1)- Fp4 (1, 1) l ( ')- Fq3(1, 1)- F<T(l, 1) + Fp2 (1, 1)- Fp4 (1, 1) Ft 0 (l,l)- Fql(1, 1) + Fp 1 (1, 1)- Fp3(1, 1)

A-76

Torus Point: (z, z) for all z E T such that Im(z) > 0 have non-trivial and equal isotropy subgroup.

Typical element in A(z,z):

F(z, z) =

Ft 0 (z, z) zF~3(z, z) F~(z,z) Ft 0 (z,z) F~,(z,z) ~~~(z,z) F~3(z, z) iF~,(z, z) Fp 1 (z,z) Fp,(z,z) Fp,(z, z) zFp3(z, z) Fp3(z,z) zFp,.(z,z) Fp,.(z,z) Fp 1 (z,z)

F~,(z,z) zF~3(z, z) F1 0 (z,z) zF~(z, z) zfp3(z,z) zzFp,.(z, z) zFp1 (z, z) zzFp,(z, z)

zF~(z, z) zF~,(z,z) F~3(z, z) Ft 0 (z, z) zFp,. (z, z) zFp1 (z, z) Fp,(z, z)

zzFp3(z, z)

zFp1 (z, z) Fp,(z, z)

-~zFp3 (z, z) zFp4 (z, z) F1 0 (z, z) zF~(z, z) zF~,(z,z)

F~3(z, z)

Fp,(z,z) Fp3 (z,z) Fp4 (z, z) Fp 1 (z, z) F~3(z,z) F1 0 (z, z) Fo(z, z) F17 2(z, z)


1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 2 -2 2 2 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1

2 -2 2 2 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 2 -2 2 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 2 -2 2 2 0 0 0 1 1 0 0 0 ' 0 0 0 1 1 0 0 -2 2 2 2 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 -2 2 2 2 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1

-2 2 2 2 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 -2 2 2

Change of basis used to block diagonalize A(z,z) with respect to these projections:

12.!. 2 2 0 0 0 0 0 0 12.!. 2 2

0 12.!. 2 2 0 0 0 121 2 0 0

0 0 121 2 0 0 0 12.1 2 2 0

0 0 0 12-! 2

12-! 2 0 0 0

0 0 0 -!2; 12-! 2 0 0 0

-!21 0 0 0 0 0 0 1 2.1 2 2

0 -!2; 0 0 0 121 2 0 0

0 0 -!2; 0 0 0 12-! 2 0

Lifted Representations for C*(p4mg) at (z, z):

[ F, 0 (z, z)- F,,(z, z) zF173(z, z)- zFp3(z, z) F172(z,z)- Fp,.(z,z)

ll{z,z)(F) = Fo(z, z)- Fp3(z, z) F1 0 ("z, z)- zFp,.(z, z) zF173(z, z)- zFp 1 (z, z) F~2(z, z)- Fp,.(z, z) F~(z, z)- Fp 1 (z, z) F1o (z, z)- Fp2 (z, z) F17 3(Z, z)- Fp 1 (z, z) zF172(z, z)- Fp2(z, z) zFo(z,z)- zFp3(z,z)

[ F,0 (z, Z) + zF,, (z, Z) zF~2(z, z) + Fp2(z, z) zF~(z, z) + zFp3(z, z) II2 (F)= zFo2(Z, z) + FP2(z, z) Ft 0 (z, z) + zFp,.(z, z) zF173(z, z) + zFp 1 (z, z)

(z,z) - F173(z, z) + Fp3(z, z) F~(z, z) + Fp 1 (z, z) Ft 0 (z, z) + Fp2 (z, z)

zF17 (z, z) + zFp 1 (z, z) zF173(z, z) + zFp3(z, z) F~2(z, z) + Fp" (z, z)

0

0 1 2 0 0

0 0 1 2

zFp3(z,z) zFp,.(z, z) Fp 1 (z, z) Fp,(z,z) zF172(z, z) zF~3(z, z) F1 0 (z, z) zzF0 (z, z)

zzFp,. (z, z) ~fP1 (z, z) zzFp2(z,z) zFp3(z, z) zFo(z, z) F17 2(z, z)

zF173(z,z) F1 0 (z, z)

:_F.(z, Z~- zF,, (z, ~)] zF172(z, z)- Fp2(z, z) Fo3(Z, z)- Fp3(z, z)

F1 0 (z, z)- zFp4 (z, z)

F •' ( z, z) + F,, ( z, z) ] F17 (z, z) + Fp3 (z, z) F17 2(z, z) + Fp 4 (z, z) F1 0 (z, z) + Fp2(z, z)

A-77

Torus Point: (-1,-1).

Typical element F( -1, -1) in A(-l,-l):

Flo -F~3 F~, -F~ -FP• Fp, -FP3 Fp,. F~ Flo -F~3 -F~, Fp, FP3 -Fp,. -Fp. F~, F~ Flo F~3 FP3 Fp,. Fp• Fp, F~3 -F~, -F~ Flo -Fp,. FP• Fp, -FP3 (-1, -1) Fp. Fp, -Fp3 -Fp,. Flo F~3 -F~, -F~ Fp, -FP3 Fp,. -FPt -F~ Flo -F~3 F~,

FP3 -Fp,. -FPt Fp, -F~, F~ Flo -F~3 Fp,. FPl FP2 Fp3 F~3 F~, F~ Flo


1 1 • 1 1 . 1 . 1 1 • 1 8 81 -8 81 -81 -8 -81 8 1 . 1 1 • 1 1 1 . 1 1 .

-81 8 81 8 -8 81 -8 -81 1 1 • 1 1 . 1 • 1 1 . 1

-8 -81 8 -8z 8~ 8 8~ -8 1 • 1 1 . 1 1 1 • 1 1 .

-81 8 8~ 8 -8 8' -8 -81 1 • 1 1 • 1 1 1 . 1 1 • 8' -8 -8z -8 8 -8z 8 8'

1 1 . 1 1 . 1 . 1 1 . 1 -8 -8z 8 -8z 8' 8 8' -8 1 . 1 1 . 1 1 1 . 1 1 . 8' -8 -8z -8 8 -8z 8 8~ 1 1 . 1 1 . 1 . 1 1 . 1 8 8' -8 8' -8z -8 -8z 8

1 1 . 1 1 . 1 . 1 1 . 1 8 -8z -8 -81 -8z 8 -8z -8 1 . 1 1 . 1 1 1 . 1 1 . 8' 8 -8z 8 8 8' 8 -8z

1 1 . 1 1 . 1 . 1 1 . 1 -8 8' 8 8' 8' -8 8' 8 1 . 1 1 . 1 1 1 . 1 1 . 8' 8 -8z 8 8 8' 8 -8z 1 . 1 1 . 1 1 1 . 1 1 • 8' 8 -8z 8 8 8' 8 -8z 1 1 . 1 1 . 1 . 1 1 . 1 8 -8z -8 -8z -8z 8 -8z -8 1 . 1 1 • 1 1 1 . 1 1 . 8' 8 -8z 8 8 8' 8 -8z

1 1 . 1 1 . 1 . 1 1 . 1 -8 8' 8 8' 8~ -8 8' 8

1 1 • -l 1 • 1 . 1 1 . -i 8 8' 8' 8' 8 8' 1. 1 1 • 1 1 1 • 1 1 .

-8z 8 8~ 8 8 -8z 8 8~ 1 1 . 1 1 . 1 . 1 1 . 1 -8 -81 8 -81 -8z -8 -81 8 1 . 1 1 . 1 1 1 • 1 1 .

-81 8 8' 8 8 -81 8 81 1 . 1 1 . 1 1 1 . 1 1 .

-81' 8 8' 8 8 -81 8 8' 1 1 . 1 1 . 1 . 1 1 . 1 8 81 -8 81 81 8 81 -8 1 . 1 1 . 1 1 1 . 1 1 .

-81 8 81 8 8 -81 8 81 1 1 . 1 1 . 1 . 1 1 . 1 -8 -81 8 -8z -81 -8 -81 8

1 0 1 0 1 . 0 1 . 0 4 4 -41 41 0 1 0 _1 0 1 • 0 1 •

4 4 4' 4' 1 0 1 0 1 . 0 1 . 0 4 4 -41 41 0 _1 0 1 0 1 .

0 1 .

4 4 -41 -41 1 .

0 1 . 0 1 0 1 0 4' 41 4 -4 0 -ti 0 1• 0 1 0 1

4' 4 4 1 . 0 1 . 0 1 0 1 0 -4z -41 -4 4

0 1 . 0 1• 0 1 0 1 -41 4' 4 4

A-78

t 1 • 1 1 • 1 . 1 1 • 1 -g% -i -g% g% -s g% i 1 • 1 1 • 1 1 1 • 1 1 • i% i -i% i -i -i% -i gZ

1 1 . 1 1 . 1 . 1 1 . 1 -s g% 8 g% -gl i -g% -s 1 . 1 1 . 1 1 1 . 1 1 . g% i -gl i -s -gl -s g% 1 . 1 1 . 1 1 1 . 1 1 . ' -~% -i 81 -~ i Jl i -wz 1 1 . 1 1 . 1. 1 1 . 1 -s g% i gl -g% i -g% -i 1 . 1 1 . 1 1 1 . 1 1 . -wz -~ Jl -~ i Jl i -g% 1 1 . 1 1 . 1 . 1 1 . 1 i -sl -s -g% 81 -8 8% 8 1 0 1 0 1 . 0 1 . 0 4 4 41 -4% 0 1 0 1 0 1 . 0 1 .

4 -4 -41 -41 1 0 1 0 1 . 0 1 . 0 4 4 41 -41 0 -t 0 1 0 1 . 0 1 •

4 4' 4% 1 . 0 1 . 0 1 0 1 0 -41 -4% 4 -4

0 1 . 0 1 . 0 1 0 1 41 -41 4 4 1 . 0 1 . 0 1 0 1 0 4' 4' -4 4 0 1 . 0 1 . 0 1 0 1

4' ..;..4z 4 4

Change of basis used to block diagonalize Ac _1,-1) with respect to these projections:

121 121 121 1 0 121 1 0 4 4 4 2 4 2

-t2!i 12!i -t2!i 0 1 12!i 0 1 4 2 4 2

-!2; -!2; -t2! 1 0 -!2; 1 0 2 2 -t2!i 12!i 121 . 0 1 121i 0 1

4 -4 ~% -2 4 -2 12!i 12!i -t2!i 1 . 0 -t2!i 1 . 0 4 4 2% -2% -!2; 12! 12! 0 1 • -!2; 0 1 .

4 4 -21 2z 12!i 121 . -t2!i 1 . 0 121 . 1 • 0 4 4 ., z -2z -4 "z 2' 12! -!2; -!2; 0 1 . 12! 0 1 . 4 -21 4 2'


Lifted Re12resentations for C*(p4mg) at ( -1, -1):

rrt-1.-1)(F) =Fp4 (-1, -1)- F11,(-1, -1) + Ft0 (-1, -1)- Fp,(-1, -1)

+ Fa3( -1, -1)i + F11 ( -1, -1)i- Fp 1 ( -1, -1)i- Fp3( -1, -1)i

IIf_ 1,-1)(.E) =F10 ( -1, -1)- F113( -1, -1)i- F112( -1, -1)- F11 ( -1, -l)i

- Fp1 ( -1, -1)i + Fp,( -1, -1)- Fp3( -1, -1)i- Fp 4 ( -1, -1)

II( -1,-l)(F) =F10 ( -1, -1) + Fa3( -1, -1)i- Fa,( -1, -1) + F11 ( -1, -1)i

+ Fp1(-1, -1)i + Fp,(-1, -1) + Fp3(-1, -1)i- Fp4 (-1, -1)

II(_ 1,- 1)(F) =Fp4 (-1, -1)- Fa,{-1, -1) + F10{-1, -1)- Fp,(-1, -1)

- Fq3( -1, -1)i- Fl1( -1, -1)i + FP1 ( -1, -1)i + Fp3( -1, -1)i

A-79

Torus Point: ( -1, w) for all w E T such that Im( w) > 0 have non-trivial and equal isotropy subgroup.

Typical element F( -1, w) in .A(-l,w):

-F6~(-m, -1) Ft0 (lD,-1)

-lDF6(lD, -1) -F62(m,-l) FP2(lD,-1}

-FPs(lD, -1} TDFp4 (TD, -1) Fp1 (m,-1)

F62(-1,TD) -F~(-1,TD) Ft0 (-l,-1D) -F6(-1,1D) -Fp3 ( -1, 1D)

-wFp4 ( -1, 1D) -Fp1 ( -1, -m)

-wFP2( -1, 1D)

wF6(w,-1) -F62(w,-1) F6,(w,-1) Ft0 (w,-1)

wFp4 (w,-1) wFPt(w, -1) Fh(w,-1)

-wFp3 ( w, -1)


1 0 0 0 0 0 1w! 0 2 2

0 1 0 0 0 0 0 1-.l. 2 -2W2

0 0 1 0 1-; 0 0 0 2 2w 0 0 0 1 0 1-; 0 0 2 2w 0 0 .!.w1 0 1 0 0 0 2 2 0 0 0 .!.w! 0 1 0 0 2 2

.!.U}i 0 0 0 0 0 1 0 2 2 0 -!w! 0 0 0 0 0 1

2

1 0 0 0 0 0 1 J. 0 2 -2W2 0 1 0 0 0 0 0 1-.l

2 2W2

0 0 1 0 1-1 0 0 0 2 -2w 0 0 0 1 0 -!w1 0 0 2 0 0 -!w1 0 1 0 0 0 2 0 0 0 -!w1 0 1 0 0 2 1-; 0 0 0 0 0 1 0 -2w 2 0 .!.w1 0 0 0 0 0 1

2 2

wFp3 (-1,w) -Fp4 (-1,w) Fp1 (-1,w) FP2(-1,w)

wF62(-1,w) wFcr1( -1, w) F10 (-1,w)

-wF6 (-1,w)

Change of basis used to block diagonalize .A(-l,w) with respect to these projections:

.!.2-i 2 0 0 0 .!.2-i

2 0 0 0

0 ~2; 0 0 0 .!.2-i 2 0 0

0 0 12-i 2 0 0 0 12-i

2 0

0 0 0 12-i 2 0 0 0 !2;

2

0 0 1 J. J. 222W2 0 0 0 1 J. J. -222W2 0

0 0 0 1 J. J. 222W2 0 0 0 1 J. J. -222 W2 12~w~ 2 0 0 0 121-.l. -2 2W2 0 0 0

0 1 J. J. -222W2 0 0 0 1 J. J. 222W2 0 0

Li(t~d Re12Ie~~n tat iQD~ for C*(p4mg) at ( -1, w):

rrt -l,w) (.E) =

-lDFp4 (lD, -1) lDFp1 (m,-1)

-1DFP2(1D, -1) 1DFp3 (1D, -1) lDF~(lD, -1) F~2(TD, -1)

TDF~.l(TD, -1) Ft0 (UJ,-1)

A-80

FC7(-1,w)- Fp4 (-1,w)~ Ft0 (m,-1) -lD~Fp1 (m,-1) -FC7,(-1,lD) +Ffj(-1,lD)wi -FC7,(w, -1) +Fp3 (w, -1)wL

[

F10(-1, w) +wiFp3 ( -1, w) -FC73(tD,-1) +Fp4 (1D, -1)mi Fe1,( -1,1D)- Fp1 ( -1, lD)wi wFe1(w, -1) + Fp,(w, -1)wt~ l FC7,( -1, w) + Fp1(-1, w)mi -lDFC7(lD,-1) +~FP2(lD, -1) Ft0 (-1, lD} -lD Fp3 ( -1, w). F~(w, -1) + Fp4 (w, -1)w! F~(-1,w) +FP2(-1, w)mi -FC7,(m,-1) -~Fp,(lD,-1) -FC7(-1, lD} -Fp4 (-1,1D)wi Ft0 (w, -1) + Fp1(w, -1)w!

FC7(-1, w) +Fp4 (-1, w)lDi Ft0 (1D,-1) +~Fp1 (1D,-1) -FC7,(-1, lD)- Fp,(-1, lD)w! -FC1,(w, -1)- Fp,(w, -1)wl

[

F10( -1, w)- w! Fp3 ( -1, w) -FC7,(1D, -1)- Fp4 {lD, -1}1Di Fe1,( -1, lD) + Fp1 ( -1, lD)w! wFe1( w, -1)- F p,( w, -l)w~ l Fe12(-1, w)- Fp1 (-1, w)mi -lDFC7(1D,-1) -~FP2(1D, -1) F10(-1, lD) +~Fp3(-1;m) Fe1"(w, -1)- Fp4 (w, -1)w! · F~( -1, w) -Fp,(-1, w)lDi -FC7,(tD, -1) +-miFp3 (1D,-1) -Fe1( -1, lD) +Fp4 (-1,1D)wi Ft 0 (w, -1)- Fp1(w, -1)wi


Typical element F( -1, 1) in A(-l,l):

Ft0 (-1, 1) -FC73(l, -1) FC7,(-1, 1) Fe1(1, -1) -Fp1(-1, 1) Fp,(1, -1) Fp,( -1, 1) -Fp4 (1, -1) Fe1(-1, 1) Ft0 (1, -1) -Fe1"( -1, 1) -Fe12(1, -1) FP2( -1, 1) Fp,(1, -1) -Fp4 ( -1, 1) Fp1 (1, -1) F(72(-1, 1) -Fe1(1, -1) Ft0 (-1, 1) Fe13(1, -1) -Fp,(-1, 1) Fp4 (1, -1) Fp1 ( -1, 1) -FP2(1, -1) Fe1"( -1, 1) -FC7,(1, -1) -FC7(-1, 1) Ft0 (1,-1) -Fp4 (-l, 1) Fp1 (1, -1) FP2( -1, 1) Fp,(1, -1) Fp1 ( -1, 1) Fp,(1, -1) -Fp3 (-1, 1) Fp4 (1, -1) Ft0 (-1, 1) FC7,(1, -1) FC7,( -1, 1) Fe1(1, -1) Fp,( -1, 1) -Fp3 (1, -1) -Fp4 (-1, 1) FPl (1, -1) -Fe1( -1, 1) Ft0 (1, -1) Fe1"( -1, 1) Fe12(1, -1) Fp3 (-1, 1) Fp4 (1, -1) -Fp1(-1, 1) FP2(1, -1) FC7,( -1, 1) Fe1(1, -1) Ft0 ( -1, 1) FC73(1, -1) Fp4 ( -1, 1) Fp1 (1, -1) -Fp,(-1,1) -Fp,(1, -1) Fe1"( -1, 1) FC7,(1, -1) -Fe1( -1, 1) Ft0 (1, -1)


1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 2 -2 2 2 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 2 2 2 -2

1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 -2 2 2 2 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 2 2 -2 2 0 0 0 0 1 0 1 0 ' 0 0 0 0 1 0 1 0 2 2 2 -2 0 0 0 0 0 l 0 1 0 0 0 0 0 1 0 1

2 2 2 -2 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 2 2 -2 2 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1

2 2 -2 2

Change of basis used to block diagonalize .A(-l,l) with respect to these projections:

0 0 0 121 0 0 121 0 2 2

0 0· 121 0 0 0 0 121 2 2

0 0 0 -!21 0 0 12.1 0 2 ,

0 0 12.1 0 0 0 0 -!21 2 ,

12.1 2 , 0 0 0 0 12.1 2 , 0 0

0 121 2 0 0 121

2 0 0 0

121 2 0 0 0 0 -t21 0 0

0 121 2 0 0 -t21 0 0 0


Lifted Representations for C*(p4mg) at ( -1, 1):

II(-l,l)(F) =

Fq3(1, -1) + F()'(1, -1) F1 0 (1, -1) + F()',(1, -1) Fp3(1,-1) + Fp1 (1,-1) Fp,(l,-1)- Fp4 (1,-1)

A-81



F10 (z, I) zFq3(I, z) F()',(Y, I) F()'(1,z) zFp1 (z,1) Fp,(I, z) Fp3(Y, I) zFP• (1, z) F()'(z,1) F1 0 (1,z) zFq3(z, 1) zF()',(1, z) Fp,(z, I) Fp3(1,z) zFp.(z, 1) Fp

1 (1, z)

Fql(Z, 1) zF0'(1,z) F1 0 (z, 1) Fu3{1, z) zFp3 (z, 1) Fp4(1,z) Fp 1 (z, 1) zFp2 (1, z)

F(z, 1) = Fu3(z, I) zFu2(1, z) zFu(z, 1) F1 0 (1,z) zFp4 (z, 1) Fp 1 (1, z) Fp2 (z, 1) Fp3 (1, z) Fp1 (z, 1) Fp,(1, z) zFp3 (z, 1) Fp. (1, z) F10 (z, 1) Fu3{1,z) Fu:z(z, 1) Fu(1, z) Fp,(z, 1) zFp3 (1, z) zFp.(z, 1) Fp1 (1, z) zFu(z, 1) F1 0 (1,z) Fu3(z, 1) Fu:z(1, z) Fp3 (z, 1) Fp4 (1, z) zFp 1 (z, 1) Fp,(l, z) Fu,(z, 1) Fu(1,z) F10 (z, 1) Fu3(1, z) Fp,. (z, 1) Fp 1 (1,z) zFp,(z, 1) zFp3 (1, z) Fu3(Z, 1) Fu,(1,z) zFu(z, 1) F1 0 (1, z)


1 0 0 0 1z1 0 0 0 2 2

0 1 0 0 0 1-1 0 0 2 2% 0 0 1 0 0 0 1z1 0 2 2 0 0 0 1 0 0 0 1z1 2 2

1-.l 0 0 0 1 0 0 0 2%2 2 0 1z1 0 0 0 1 0 0 2 2 0 0 1-.l 0 0 0 1 0 2Zl 2 0 0 0 1-.l. 0 0 0 1 2z2 2

1 0 0 0 -!z! 0 0 0 2 0 1 0 0 0 1-~ 0 0 2 -2z 0 0 1 0 0 0

1 1 0 2 -2z' 0 0 0 1 0 0 0 -!z! 2

-j:z! 0 0 0 1 0 0 0 2 0 -!z! 0 0 0 1 0 0 2 0 0 1-.l. 0 0 0 1 0 -2Zl 2 0 0 0 1-1 0 0 0 1 -2z 2


l2.l 2 2 0 0 0 121 2 0 0 0

0 121 2 0 0 0 121

2 0 0

0 0 121 2 0 0 0 121

2 0

0 0 0 121 2 0 0 0 121

2 12.l.-.l. 2 2%2 0 0 0 121-.l. - 2 z2 0 0 0

0 !21z1 0 0 0 -!21z1 0 0

0 0 12.l.-.l. 2 2%2 0 0 0 12.1-.l. -2 :zz2 0

0 0 0 12.1-.l 2 2Z2 0 0 0 12.l.-.l. -2 2%2

A-82

Lifted Representations for C*(p4mg) at (z, 1):

ntz,l)(F) =

[

Ft 0 (z, 1) + z~ Fp1 (z, 1) F(T(z, 1) + Fp,(z, 1)zi F(12(z, 1) + z~Fp3 (z, 1) F(13(z, 1) + z~ Fp4 (z, 1)

II~z,l)(F) =

zF(13(1,z)+Fp,(1,z)z~ F(1,(z,1)+Fp3 (z,1)z~ F(1(1,z)+z1Fp.(1,z) l Ft 0 (1, z) + Fp3 (1, z)z~ zF(13(z, 1) + -zi Fp.(z, 1) zF(1,(1, z) + Fp 1 (1, z)z1' zF(1(1, z) + Fp4 (1, z)z~ F10 (z, 1) + Fp 1 (z, 1)z1 F()'3(1, z) + z! Fp,(1, z) zF()'2(1, z) + Fp 1 (1, z)zi zFO'(z, 1) + Fp,(z, 1)z1 Ft0 (1, z) + Fp3 (1, z)zi

[

F10 (z, 1)- z1 Fp 1 (z, 1) zF()'3(1, z)- Fp, (1, z)zi FO',(z, 1)- Fp3(z, 1)z! F(1(1, z)- z1 Fp. (1, z) l F()'(z, 1)- Fp,(z, 1)z1 F10(1, z)- Fp

3(1, z)z! zF(13(z, 1)- -zi Fp.(z, 1) zF()'2(1, z)- Fp

1 (1, z)'zi

F(12(z, 1)- z1 Fp3 (z, 1) zF()'(1, z)- Fp. (1, z )z!- F10 (z, 1) - Fp 1 (z, 1)z! F()'3(1, z) - z1 Fp2 (1, z) F(13(z, 1)- z! Fp4 (z, 1) zF()'2(1, z)- Fp

1 (1, z)z! zF()'(z, 1) - Fp,(z, 1 )z-1 F10 ( 1, z) - Fp3 (1, z)z1

GROUP p6mm

General element Fin C*(p6mm) :

where

A=

C=

B=

D=

F10 (z, w) Ft~(z, w) Ft~l(z, w) Ft~.s(z, w) Ft7c(z, w) Ft7s(z, w)

Fp1 (z, w) Fp,(z, w) Fp,(z, w) Fp4 (z, w) Fp,(z, w) Fp.(z, w)

Fp1 (z, zw) Fp,(z, zw) Fp,(z,zw) Fp4 (z, zw) Fp,(z, zw) Fp.(z, zw)

F1 0 (z, zw) Ft1(z, zw) Ft12(z, zw) Ft1.s(z, zw) Ft7c(z, zw) Ft1s(z, zw)

Ft1s(zw,z) F1 0 (zw, z) Ft~(zw, z) Ft1,(zw,z) Ft1:s(zw, z) Ft1c(zw, z)

Fp,(zw, z) Fp,(zw, z) Fp.(zw,z) Fp,.(zw, z) Fp.(zw,z) Fp 1 (zw,z)

Fp,(w, z) Fp3 (w, z) Fp.(w, z) Fp,. (w, z) Fp.(w, z) Fp1 (w, z)

Ft7s(w, z) F1 0 (w, z) Ft~(w,z) Ft7,(w,z) Ft~"(w, z) Ft7c(w,z)

Ft1c(w, zw) Ft1s(w,zw) F1 0 ("W,zw) Ft1(w,zw) Ft1,(w,zw) Ft1:s(w,zw)

Fp,(w,zw) Fp.(w,zw) Fp5 (w,zw) Fp.(w,zw) Fp 1 (w,zw) Fp,(w,zw)

Fp,(zw,w) Fp4 (zw,w) Fp5 (zw, w) Fp.(zw, w) Fp 1 (zw, w) Fp,(zw, w)

Ft~•(zw, w) Ft1a(zw, w) F1 0 (zw, w) Ft1(zw, w) Ft1,(zw, w) Ft1.s(zw, w)

Ft1:s(z, w) Ft1c(z, w) Ft1a(z, w) Ft 0 (z,w) Ft1(z, w) Ft7,(z, w)

Fp4 (z, w) Fp 5 (z, w) Fp.(z,w) Fp 1 (z, w) Fp,(z, w) Fp,(z, w)

&. (!,.zw) F:a(z, zw) Fp.(z, zw) FPl (z, zw) Fp,(z,zw) Fp,(z, zw)

Ft~"(z, zw) Ft1c(z, zw) Ft1a(z, zw) F10 (z,zw) Ft1(z, zw) Ft1,(z, zw)

Ft1,(zw,z) · Ft1:s(zw, z) Ft~•(zw, z) Ft1a(zw,z) F10 (zw,z) Ft1(zw, z)

Fp5 (zw,z) Fp.(zw, z) Fp 1 (zw, z) Fp,(zw, z) Fp,(zw, z) Fp.(zw, z)

Fp5 (w,z) Fp.(w, z) Fp 1 (w,z) Fp,(w,z) Fp3 (w, z) Fp.(w,z)

Ft1,(w,z) Ft1,(w,z) Ft1·(w, z) Ft1a(w, z) F1 0 (w,z) Ft1(w,z)

Ft1(w,zw) Ft1,(w,zw) Ft7:s(w, zw) Ft~•(w, zw) Ft1a(w,zw) Ft 0 (w,zw)

Fp.(w,zw) Fp 1 (w,zw) Fp,(w,zw) Fp 3 (w,zw) Fp.(w, zw) Fp 5 (w,zw)

Fp.(zw,w) Fp 1 (zw,w) Fp,(zw,w) Fp3(zw,w) Fp 4 (zw,w) Fp5 (zw,w)

Ft1(zw, w) Ft1,(zw,w) Ft13(zw, w) Ft1·(zw, w) Ft1a(zw, w) Ft 0 (zw,w)

Torus Point: (z, 1) for z E T such that Im(z) > 0 have non-trivial, equal isotropy subgroup.

Typical element in .A(.z,l):

where

A=

C=

Ft 0 (z, 1) Ft7(z,1) Ft12(Z, 1) Ft1:s(z, 1) Ft~•(z, 1) Ft7r.(z,1)

Fp 1 (z, 1) Fp,(z, 1) Fp3 (z, 1) Fp4 (z, 1) Fp,.(z, 1) Fp.(z, 1)

F(z,l)=l~ ~]

Ft~a(z, z) F1 0 (z, z) Ft1(z,z) Ft12(z, z) Ft1,(z, z) Ft~•(z,z)

Fp,(z, z) Fp3 (z, z) Fp4 (z, z) t:es c~ ~i!) Fp.(z, z) Fp 1 (z, z)

Ft~•(1, z) Ft7r.(1, z) Ft 0 (1,z) Ft7(1,z) Ft7,(1,z) Ft7:s(l,z)

Fp3 (1,z) Fp4 (1, z) Fp11 (1,z) Fp.(1, z) Fp1 (1, z) Fp2 (1, z)

Ft1:s(z, 1) Ft1c(z, 1) Ft1,(z, 1) F10(z, 1) Ft1(z, 1) Ft~,(z, 1)

Fp.(z, 1) Fp 11 (z, 1) Fp.(z, 1) Fp 1 (z, 1) Fp,(z, 1) Fp,(z, 1)

Ft7,(z, z) Ft1:s(z, z) Ft7c(z, z) Ft1,(z, z) F10 (z, z) Ft1(z,z)

Fp 11 ('Z, z) Fp.(z, z) Fp1 (z, z) Fp,(z, z) Fp,(z, z) Fp.(z, z)

Ft7(1, z) Ft72(1, z) Ft73(1, z) Ft7·(1, z) Ft7,(1, z) F1 0 (1, z)

Fp.(1, z) Fp 1 (1, z) Fp,(l, z) Fp3 (l, z) Fp.(1,z) Fp11 (1, z)

A-83

A-84

Fp1 (z,z) Fp2(1, z) Fp3('z, 1) Fp.(z, z) Fp5 (1, 'Z) Fp6 (z, 1) Fp2 (z, z) Fp3(1, z) Fp.(z, 1) Fp5 (z, z) Fp6 (1, z) Fp 1 (z,1)

B= Fp3(z,z) Fp4 (1,z) Fp5 (Z', 1) Fp6 (z, z) Fp 1 (1, z) Fp2 (z, 1) Fp4 (z, z) Fp5 (1,z) Fp6 {'z, 1) Fp1 (z, z) Fp2(1, z) Fp3(z, 1) Fp5 (z, z) Fp6 (1,z) Fp1 (z, 1) Fp2 (z, z) Fp3(1, z) Fp4 (z, 1) Fp6 (z, z) Fp1 (1,z) Fp2 (z, 1) Fp3(z, z) Fp. (1, z) Fp5 (z, 1)

F1 0 (z, z) F0 s{1, z) Fo•(z, 1) Fo3(z, z) F0 2(1, %) Fo(z, 1) F(1(z,z) F10 (1,z) F(1s(z, 1) F(1·(z, z) F(13(1, z) F(12(Z, 1)

D= F(12(z, z) F(1(1,z) F10 ('Z, 1) Fo!!J(z, z) F(1. (1, z) F(13(Z, 1) F(13(z, z) F(12(1,z) F(1(z, 1) F10 (z, z) F(1!!J(1, z) F(1t(Z, 1) F(1•(z, z) F(13(1,z) F(12('Z, 1) F(1(z, z) Ft 0 (1, z) F(15(Z, 1) F(15(z, z) F(1•(1,z) F(13(z, 1) F0 2(z, z) F(1(1, z) Ft 0 (z, 1)


1 0 0 0 0 0 0 0 0 0 0 1 2 2 0 1 0 0 0 0 1 0 0 0 0 0 2 2 0 0 1 0 0 0 0 1 0 0 0 0 2 2 0 0 0 1 0 0 0 0 1 0 0 0 2 2 0 0 0 0 1 0 0 0 0 1 0 0 2 2 0 0 0 0 0 1 0 0 0 0 1 0 2 2 0 1 0 0 0 0 1 0 0 0 0 0 ' 2 2 0 0 1 0 0 0 0 1 0 0 0 0 2 2 0 0 0 1 0 0 0 0 1 0 0 0 2 2 0 0 0 0 1 0 0 0 0 1 0 0 2 2 0 0 0 0 0 1 0 0 0 0 1 0 2 2 1 0 0 0 0 0 0 0 0 0 0 1 2 2

1 0 0 0 0 0 0 0 0 0 0 1 2 -2 0 1 0 0 0 0 1 0 0 0 0 0 2 -2 0 0 1 0 0 0 0 1 0 0 0 0 2 -2 0 0 0 1 0 0 0 0 1 0 0 0 2 -2 0 0 0 0 ! 0 0 0 0 1 0 0 2 -2 0 0 0 0 0 1 0 0 0 0 1 0 2 -2 0 1 0 0 0 0 1 0 0 0 0 0 -2 2 0 0 1 0 0 0 0 1 0 0 0 0 -2 2 0 0 0 _! 0 0 0 0 1 0 0 0 2 2 0 0 0 0 -t 0 0 0 0 ! 0 0 2 0 0 0 0 0 1 0 0 0 0 1 0 -2 2

_! 0 0 0 0 0 0 0 0 0 0 ! 2 2

Olange of basis used to block diagonalize A(z,l) with respect to these projections:

0 0 !2~ 0

0 ~2! 0 0 0 0 0 0

~2! 0 0 ~2! 0 0 0 0 0 0 0 0

0 0 0

~2! 0 0 0 0

~2! 0 0 0

0 0 0 0 0

!2! 0 0 0 0

121 '2 .,

0

!2! 0 0 0 0 0 0 0 0 !2! 0 0 0 0 0 0 0 0 0 -~2! 0 0

~2! 0

Lifted Representations for C*(p}mm) at (z, 1):

ntz,l)(.E) =

0 0

121 '2 .,

0 0 0 0

-~2! 0 0 0 0

0

!2! 0 0 0 0 121 -'2 .,

0 0 0 0 0

0 0 0 0 0

~2! 0 0 0 0

-~2! 0

0 0 0

121 '2 .,

0 0 0 0

-~2! 0 0 0

121 '2 "2

0 0 0 0 0 0 0 0 0 0 121 -'2 "2

A-85

(Fq2 + Fp.)(z, z) (Fe1 + Fp6 )(1, z) (F10 + FP6)(2', 1) (Fq& + Fp1 )(2', z) (Fe1• + Fp2 )(1, z) (Fq3 + Fp3 )(z, 1)

[

(Flo+ Fp2)(z, z) (Fq& + Fp3 )(1, z) (Fe1• + Fp4 )(!, 1) (Fq3 + Fp6 )(!, 2') (Fe12 + Fp6 )(1, z) (Fe1 + Fp1 )(z, 1) (Fe1 + Fp3 )(z, z) (F10 + Fp4 )(1, z) (Fq& + Fp6 )(2', 1) (Fe1• + Fp6 )(2', z) (Fq3 + Fp1)(1, z) (Fq2 + F P2)(z, 1)

(Fq3 + Fp5 )(z, z) (Fe12 + F P6)(1, z) (Fe1 + Fp1 )(:, 1) (F10 + Fp2 )(':, ~) (Fq& + Fp3 )(1, z) (Fe1• + Fp.)(z, 1) (Fe1• + FP&)(z, z) (Fq3 + Fp1 )(1, z) (Fq:~ + FP2)(~, 1) (Fe1 + Fp3 )(~, z) (Flo+ Fp4 )(1, z) (Fq& + Fp5 )(z, 1) (Fq& + Fp1 )(z, z) (Fe1• + F P2)(1, z) (Fq3 + Fp3 )(!, 1) (Fq:~ + Fp4 )(2', z) (Fe1 + Fp5 )(1, z) (Flo+ Fp6 )(z, 1)

TI[z,l)(F) =

Torus Point: (z, z) for z E T such that Im(z) > 0 and Re(z) > -! have non-trivial, equal isotropy

subgroup.

Typical elerrent in A(z.~):

where

A-86

Fp 1 (z, z) Fp,(z 2 ,z) Fp3 (z, z2 ) Fp4 (z, z) Fp5 (z2 , z) Fp6(z, z-2)

Fp,(z, z) Fp3 (z 2 ,z) Fp4 (z, z2 ) Fp 5 (z, z) Fp6(z2 , z) Fp.(z, z 2)

C= Fp3 (z, z) Fp4 (z 2 , z) FPt>(z, z2) Fp6(z, z) Fp. (z2

, z) Fp,(z, z 2)

Fp4 (z, z) Fp5 (z 2 ,z) Fp6(z, z2 ) Fp 1 (z, z) Fp,(z2 ,z) Fp 3 (z, z2)

Fp5 (z, z) Fp6(z 2 , z) Fp 1 (z, z2 ) Fp,(z, z) Fp3 (z2 ,z) Fp.(z, z 2)

Fp6(z, z) FP• (z2, z) Fp,(z, z2 ) Fp3 (z, z) Fp.(z2 ,z) Fpa(z, z-2)

Fp1 (z, z2 ) Fp,(z, z) Fp3 (z2 ,z) Fp.(z, z 2) Fp5 (z,z) Fp6(z 2 , z)

Fp,(z, z2 ) Fp3 (z, z) Fp4 (z2 ,z) Fpa(z, z-2) Fp6(z,z) Fp1 (z2 , z)

B= Fp3 (z, z2 ) Fp4 (z,z) Fp5 (z2 ,z) Fp6(z, z-2) Fp1 (z, z) Fp,(z2, z)

Fp4 (z, z2 ) Fp5 (z, z) Fp6(z2 ,z) Fp 1 (z, z2) Fp,(z,z) Fp3 (z 2 ,z) Fp5 (z, z 2 ) Fp6(z, z) Fp.(z2 ,z) Fp,(z, z2) Fp3 (z, z) Fp4 (z 2 , z) Fp6(z, z2 ) Fp1 (z, z) Fp,(z2

, z) Fp3 (z, z 2) Fp4 (z,z) Fpt>(z 2

, z)

F1 0 (z, z 2) F(1r.(z, z) F(1.(z2 , z) F(13(z, z2) F(1,(z, z) F(1(z 2 ,z) F(1(z, z 2 ) F1 0 (z, z) F(1r.(z2 , z) F(1·(z, z2) F(13(z, z) F(1-:~(z 2 ,z)

D= Ft12(z, z2 ) F(1(z, z) F1 0 (z2 ,z) F(1s(z, z 2 ) F(1·(z, z) F(1:s(z 2 ,z) F(13(z, z2 ) F(1-:~(z, z) F(1(z2 , z) F1 0 (z, z 2

) F(1r.(z, z) F(1•(z 2 ,z) F(1•(z, z2 ) F(13(z, z) F(1-:~(z2 , z) F(1(z, z2) F1 0 (z, z) F(1r.(z 2 ,z) F(1r.(z, z 2) F(1•(z, z) F(13(z2 , z) F(1,(z, z 2 ) F(1(z, z) Ft 0 (z 2 , z)


1 0 0 0 0 0 0 0 0 0 1 0 2 2 0 1 0 0 0 0 0 0 0 0 0 1

~ ~ 0 0 1 0 0 0 1 0 0 0 0 0 2 2 0 0 0 1 0 0 0 1 0 0 0 0 2 2 0 0 0 0 1 0 0 0 1 0 0 0 2 2 0 0 0 0 0 1 0 0 0 1 0 0 2 2 0 0 1 0 0 0 1 0 0 0 0 0

, 2 2

0 0 0 1 0 0 0 1 0 0 0 0 2 2 0 0 0 0 1 0 0 0 1 0 0 0 2 2 0 0 0 0 0 1 0 0 0 1 0 0 2 2 1 0 0 0 0 0 0 0 0 0 1 0 2 2 0 1 0 0 0 0 0 0 0 0 0 1

2 2

1 0 0 0 0 0 0 0 0 0 1 0 2 -2 0 1 0 0 0 0 0 0 0 0 0 1

2 -2 0 0 1 0 0 0 1 0 0 0 0 0 2 -2 0 0 0 1 0 0 0 1 0 0 0 0 2 -2 0 0 0 0 1 0 0 0 1 0 0 0 2 -2 0 0 0 0 0 1 0 0 0 1 0 0 2 -2 0 0 1 0 0 0 1 0 0 0 0 0 -2 2 0 0 0 1 0 0 0 1 0 0 0 0 -2 2 0 0 0 0 1 0 0 0 1 0 0 0 -2 2 0 0 0 0 0 _1 0 0 0 1 0 0 2 2

1 0 0 0 0 0 0 0 0 0 1 0 -2 2 0 _l 0 0 0 0 0 0 0 .o 0 l

2 2

Change of basis used to block diagonalize A(z,::r) with respect to these projections:

0 0 0 0 0 121 0 12~ 0 0 0 0 '2 ., '2 0 ~2! 0 0 0 0 0 0 ~2~ 0 0 0 0 0 12~ 0 0 0 0 0 0 12! 0 0 ~ ~ 0 0 0 0 121 0 0 0 0 0 121 0 ~ ., ~ ., 0 0 0 12! 0 0 0 0 0 0 0 12~ '2 '2

~2! 0 0 0 0 0 ~2! 0 0 0 0 0 0 0 121 0 0 0 0 0 0 121 0 0 ~ ., -~ '2 0 0 0 0 12! 0 0 0 0 0 -~2! 0 '2 0 0 0 12! 0 0 0 0 0 0 0 -~2! '2

~2; 0 0 0 0 0 -~2; 0 0 0 0 0 0 0 0 0 0 121 0 1 1 0 0 0 0 '2 '2 -~2'2 0 12! 0 0 0 0 0 0 1 1 0 0 0 '2 -'22'2

Lifted Representations for G-(p)mm) at (z, -z):

IT(z,::r)(E) =

(F10 + Fp3 )("Z, '%2) (F11• + Fp5 )(z2, z) (F113 + FP&)(z, z2) (Fu + Fp,)('%2, '!) (F11, + Fp1 )('%, z) (F11 , + Fp5 )(-z, ~2 ) (F10 +Fp1 )(z2,z) (F11s + Fp,)(z, z2) ( F113 + Fp4 )(-z2 , -z) (Fu• + Fp3)(!, z) (F113 + FP&)(-z, z2) (Fu +Fp,)(z2,z) (F1 0 + Fp3 )(z, z2) (F11• + Fp5 )(z2, -z) (F11s + Fp4 )(z, z) ( F11s + F p, )('!, '!2) ( F113 + Fp4 )( z2, z) (Fu, + Fp5 )(z, z2) (F1 0 + FpJ('%2, '!) (F11 + FP&)(-z,z) (Fu• + Fp1 )('!, z

2) (Fu, +Fp3 )(z2,z) (Fu + Fp4 )(z, z2) (Fus + Fp6 )(z2, '%) (Ft0 + Fp5 )('%, z) (Fu + Fp4 )('%, '!2) (F11s + Fp6 )(z2, z) (F11• + Fp1)(z,z2) (F11, + Fp3)('!

2 , z) (F113 + Fp,)(-z, z)

Jifz,::r)(E) =

(F1 0 - Fp3 )('!, z2) (F1111- Fp4 )(z, '!) (F11•- Fp5 )(z2, z) (F113- FP&)(z, z2) (F11,- FpJ(-z, z) ( Fu - Fp4 )(z, '%2) (Fto- Fp5 )(z, z) (F115- Fp6 )(z2, z) (F11•- Fp1 )(z, z2) (F113 - Fp,)(z, z) (F11,- Fp5 )('!, -z2) (F11 - Fp6 )(z, z) (Ft0 - Fp1 )(z

2, z) (F115- Fp,)(z, z2) (F11• - Fp3 )(-z, z) ( F173 - F P& )(z, z2) (F11,- Fp1 )(z, z) (F11 - Fp,)(z2 ,z) (F10 - Fp3 )(z, z2) (F115 - Fp4 )('!, z) ( F11• - Fp1 )('!, '%2) (F113- Fp,)(z, '%) (Fu,- Fp3 )(z2, z) (F11 - Fp4 )(z, z2) (Ft0 - Fp5 )(z, z) (Fu5- Fp,)(z,z2) (F11• - Fp3 )(z, z) (F113- Fp4 )(z2, z) (F11, - Fp5 )(z, z2 ) (F11 - Fp6 )(-z, z)

Torus Point: (z, z2) for z E T such that Im(z) > 0 and Re(z) < -! have non-trivial, equal isotropy

subgroup.

Typical elerrent in A(z,z2):

where

A-87

(F11 11 + Fp4 )(z, '!) (F11 + Fp6 )(z, z) (Fu, + Fp1 )(z, z) (F11 • + Fp3 )(z, '!) (Fu3 + Fp,)(z,z) (F1 0 + Fp5)(z, -z)

(F11 -Fp,)(-z2 ,-z) (Fu,- Fp3 )(z2 ,z) (F113 -Fp.)('!2 ,z) (F114 - Fp5)('!2 , z) (Fu5 -Fp6 )(z2 ,z) (Ft 0 -FpJ(:z2,z)

A-88

Fp 1 {z, z2) Fp2(z, z) Fp3(z2, z) Fp.(z, z2) Fp5 (z, z) Fp6 (z2, z) Fp2(z, z2) Fp3(z, z) Fp.(z2

, z) Fps(z, z-2) Fp6(z, z) Fp 1 (z2, z)

C= FP3(z, z2) Fp.(z, z) Fp5 (z2, z) Fp6(z, -z2) FPl {z, z) Fp2 (z2, z) Fp4 (z, z2) Fp 5 (z, z) Fp6 (z2 , z) Fpl(z, -z2) Fp2(z, z) Fp3(z2, z) Fp5 (Z, z2) Fp6 (z, z) Fp 1 (z2, z) Fp2(z, -z2) Fp3(z, z) Fp 4 (z2, z) Fp6 (z, z2) Fp 1 (z, z) Fp2(z2, z) Fp3(z, -z2) FP• (z, z) Fp5 (z2, z)

Fp1 (z, z) Fp,(z2, z) Fp3(z, z2) Fp4 (z, z) Fp5 (z2,z) Fp.(z, z2) Fp,(z, l') Fp3(z2, z) Fp4 (z,z2) Fp5 (l', z) Fp.(z2

, Y) FPl (z, z2)

B= Fp3(z, z) Fp4 (z2, z) Fp5 (z, z2) Fp6 (z, z) Fpl(z-2, z) Fp2(z, z-2) Fp4 (z, z) Fp5 (z2 , z) Fp6 (z, z2) Fp1 (z, z) Fp,(z2,z) Fp3(z, -z2) Fp5 (z, z) Fp6 (z2 , z) FPl (z, z2) Fp,(z, z) Fp3(z2, Y) Fp,.(z, z

2) Fp6 (z, z) Fp 1 (z2, z) Fp,(z, z2) Fp3(z, z) Fp,.(z2

, z) Fpa(z, -z2)

F10 (z, z) F11 a(z2, z) F11.(z, z2) F113(z, z) F11,("z2, z) F11 (z, z2 )

F11(z, z) F1 0 {z2,z) F11a(z, z2) F11•(z, z) F113(z2, z) F11 2(z, z

2 )

D= F11,(z, z) F11 (z2, z) F10 (z, z2) F11a(z, z) F11• (z2

, z) F113(z, z2) F113(z, z) F112(z2, z) F11 (z, z2) F10 (z, z) F11a(z2 , z) Fq·(z, z 2 )

F11•(z, z) F11 3(z2, z) F112(z, z2) F11 (z, z) F1 0 (z2, z) F11a(z, z2) F11a(z, z) F11•(z2, z) F113(z, z2) F11,(z, z) F11(z2

, z) F1 0 (z, z2)


1 0 0 0 0 0 0 0 1 0 0 0 2 -2 0 1 0 0 0 0 0 0 0 1 0 0 2 -2 0 0 1 0 0 0 0 0 0 0 1 0 2 -2 0 0 0 1 0 0 0 0 0 0 0 1

2 -2 0 0 0 0 1 0 1 0 0 0 0 0 2 -2 0 0 0 0 0 1 0 1 0 0 0 0 2 -2 0 0 0 0 1 0 1 0 0 0 0 0 -2 2 0 0 0 0 0 1 0 1 0 0 0 0 -2 2

1 0 0 0 0 0 0 0 1 0 0 0 -2 2 0 1 0 0 0 0 0 0 0 1 0 0 -2 2 0 0 1 0 0 0 0 0 0 0 1 0 -2 2 0 0 0 1 0 0 0 0 0 0 0 1

-2 2

1 0 0 0 0 0 0 0 1 0 0 0 2 2 0 1 0 0 0 0 0 0 0 1 0 0 2 2 0 0 ..!. 0 0 0 0 0 0 0 1 0 2 2 0 0 0 1 0 0 0 0 0 0 0 1

2 2 0 0 0 0 ..!. 0 1 0 0 0 0 0 2 2 0 0 0 0 0 ..!. 0 1 0 0 0 0

2 2 0 0 0 0 1 0 1 0 0 0 0 0 2 2 0 0 0 0 0 1 0 1 0 0 0 0 2 2 1 0 0 0 0 0 0 0 1 0 0 0 2 2 0 ..!. 0 0 0 0 0 0 0 ..!. 0 0 2 2 0 0 1 0 0 0 0 0 0 0 1 0 2 2 0 0 0 ..!. 0 0 0 0 0 0 0 1

2 2

A-89

Change of basis used to block diagonalize A(z,z::l) with respect to these projections:

0 0 0 0 0 12! 121 0 0 0 0 0 ~ ~ ~

121 0 0 0 0 0 0 12! 0 0 0 0 ~ ~ ~ . 0 12! 0 0 0 0 0 0 12! 0 0 0

~ ~ 0 0 0 0 121 0 0 0 0 121 0 0 ~ ~ ~ ~

0 0 0 12! 0 0 0 0 0 0 121 0 ~ ~

0 0 ~21 0 0 0 0 0 0 0 0 ~2; 0 0 0 -~21 0 0 0 0 0 0 121 0 ~ 0 0 -~21 0 0 0 0 0 0 0 0 121

~

0 0 0 0 0 -~21 12! 0 0 0 0 0 ~

-~2; 0 0 0 0 0 0 ~21 0 0 0 0 0 1 1 0 0 0 0 0 0 121 0 0 0 -~2~ ~ ., 0 0 0 0 -~2; 0 0 0 0 12!

~ 0 0

Lifted ReQresentations for C"(]X)mm) at (z, z2):

11{z,z::li.E) =

(Flo- Fp5 )(l', z) (FO's - F P6)('!2,l') (FO'::a- Fp3 )(z2 , z) ( FO':s - F P'l )( z, '!) (FO'•- Fp1 )(l', '!2) (FO'- Fp4 )(z, z2)

(FO'- Fp6 )(%, z) (F1 0 - Fp1)(!2 , -:) (FO':s - Fp4 )(z2, z) (FO' .. - Fp3 )(z, -:) (FO's - Fp:~)(-:, -:2) (FO'::a- Fp5 )(z, z2)

(FO' .. - FP3)(l', z) (FO':s - Fp4 )(l'2, '!) (Flo- Fp1 )(z2, z) (FO'- Fp6 )(z, '!) (FO'::a - Fps)(l', '!2) (FO's- FP'l)(z, z2) ( FO':s - F P'l )('%, z) (FO'::a - Fp;s)(-:2 I'%) (FO's- Fp6 )(z2 ,z) (Flo- Fp5 )(z, '%) (FO' - Fp .. )(-:, '!2) (FO' .. - FpJ(z, z2)

(FO'::a- Fp1 )(-:, z) (FO' - F p::a)(-:2, -:) (FO'.- Fp5 )(z2, z) (FO'a- Fp4 )(z, '%) (Fto - Fp3 )(-:, -:2) (F0'3- FP6)(z, z2)

(FO's- Fp,.)(l', z) (FO' .. - Fps)(l'2' '!) (FO'- F P'l)(z2, z) (FO':~- Fp1 )(z,l') (Fu3 - Fp6 )(l', z2) (Flo- Fp3)(z, z2)

II[z,z2 )(.E) =

(Flo+ Fp3 )(z, z2) (FO's + Fp4 )(-:, z) ( FO' .. + Fps )(-:2 ''%) (Fq3 + Fp6 )(-:, -:2) (FO'::a + FpJ(z, '!) (Fu + FP'l)(z2, z)

(FO' + Fp4 )(z, z2) (Flo+ Fp5 )(l', z) (FO's + Fp6 )('!2, '!) ( FO' .. + FP1 )('!, '!2) (F0'3 + Fp:~)(z, z) (Fu::~ +Fp3)(z2 ,z) (FO'::a + Fp5 )(z, z2) (FO' + FP6)(-:, z) (Fto + FP1 )(%2

, '!) (FO's +FP'l)(%,z2) (FO' .. + Fp 3 )(z, z) (Fq3 + Fp4 )(z2 , z) (Fu3 + Fp6 )(z, i 2) (FO', + Fp1 )(-:, z) (FO' + FP::a)('%2,-:) (Fto + FP3)('%, -:2) (Fus + Fp,.)(z, '%) (Fu• +Fp5 )(z2 ,z) ( FO' .. + Fp1 )( z, z2) (Fq3 + Fp:~)(l', z) (Fu::a + Fp3 )(z2,z) (FO' + Fp,.)(l', z2) (flo +Fp5 )(z,z) (Fus + Fp6 )(z2

, z) (FO's + FP'l)(z, z2) (FO' .. + Fp3 )(-:, z) (F0'3 + Fp4 )(-:

2, -:) (F0'2 + Fp5 )(-:, -:2) (Fu + Fp6 )(z, '%) (F1 0 + Fp1 )(z2 , z)

A-'0 Torus Point: (1, 1).

Typical element in A(l,l):

Flo F~s F~· F~:s F~2 F~ FP1 FP2 Fp:s FP• Fp,. Fpe F~ Flo F~s F~· F~:s F~2 FP2 Fp:s FP• Fp,. Fps FP1 F~2 F~ Flo F~s F~· F~:s Fp:s Fp. Fp,. Fpa FP1 FP2 F~:s F~2 F~ Flo F~a F~· Fp,. Fp,. Fps FPl Fp, Fp:s F~· F~:s F~, F~ Flo F~a Fp,. Fpa Fpl Fp, Fp:s Fp,.

F(1, 1) = F~a F~· F~:s F~, F~ FlD FP• FPt Fp, Fp:s Fp,. Fp,. (1, 1) FP1 FP2 FP3 Fp,. Fp,. FP• Flo F~a F~· F~:s F~, F~

FP2 FP:s Fp,. Fp,. Fp• FPt F~ FlD F~s F~· F~:s F~,

Fp:s Fp,. Fp,. Fpa FPt FP2 F~, F~ Flo F~a F~· F~:s

FP• Fp,. FP• FPt FP2 Fp:s F~:s F~2 F~ FlD F~s F~· Fp,. Fpa FP1 FP2 FP:s Fp,. F~· F~:s F~, F~ F1 0 F~s

Fps FPl Fp, Fp:s Fp,. Fp,. F~a F~· F~:s F~, F~ Flo

A-?\


1 1 1 1 1 1 1 1 1 1 1 1 i2 i2 i2 i2 i2 i2 i2 i2 i2 i2 i2 i2 1 1 1 1 1 1 1 1 1 1 1 1

i2 12 i2 i2 i2 i2 i2 i2 12 i2 12 i2 1 1 1 1 1 1 1 1 1 1 1 1

12 i2 i2 i2 i2 i2 i2 i2 112 i2 i2 i2 1 1 1 1 1 1 1 1 1 1 1

T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 i2 1 1 1 1 1 1 1 1 1 1 1 1 i2 i2 T2 i2 i2 T2 12 T2 T2 T2 T2 T2 1 .1. 1 1 1 1 1 1 1 1 1 1

12 12 T2 T2 i2 T2 T2 T2 T2 T2 F2 i2 1 1 1 1 1· 1 1 1 1 1 1 1

T2 T2 T2 T2 i2 T2 T2 T2 i2 T2 T2 12 1 .1. .1. 1 .1. 1 1~ .1. .1. 1 .1. 1

12 12 12 T2 12 T2 12 12 T2 12 T2 1 1 1 1 1 1 1 1 1 1 1 1

T2 T2 T2 T2 T2 T2 T2 T2 12 T2 T2 T2 .1. 1 .1. 1 .1. 1 1 1 1 1 1 1 12 T2 12 T2 12 ·T2 12 T2 T2 T2 T2 T2 1 1 1 1 1 1 1 1 1 1 1 1

T2 T2 T2 T2 12 T2 T2 T2 T2 T2 T2 T2 1 1 1 1 1 1 1 1 1 1 1 1 12 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2

1 1 1 1 1 1 1 1 1 1 1 1 T2 T2 T2 T2 T2 T2 -12 -12 -n -n -12 -n 1 1 1 1 1 1 1 1 1 1 1 1

T2 12 12 12 12 12 -n -n -n -12 -12 -12 1 1 1 1 1 1 1 1 1 1 1 1 12 T2 T2 T2 T2 T2 -12 -n -12 -n -12 -n 1 1 1 1 1 1 1 1 1 1 1 1

T2 T2 T2 12 T2 12 -12 -n -12 -n -n -12 1 1 1 1 1 1 . 1 1 1 1 1 1

T2 T2 T2 T2 T2 12 -n -n -n -n -n -n 1 1 1 1 1 1 1 1 1 1 1 1 12 T2 T2 12 12 T2 -12 -12 -12 -n -n -12

1 1 1 1 1 1 1 1 1 1 1 1 -12 -n -n -n -n -n T2 T2 T2 T2 T2 T2

1 1 1 1 1 1 1 1 .1. 1 1 1 -12 -n -n -n -12 -n 12 T2 12 12 T2 T2

1 1 1 1 1 1 1 1 1 1 1 1 -n -n -n -n -n -n T2 T2 T2 T2 T2 T2 1 1 1 1 1 1 1 1 1 1 1 1 -n -n -n -n -n -n T2 T2 T2 T2 T2 T2 1 1 1 1 1 1 1 1 1 1 1 1 -n -n -n -n -n -12 T2 T2 T2 T2 T2 T2 1 1 1 1 1 1 1 1 1 1 1 1 -n -n -12 -n -n -12 T2 T2 T2 12 12 12

1 1 , ... --ie-3' 1 1elfi leii 0 0 0 0 0 0 0 - 0e7' 6 t) f , ...

~eii 1 1 'fi --%eii 1 0 0 0 0 0 0 0 -oe 6 0e7' 1 , ... le-ii 1 1 'f"

ie!'. 1 0 0 0 0 0 0 0 e71

f ,,... 0 6e • 6 1 leii 1 -~eii 0 0 0 0 0 0 -0 oe~• t) t) 6e~'

--%eii 1 1 ,,.. . le-3' 1 1 'f" 0 0 0 0 0 0 -0 oe-.r• fe'fi

0 6e • -te'fi -~e-3' ~ ieii 1 0 0 0 0 0 0 1; 1

0 0 0 0 0 0 1 %e-3i 1e':fi 1 -%eii 1 ,,... ,

1o'f. 0 6 -oe-.r• 0 0 0 0 0 0 1 ie-3' 1e'fi 1 -%eii 6e • 0 0 -0 0 0 0 0 0 0 ~eii ~e'fi 1 ieii te'fi -i 11,,... 1 0 0 0 0 0 0 1 --%ei' 1 1e-3'

1 ,,.. .

lJ:i 6e~•

1!¥• 0 oe-.r•

0 0 0 0 0 0 1 --ie-3' 1 ie-3' 0 6 6 0 0 0 0 0 0 0 ie-ii

1 ,,.. . 1 -ieii 1 , ... 1 1e~• 6 -ae-.r• 1

1 leii le'fi 1 le-3' le'fi 0 0 0 0 0 0 0 t) 0 0 t) 0 1 ,,... 1 leii le'fi 1 1eii 0 0 0 0 0 0 oe~• te\.•

0 0 0 0 ie-3' 1 %e-3i le'fi 1 0 0 0 0 0 0 0 0 0

te\• 1 ieii le'fi 1 1eii 0 0 0 0 0 0 te\i

a ie~i 1 1 1 ~ei' 1 ieii 0 0 0 0 0 0 0 1 ,,...

te\• 1e-3i oe-.r• 1 ie-ii 1 0 0 0 0 0 0 0 t) 0 0 0 0 0 0 0 0 1 1 ,,.. .

pe!'. 1 1e'fi pe!'. a t>e~• 1 1

0 0 0 0 0 0 ie-3' 1 1ei' 1 0 oe~•

fe'fi 0 oe-.r•

0 0 0 0 0 0 le'fi 1e-3i 1 1eii 1 0 f ';" 0 0 0 0

0 0 0 0 0 0 i p;~', 1 ie'fi p;~·. 1e • a

0 0 0 0 0 0 ie-3' 1 1ei' 1 0 0

fe'fi 0 0

0 0 0 0 0 0 le'fi ieii 1 1eii 1 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 i2 -12 i2 -12 i2 -n -n 12 -n i2 -n i2

1 1 1 1 1 1 1 1 1 1 1 1 -12 i2 -n i2 -12 i2 i2 -12 12 -12 i2 -n

1 1 1 1 1 1 1 1 1 1 1 1 i2 -n i2 -n i2 -n -n i2 -n i2 -n i2

1 1 1 1 1 1 1 1 .l.. 1 1 1 -n i2 -n i2 -n i2 i2 -n 12 -n i2 -n 1 1 1 1 1 1 1 1 1 1 1 1

i2 -n T2 -n T2 -n -n i2 -n T2 -n i2 1 1 1 1 1 1 1 1 1 1 1 1 -n T2 -n T2 -n i2 T2 -n T2 -12 i2 -n 1 1 1 1 1 1 1 1 1 1 1 1 -n T2 -n T2 -n i2 T2 -n T2 -n i2 -n

1 -1

12

1 1 1 1 1 1 1 1 1 1 T2 T2 -n T2 -n -12 T2 -n T2 -n T2

1 1 1 1 1 1 1 1 1 1 1 1 -n T2 -n T2 -n i2 T2 -n T2 -n T2 -n 1 1 1 1 1 1 1 1 1 1 1 1

T2 -n T2 -n T2 -n -n 11 -n T2 -n T2 1 1 1 1 1 1 1 1 1 1 1 1 -n i2 -12 T2 -12 i2 T2 -n T2 -n i2 -n

1 1 1 1 1 1 1 1 1 1 1 1 12 -n 12 -n T2 -n -n T2 -n T2 -12 12

1 1 1 1 1 1 1 1 1 1 1 1 12 -n 12 -n T2 -n T2 -12 12 -12 T2 -n

1 1 1 1 1 1 1 1 1 1 1 1 -n T2 -n 12 -n T2 -n T2 -12 T2 -12 12 1 1 1 1 1 1 1 1 1 1 1 1

T2 -12 T2 -12 T2 -12 T2 -n T2 -n 12 -12 1 1 1 1 1 1 1 1 1 1 1 1 -n 12 -n 12 -n i2 -12 T2 -12 T2 -n T2

1 1 1 1 1 1 1 1 1 1 1 1 12 -n T2 -n T2 -n T2 -n T2 -n T2 -n

1 1 1 1 1 1 1 1 1 1 1 1 -n I2 -n T2 -n T2 -n T2 -n T2 -12 T2 1 1 1 1 1 1 1 1 1 1 .l.. 1

T2 -n T2 -n T2 -n i2 -12 i2 -n 12 -n 1 1 1 1 1 1 1 1 1 1 1 1 -n 12 -n i2 -n i2 -n T2 -n i2 -n i2

1 1 1 1 1 1 1 1 1 1 1 1 i2 -n 12 -n i2 -n T2 -n T2 -n i2 -12

1 1 1 1 1 1 1 1 1 1 1 1 -n i2 -n T2 -12 i2 -12 T2 -n 12 -n T2 1 1 1 1 1 1 1 1 1 1 1 1

i2 -n T2 -n i2 -n i2 -n i2 -n i2 -n 1 1 1 1 1 1 1 1 1 1 1 1 -n i2 -12 T2 -12 12 -n i2 -n 12 -12 i2

1 1 ,,.. 1e-Ji 1 1 ,,..

1e-Ji 0 0 0 0 0 0 ~ ~e-:r• f ,,.. ~ ~e-:r' f ,,..

1e-Ji 1 ~e-:r' 1e-Ji 1 ~e-:r' 0 0 0 0 0 0 ~ ·~ f ,,.. ~ 1 ,,..

1e-Ji 1 1e-Ji 1 0 0 0 0 0 0 ~e'"'3"'' felfi

~ ~e-:r' fe¥i

~ 1 1e-Ji 1 %e-Ji 0 0 0 0 0 0 ~ ~

fe¥i ~ ~

1e-Ji 1 1e-Ji 1 1 ,,.. 0 0 0 0 0 0

~ ~ ~ ~ ~ ~e-:r' 1 ,,..

!e-Ji 1 1 ,,. . !eii 1 0 0 0 0 0 0 ~e'"'3"'' ~ ~e'"'3"'' ~

0 0 0 0 0 0 1 %eii 1elfi 1 1e-Ji 1 , ... '

1e"\• '8

1e\i '8 'Se'"'3"''

0 0 0 0 0 0 1 1e-Ji 1 1e-Ji ~ 1 , ... ~ ~

1e\• '8

0 0 0 0 0 0 !eii ~e'"'3"'' 1 !e-Ji 1 1 ,.,. ~ 1 ,.,.

0 0 0 0 0 0 1 1eii 1 1e-Ji

1e\;.i '8 'Se'"'3"''

1e\.i '8 'Se'"'3"''

0 0 0 0 0 0 1 1e-Ji 1 !e-Ji '8 !e\.• '8 ~ '8

0 0 0 0 0 0 !eii 1 !eii !e'fi 1 '8 '8

1 %e!i 1elfi 1 -%e!i 1 ¥' 0 0 0 0 0 0 ~ ~ , ,e '

1 ':fi 1 lei• 1elfi 1 -!eii 0 0 0 0 0 0 -~e 1'8lfi

'8 , -!e-Ji 1 !e-Ji 1elfi 1 0 0 0 0 0 0 -'Be ~ '8 -'8

1 -!e-Ji 1 , ... 1 !e-Ji

1 ,,... 0 0 0 0 0 0

-~ -~e'"'3"''

-t!¥•

~e-:r•

1e2Ji 1 -!eii 1 1e-Ji 0 0 0 0 0 0 ~ -~

-t!¥• ~

!eii 1elfi 4 4e-Ji 1 0 0 0 0 0 0

~ '8 1 , .... ie!•. -1

1 ,,.·. 1e-Ji 0 0 0 0 0 0 1

'8 -'8e'"'3"'' ~e'"'3"'' P 2r•

0 0 0 0 0 0 1e-3i 1 -%e-3i 1

fe':fi '8 ,e-:r' -~ ~e-:r'

1e-Ji 1 1 2r • 4e3i 1 0 0 0 0 0 0 '8

fe':fi ~ ,e'"'3"'' ,

0 0 0 0 0 0 -% p;~ii 1 1 J;' -!eii

'8 ~ ,e '

0 0 0 0 0 0 -!e-li 1 1e-Ji 1 1elfi , ~ fe':fi

'8 , 0 0 0 0 0 0 1e¥i -!e-li 1 !e-li 1 , , '8 '8

A·-93

Change of basis used to block diagonalize .A(l,l) with respect to these projections:

:& :& tv'6 0 tv'6 0 :& :& tv'6 0 tv'6 0 6 6 6 6 :& :& :l!efi 0 :1! :l•i 0 _:& -~ ~efi 0 ::f1 211' i 0 6 6 6 6e'T 6 6 6 - 6 ea

4 :& ~ 2•i 0 4-efi 0 :& :& :l!elfi 0 _::f1.efi 0 6 ea 6 6 6 6

~ ~ -iv'6 0 lv'6 0 ~ ~ tJ6 0 -iv'6 0 6 6 - 6 - 6 ~ ~ ~efi 0 ~elfi 0 ~ ~ =iiefi 0 =iieAf-i 0 6 6 - 6 6 6 6 6 6 ~ ~ ~elfi 0 :l!efi 0 ~ ~ ~elfi 0 ~efi 0 6 6 - 6 6 - 6 - 6 6 6 :& _:& 0 t¥'6 0 tv'6 ~ ~ 0 tv'6 0 i/6 6 6 - 6 6 :& _:& 0 _:f!elfi 0 ~efi :& -~ 0 :l!elfi 0 :l!efi

6 6 6 6 6 6 6 6 ~ _:& 0 _:f!efi 0 ~elfi _:& ~ 0 :l!efi 0 ::f1 211'i

6 6 6 6 6 6 6 6 eT

~ -~ 0 -lv'6 0 !v'6 ~ -~ 0 !v'6 0 -!/6 6 6 6 6 ~ -~ 0 :11. 211' i 0 :l!efi -~ ~ 0 :IJ.. 2lr i 0 _.i§:eii 6 6 6e'T 6 6 6 6

eT 6

~ -~ 0 :IJ.efi 0 :l!elfi ~ -~ 0 :l!eii 0 :11 2£i 6 6 6 6 6 6 6 - 6 e 3



Lifted Representations for c• (p6mm)

IIh,1)(F) =F1 0 (1, 1) + Fu&(1, 1) + Fu•(1, 1) + F173(1, 1) + Fu,(1, 1) + Fu(1, 1)

+ Fp 1 (1, 1) + Fp:l(1, 1) + Fp3(1, 1) + Fp4 (1, 1) + Fp&(1, 1) + Fp6 (1, 1)

IIf1,1)(F) =F1 0 (1, 1) + Fu&(1, 1) + Fu•(l, 1) + Fu3(l, 1) + Fu:l(1, 1) + Fu(1, 1)

- Fp 1 (1, 1)- Fp2 (1, 1)- Fp3(1, 1)- Fp 4 (1, 1)- Fp&(1, 1)- Fp6 (1, 1)

nf,,l)(F) = [: ! ] (1, 1)

where

where

1 1 J3. J3. J3. J3. 1 1 a=- 2Fa- 2Fa• - 2'Fa + T'Fas - T'Fa• + T'Fa2 + F1v - '2Fas + Fa3 - '2Fu2

1 1 1 1 v'3. -13. v'3. J3. b =- 2FP3 - 2Fpe + Fp4 + FPt - 2FPs - 2FP2 - T1FP2 - 21Fpa + T 1FP3 + T 1FP6

1 1 1 1 v'3. v'3. v'3. v'3. c =- 2FP3- 2FPe + Fp. + FP•- 2FPs- 2FP2 + 2aFP2 + 2'Fp,.- 2'FP3- 2zFP6

1 1 v'3. v'3. v'3. v'3. 1 1 d =- 2Fa- 2Fa• + T'Fa- T'Far. + 21Fa• - T'Fa2 + Ftv - 2Far. + Fa3 - 2Fa2

5 . ll(t,t)(F) =Ftv(1, 1)- Far.(1, 1) + Fa•(1, 1)- Fa3(1, 1) + Fu2(1, 1)- Fa(1, 1)

- Fp 1 (1, 1) + Fp2(1, 1)- Fp3(1, 1) + Fp4(1, 1)- Fp5 (1, 1) + Fp6(1, 1)

ll(1,1)(F) =Ftv(1, 1)- Fa5(1, 1) + Fa•(l, 1)- Fa3(1, 1) + Fu2(1, 1)- Fa(1, 1)

+ Fp 1 (1, 1)- Fp2(1, 1) + Fp3(1, 1)- Fp4 (1, 1) + Fp5 (1, 1)- Fp6 (1, 1)

A-95 Torus Point: (-1, 1)

Flo Fqa Fqt Fq:s Fq'l Fq Fp. FP'J Fp:s Fp. Fpa Fps Fq Flo Fqa Fqt Fq:s Fq'l FP'J Fp:s Fp. Fpa Fps Fp. Fq'l Fq Flo Fqa Fqt Fq:s Fp:s Fp" Fpa Fps Fp. Fp'J Fq:s Fq'l Fq Flo Fqa Fq• Fp,. Fpa Fp. FP• FP'J FP:s Fqt Fq:s Fq'l Fq F1 0 Fqa Fps FPs Fp. Fp'J FP:s Fp,. Fqs Fqt Fq:s Fq'J Fq Flo Fps Fp. Fp'J Fp:s Fp,. Fpa Fp. FP'J Fp:s Fp,. Fps Fps F10 Fqa Fq" Fq:s Fq'l Fq FP'J Fp:s Fp,. Fpa Fps FP1 Fq Flo Fq5 Fq" Fq:s Fq'l FP:s Fp,. FP5 FP& FPl FP2 Fq2 Fq Flo Fqa Fq" Fq:s Fp,. Fps FP• Fp• Fp, FP:s Fq:s Fq'l Fq F1 0 Fqa Fq" Fpa Fps FP• FP2 Fp:s Fp,. Fq" Fq:s Fq'l Fq Flo Fqa Fp,. Fp. FP'l FP:s Fp,. Fpa Fqa Fq" Fq:s Fq, Fq F10

where columns number 1, 4, 9, 12 are evaluated at ( -1, 1), and columns number 2, 5, 7, 10 are

evaluated at ( -1, -1), and columns number 3, 6, 8, 11 are evaluated at (1, -1).


1 0 0 1 0 0 0 0 1 0 0 1 4 -4 4 -4 0 1 0 0 1 0 1 0 0 1 0 0 4 -4 -4 4 0 0 1 0 0 1 0 1 0 0 1 0 4 -4 -4 4

1 0 0 1 0 0 0 0 1 0 0 1 -4 4 -4 4 0 1 0 0 1 0 1 0 0 1 0 0 -4 4 4 -4 0 0 1 0 0 1 0 1 0 0 1 0 -4 4 4 -4 0 1 0 0 1 0 1 0 0 1 0 0 -4 4 4 -4 0 0 1 0 0 1 0 1 0 0 1 0 -4 4 4 -4 1 0 0 1 0 0 0 0 1 0 0 1 4 -4 4 -4 0 1 0 0 1 0 1 0 0 1 0 0 4 -4 -4 4 0 0 ! 0 0 1 0 1 0 0 1 0 4 -4 -4 4

1 0 0 1 0 0 0 0 1 0 0 1 -4 4 -4 4

1 0 0 1 0 0 0 0 1 0 0 1 4 4 4 4 0 1 0 0 1 0 1 0 0 1 0 0 4 4 4 4 0 0 1 0 0 ! 0 ! 0 0 1 0 4 4 4 4 1 0 0 1 0 0 0 0 1 0 0 1 4 4 4 4 0 1 0 0 1 0 1 0 0 1 0 0 4 4 4 4 0 0 1 0 0 1 0 1 0 0 1 0 4 4 4 4 0 1 0 0 1 0 1 0 0 1 0 0 4 4 4 4 0 0 1 0 0 1 0 1 0 0 1 0 4 4 4 4 1 0 0 1 0 0 0 0 1 0 0 1 4 4 4 4 0 1 0 0 1 0 1 0 0 1 0 0 4 4 4 4 0 0 1 0 0 1 0 1 0 0 1 0 4 4 4 4 1 0 0 1 0 0 0 0 1 0 0 1 4 4 4 4

1 4 0

0 1 -4

0 0

0

0 _1

4 0

0 1 4

1 4 0 0 1 4 0 0

0 0

0 1 4 0

0

-i 0 1 4 0

0

0 1 4 0 0 1 4 0

0

0 1 4 0

0 1 -4

0 1 4 0

0

0 0 1 4 0

0 1 4 0

1 -4 0

0 1 4 0 0

0 0 1 4 0

0

1 4 0 0 1 4 0 0

0 0

0 _1

4 0 0 1 4 0

1 -4 0

0 1 4 0

0

0 1 4 0 0 1 4 0

0

0 1

-4 0 0 1 4 0

1 -4 0 0 1 4 0

0 0 1 4 0 0 1 4 0

0 1 4 0 0

1 -4 0 1 4 0

0

0

1 -4 0 1 4 0

0 1 4 0 0

0

0 1 4 0

0 1

-4 0 1 4 0 0

0

0 .1

-4 0 0

-! 0 1 4 0 0 1 4 0

1 -4 0

0 1 4 0 0

0

0 1 4 0 0

_1 4

1 -4 0

0

0 0 1 4 0

0 1 4

0 _!

4 0 0 1 4 0

-i 0

0 1 4 0 0

0

1 -4 0 1 4 0

0 1 4 0 0

0

0 1

-4 0

0 1 4 0

1 -4 0 0 1 4 0

0

0

1 -4 0 1 4 0 0 1 4 0

1 4 0

0

1 -4 0 0

0 0 1 4 0 0 1 4

Change of basis used to block diagonalize A( _1,1) with respect to these projections.

1 2 0 0

1 -2 0 0

0 0 1 2 0

0

0

i 0 0

0

0 1 2 0

0 0 ~ i 0 0 0 ~ 0 0 0 1

2 0 ~ 0 0 -i 0

0 ~ i 0 0 0

-i 0 ! 0 0 0 0 ~ 0 1 0 0 2 ~ 0 ~ 0 0 0 0 ~

0 1 2 0

0 1

-2 0 1 2 0 0

0 0 1 2 0 0

1 2 0 0

1 -2 0

0

0 0

1 2 0 0 1 2 0 0

0

0

0 1 2 0 0 1 2 0

0

0 1 2 0 0 1 2 0

A-71

Lifted Representations for C* (p6mm) :

TI(-l,l)(F) =

[

(Flo- Fu3 + Fp3 - Fp6 )(-1, 1) (Fu- Fu• + Fp4 - Fp 1 )( -1, 1) (Ful- Fu:s + Fp 5 - Fp2 )( -1, 1)

n[_1,l)(F) =

(Fu:s- Fql- Fp 1 + Fp4 )( -1, -1) (Fu•- Fa- Fp2 + Fp5 )(1, -1) ] (Flo- Fu3- Fp2 + Fp5 )(-1, -1) (Fu:s- Ful- Fp3 + Fp6 )(1, -1) (Fu- Fu•- Fp3 + Fp6 )(-1, -1) (F1 0 - Fa3- Fp4 + FpJ(1, -1)

[

(Flo +Fu3 +Fp2 +FpJ(-1,-1) (Fa:s +Ful +Fp3 +Fp6 )(1,-1) (Fu +Fu• +Fp4 +FpJ(-1,1)] (Fu + Fu• + Fp3 + Fp6 )( -1, -1) (F1 0 + Fa3 + Fp4 + FpJ(1, -1) (Ful + Fu:s + Fp5 + Fp.;z)( -1, 1) (Fu:s + Ful + Fp 1 + Fp4 )( -1, -1) (Fu• + Fu + Fp2 + Fp5 )(1, -1) (Flo+ Fu3 + Fp3 + Fp6 )( -1, 1)

n~-l,l)(F) =

[

(Flo- Fu3 + Fp2 - Fp5 )(-1, -1) (Fu:s- Ful + Fp3 - Fp6 )(1, -1) (Fu- Fa•- Fp4 + FpJ(-1, 1)] (Fu- Fu• + Fp3 - Fp6 )( -1, -1) (Flo- Fu3 + Fp4 - FpJ(1, -1) (Ful- Fu:s- Fp5 + Fp2 )( -1, 1) (Fus- Fu'J + Fp 1 - Fp4 )( -1, -1) (Fu•- Fa+ Fp2 - Fp5 )(1, -1) (F1 0 - Fu3- Fp3 + Fp6 )( -1, 1)

nt-l,l)(F) = .·

A-98

Torus Point: (ei1ri ,e-i1ri)

Typical element in A j•i -j•i : (e ,e )

Flo F6 a F(l• F6 3 F6 2 F(l FPt Fp, Fp3 Fp. Fpa FP6 F(l Flo F6 a F(l. F6 3 F6 2 Fp, FP3 FP• Fpa FP6 FPl F6 2 F(l Flo F6 s F(l. F6 3 FP3 Fp. FPs Fp6 FPt Fp, F6 3 F6 2 F(l Flo F6 a F(l• Fp. Fpa FP6 FPt Fp, FP3 F(l. F6 3 F6 2 F(l F1 0 F6 s Fpa FP6 Fpl FP2 FP3 FP• F(ls F(l• F6 3 F6 2 F(l Flo FP6 FPt FP2 Fp3 FP• Fpr. FPt FP2 FP3 FP• FPr. FP6 Flo F6 a F(l. F6 3 F6 2 F(l FP'l FP3 FP• FPr. FP6 FPl F(l Flo F(lt> F(l• F6 3 F6 2 FP3 Fp. Fpa FP6 FPl FP2 F6 2 F(l Flo F(lr. F(l• F6 3 Fp. Fpa FP6 FPt FP2 FP3 F6 3 F6 2 F(l Flo F6 a F(l• Fpa FP6 FPt FP'l FP3 Fp. F(l• F6 3 F6 2 F(l Flo F(lr. FP6 FPt FP2 FP3 FP• Fpa F6 a F(l• F6 3 F6 2 F(l Flo

where columns 1,3,5,7,9,11 are evaluated at (eill'i, e-ill''), and columns 2,4,6,8,10,12 are evaluated

at ( e-ill'i, eill'').


1/3 0 -1/6 0 -1/6 0 1/6 0 -1/3 0 1/6 0 0 1/3 0 -1/6 0 -1/6 0 1/6 0 -1/3 0 1/6

-1/6 0 1/3 0 -1/6 0 1/6 0 1/6 0 -1/3 0 0 -1/6 0 1/3 0 -1/6 0 1/6 0 1/6 0 -1/3

-1/6 0 -1/6 0 1/3 0 -1/3 0 1/6 0 1/6 0 0 -1/6 0 -1/6 0 1/3 0 -1/3 0 1/6 0 1/6

1/6 0 1/6 0 -1/3 0 1/3 0 -1/6 0 -1/6 0 0 1/6 0 1/6 0 -1/3 0 1/3 0 -1/6 0 -1/6

-1/3 0 1/6 0 1/6 0 -1/6 0 1/3 0 -1/6 0 0 -1/3 0 1/6 0 1/6 0 -1/6 0 1/3 0 -1/6

1/6 0 -1/3 0 1/6 0 -1/6 0 -1/6 0 1/3 0 0 1/6 0 -1/3 0 1/6 0 -1/6 0 -1/6 0 1/3

1/3 0 -1/6 0 -1/6 0 -1/6 0 1/3 0 -1/6 0 0 1/3 0 -1/6 0 -1/6 0 -1/6 0 1/3 0 -1/6

-1/6 0 1/3 0 -1/6 0 -1/6 0 -1/6 0 1/3 0 0 -1/6 0 1/3 0 -1/6 0 -1/6 0 -1/6 0 1/3

-1/6 0 -1/6 0 1/3 0 1/3 0 -1/6 0 -1/6 0 0 -1/6 0 -1/6 0 1/3 0 1/3 0 -1/6 0 -1/6

-1/6 0 -1/6 0 1/3 0 1/3 0 -1/6 0 -1/6 0 0 -1/6 0 -1/6 0 1/3 0 1/3 0 -1/6 0 -1/6

1/3 0 -1/6 0 -1/6 0 -1/6 0 1/3 0 -1/6 0 0 1/3 0 -1/6 0 -1/6 0 -1/6 0 1/3 0 -1/6

-1/6 0 1/3 0 -1/6 0 -1/6 0 -1/6 0 1/3 0 0 -1/6 0 1/3 0 -1/6 0 -1/6 0 -1/6 0 1/3

A-99

1/6 0 1/6 0 1/6 0 -1/6 0 -1/6 0 -1/6 0 0 1/6 0 1/6 0 1/6 0 -1/6 0 -1/6 0 -1/6

1/6 0 1/6 0 1/6 0 -1/6 0 -1/6 0 -1/6 0 0 1/6 0 1/6 0 1/6 0 -1/6 0 -1/6 0 -1/6

1/6 0 1/6 0 1/6 0 -1/6 0 -1/6 0 -1/6 0 0 1/6 0 1/6 0 1/6 0 -1/6 0 -1/6 0 -1/6

-1/6 0 -1/6 0 -1/6 0 1/6 0 1/6 0 1/6 0 0 -1/6 0 -1/6 0 -1/6 0 1/6 0 1/6 0 1/6

-1/6 0 -1/6 0 -1/6 0 1/6 0 1/6 0 1/6 0 0 -1/6 0 -1/6 0 -1/6 0 1/6 0 1/6 0 1/6

-1/6 0 -1/6 0 -1/6 0 1/6 0 1/6 0 1/6 0 0 -1/6 0 -1/6 0 -1/6 0 1/6 0 1/6 0 1/6

1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6

1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6

1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6

1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6

1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6

1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6 0 1/6

Change of basis used to block diagonalize A( j.-i -j.-i) with respect to these projections: e ,e

£1. 0 0 0 £1. 0 0 0 0 ~ ~ 0 3 3 6

0 ~ 0 0 0 0 0 ~ ~ 0 0 ~ 3 3 6 6 _:/i 0 1/2 0 _:/i 0 1/2 0 0 ~ :11. 0 6 6 6 6

0 -~ 0 1/2 0 1/2 0 _£1_ ~ 0 0 :11. 6 6 6 6

_£1_ 0 -1/2 0 _£1_ 0 -1/2 0 0 :11. :11. 0 6 6 6 6

0 -~ 0 -1/2 0 -1/2 0 _£1_ :11 0 0 :11. 6 6 6 6

il 0 1/2 0 _il 0 -1/2 0 0 _:11 ~ 0 6 6 6 6

0 :a 0 1/2 0 -1/2 0 _il ~ 0 0 :11. 6 6 - 6 6 _:a 0 0 0 :fi 0 0 0 0 £§_ :11 0 3 3 - 6 6

0 -~ 0 0 0 0 0 :fi :11. 0 0 :11. 3 3 - 6 6 :a 0 -1/2 0 _:a 0 1/2 0 0 :11. :11. 0 6 6 6 6

0 :a 0 -1/2 0 1/2 0 _£1_ -~ 0 0 :11. 6 6 6 6


Lifted Reuresentations for C'" (p6mm) : 2 . 2 . _2 . 2 . • .

Let e 3 "'' = e 3 1n and e 3"" = e- 3 "'' m the followmg.

[au a12 a13 a14] . Ill F - a21 a22 a23 a24

(ei.-i,e-i"'i)(_)- a31 a32 a33 a34

a41 a42 a43 a44

au =(Flo - 1/2Fa• - 1/2Fa~ + 1/2Fp1 - Fp3 + 1/2Fpa)(eix-i, e-ill'i)

a12 =(Faa- 1/2Fa:s- 1/2Fa + 1/2Fp~ - Fp4 + 1/2Fp6 )(e-jx-i, ei11'i)

a13 = (1/2Fa•Vl- 1/2Fa~Vl + 1/2Fp1 Vl- 1/2VlFp5 )(eix-i, e-jx-i)

a14 = (1/2Fa:sVl- 1/2FaVl + 1/2Fp~Vl- 1/2VlFp6 )(e-i•i, ei•i)

a21 =(Fa - 1/2Fas - 1/2Fa:s + 1/2Fp~ - Fp4 + 1/2Fp6 )(eix-i, e-j1ri)

a22 =(Flo - 1/2Fa• - 1/2Fa~ + 1/2Fp3 - Fp5 + 1/2Fp1 )(e-jx-i, ei"'i)

a23 = {1/2FaaVl- 1/2Fa:s.J3 + 1/2Fp~ .J3- 1/2.J3Fp6 )(eix-i, e-ill'i)

a24 = {1/2Fa•Vl- 1/2Fa~.J3 + 1/2Fp3 .J3- 1/2.J3FpJ(e-ix-i, ei"'i)

aa1 = (1/2Fa~V3- 1/2Fa•V3- 1/2Fpa V§ + 1/2.J3Fp1 )(eill'i, e-ix-i)

aa2 = (1/2Fa.J3- 1/2Fa:sV3- 1/2Fp6 -J3 + 1/2-JaFp~)(e-ill'i, eill'i)

aaa =(Flo- 1/2Fa• + Fp3 -1/2Fp1 - 1/2Fa~- 1/2Fp5 )(eix-i ,e-i1ri)

a34 =(Faa - 1/2Fa:s + Fp4 - 1/2Fp~ - 1/2Fa- 1/2Fp6 )(e-ill'i, ej1ri)

a41 = (1/2Fa:sV3- 1/2FaaV3- 1/2Fp6 V§ + 1/2-J3Fp~)(ejll'i, e-j1ri)

a42 = (1/2Fa~.J3- 1/2Fa• .J3- 1/2Fp1 .J3 + 1/2-J3Fp3 )(e-ill'i, etll'i)

a4a =(Fa- 1/2Faa + Fp.- 1/2Fp~- 1/2Fa:s- 1/2Fp6 )(eill'i, e-j1ri)

a44 =(Flo - 1/2Fa• + Fp 5 - 1/2Fp3 - 1/2Fa~ - 1/2Fp1 )(e-j1ri, ej1ri)

where

a= (Flo+ Fa• +Fa~- Fp3 - FPs- FpJ(e-ill'i, eill'i)

b =(Fa+ Fas + Fa:s- Fp~- FP•- Fp6 )(eill'i, e-ill'i)

c =(Faa+ Fa:s +Fa- Fp~- FP• - Fp6 )(e-ix-i, eill'i)

d =(Flo + Fa• +Fa~ - FP1 - FP3- Fpa)(eix-i, e-jx-i)

where

a= (Flo + Fa• +Fa~ + Fp 1 + Fp3 + Fpa)(ef"'i, e-j1ri)

b = (Faa+ Fa:s +Fa + Fp~ + Fp. + Fp6 )(e-ill'i, eill'i)

c = (Fa +Faa + Fa:s + Fp~ + Fp4 + Fp6 )(efll'i, e-jx-i)

d = (F1 0 + Fa• +Fa~+ Fp3 + FPa + FpJ(e-f"'i, ef11'i)

A-100

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C*-ALGEBRAS OF THE PLANAR CRYSTAL GROUPS AND THEIR