Representations of association schemes and coherent con ...combin/ac2014/docs/hanaki.pdf · splitting p-modular systemof (X;S) if all adjacency algebras of subcon gurations and quotient

Representations of association schemesand coherent configurations

Akihide Hanaki

Shinshu University

June, 2014, Summer School

A. Hanaki (Shinshu Univ.) Representations June, 2014 1 / 52

1 Introduction

2 Coherent configurations and association schemes

3 Ordinary representations

4 Modular representationsp-Modular systemsDecomposition matrices and Cartan matricesModular irreducible representations of commutative schemesIrreducible representations of coherent configurations and their fibers

5 Examples – Coherent configurations defined by quasi-symmetric designsCharacteristic 3, 5, and 7 for 2-(15, 3, 1)-designs


Introduction

We will consider representations of association schemes and coherentconfigurations.

Usually, “representations” mean representations of adjacencyalgebras. But it is not enough. For example, it is known that thereare many strongly regular graphs with same parameters. They haveisomorphic adjacency algebras.

The standard module can contain more combinatorial informations.But standard modules over the complex number field are alwaysisomorphic.

We will consider modular representations, representations over apositive characteristic field.

As an application, we will consider p-ranks of designs.


Coherent configurations and association schemes

Let X be a finite set.

Let S be a partition of X ×X. Namely, X ×X =⋃s∈S s is disjoint

and s 6= ∅ for s ∈ S.

For s ⊂ X ×X, put s∗ = {(y, x) | (x, y) ∈ s}.A pair (X,S) is called a coherent configuration if

there is a subset {11, · · · , 1r} of S such that⋃ri=1 1i = {(x, x) | x ∈ X},

if s ∈ S, then s∗ ∈ S, andthere are nonnegative integers pust (s, t, u ∈ S) such that

pust = ]{z ∈ X | (x, z) ∈ s, (z, y) ∈ t}

when (x, y) ∈ u.

Put Xi = {x ∈ X | (x, x) ∈ 1i} (i = 1, · · · , r) and call Xi a fiber.

(X,S) is said to be homogeneous if r = 1. A homogeneous coherentconfiguration is also called an association scheme (noncommutaive).


Let (X,S) be a coherent configuration with fibers X1, · · ·Xr.

For s ∈ S, we denote by σs the adjacency matrix of s.

By definition, ZS =⊕

s∈S Zσs is a subring of MatX(Z).

σsσt =∑u∈S

pustσu

(pust : intersection number, structure constant)

For a commutative ring R with unity, define RS = R⊗Z ZS and callthis R-algebra the adjacency algebra of (X,S) over R. We often usethe notation σs for the corresponding element in RS.


For s ∈ S, there is a unique pair (i, j) such that σ1iσsσ1j = σs.

PutSij = {s ∈ S | σ1iσsσ1j = σs}.

Then S =⋃i

⋃j S

ij is a partition of S.

(Xi, Sii) is a homogeneous coherent configuration and

RSii =⊕s∈Sii

Rσs

is a subalgebra of RS (with non-common identity).


Example

Let G be a permutation group on a finite set X.

Let S be the set of G-orbits on X ×X by diagonal action of G.

Then (X,S) is a coherent configuration and the G-orbits of X arefibers.

Put G = 〈(12), (34)〉 and H = 〈(12)(34)〉.Then the coherent configurations are

0 1 4 41 0 4 45 5 2 35 5 3 2

0 1 4 51 0 5 46 7 2 37 6 3 2


Ordinary representations

Let (X,S) be a coherent configuration.

We will consider ordinary representations, representations over thecomplex number field C, equivalently representations of CS.

Lemma 3.1

Let A be a subalgebra of Matn(C) closed under transposed conjugate.Namely x ∈ A implies tx ∈ A. Then A is semisimple.

Proposition 3.2

The adjacency algebra CS of a coherent configuration (X,S) issemisimple.

We denote by Irr(S) the set of all irreducible characters of (X,S),where a character is the trace function of a representation.


For s ∈ Sii, define the valency ns = p1iss∗ .

Define a map by σs 7→ ns if s ∈ Sii and σs 7→ 0 if s ∈ Sij (i 6= j).Then the map is the character of an irreducible representation. Wecall this the trivial representation. The trivial representation hasdegree r.

The map ΓS : CS → MatX(C), ΓS(σs) = σs is also a representation.We call this the standard representation. By γS we denote thecharacter of ΓS . Then γS(σs) = |Xi| if s = 1i, and 0 otherwise.

Consider the irreducible decomposition of γS :

γS =∑

χ∈Irr(S)

mχχ.

We call mχ the multiplicity of χ.


Proposition 3.3

Let (X,S) be a homogeneous coherent configuration. For χ ∈ Irr(S), wedenote by eχ the primitive central idempotent corresponding to χ. Then

eχ =mχ

|X|∑s∈S

1

nsχ(σs∗)σs.

Proposition 3.4 (Orthogonality relation)

Let (X,S) be a homogeneous coherent configuration. For χ, ϕ ∈ Irr(S),

mχ

|X|χ(1)

∑s∈S

1

nsχ(σs∗)ϕ(σs) = δχϕ.

There are similar formulas also for non-homogeneous coherentconfigurations.Note that the multiplicity mχ is determined by character values. So,if two coherent configurations have the same intersection numbers,then the standard modules are isomorphic.


Problem 3.5

Find a good algorithm to compute character tables of (homogeneous)coherent configurations.

For finite groups, Dixon–Schneider algorithm is good.

When (X,S) is commutative, the intersections of eigenspacesdetermine the table. But it is not so easy to compute.

Question 3.6 (Bannai-Ito)

Let (X,S) be a (commutative) coherent configuration. Is it true that alleigenvalues of σs (s ∈ S) are cyclotomic numbers ?


Suppose that (X,S) be a homogeneous coherent configuration with|X| prime.

If an eigenvalue of σs (s ∈ S) is non-cyclotomic, then the fieldgenerated by the value have some strong properties.

Toru Komatsu (2006) constructed such field and possible intersectionnumbers (character table).

By the same method, Sho Teranishi (2012) constructed manyexamples.


Example : p = 2875

f1(X) = X4 +X3 − 1071X2 − 7321X − 8850,

f2(X) = X4 +X3 − 1071X2 − 4464X + 102573,

f3(X) = X4 +X3 − 1071X2 + 1250X − 279,

f4(X) = X4 +X3 − 1071X2 + 1250X + 85431

B1 =

0 1 0 0 0

714 185 162 188 1780 162 186 183 1830 188 183 166 1770 178 183 177 176

, B2 =

0 0 1 0 00 162 186 183 183

714 186 182 180 1650 183 180 177 1740 183 165 174 192

,

B3 =

0 0 0 1 00 188 183 166 1770 183 180 177 174

714 166 177 176 1940 177 174 194 169

, B4 =

0 0 0 0 10 178 183 177 1760 183 165 174 1920 177 174 194 169

714 176 192 169 176

Question 3.7

Is there an association scheme with the above intersection numbers ?


Modular representations

“Modular represntation” means representation of an adjacencyalgebra over a positive characteristic field F with non-semisimple FS.

Theorem 4.1

For a homogeneous coherent configuration (X,S) and a field ofcharacteristic p, FS is semisimple if and only if p does not divide F(S).Here F(S) is the Frame number :

F(S) = |X||S|∏s∈S ns∏

χ∈Irr(S)mχ(χ(1)2)

.

We also consider relations between representations over C and F .


p-Modular systems

Let p be a prime number.

Let R be a complete discrete valuation ring with valuation ideal πR.

Let K be the quotient field of R, and let F be the residue field R/πR.

Suppose that K is of characteristic 0 and F is of characteristic p.

Then we say that (K,R, F ) is a p-modular system.

For a coherent configuration (X,S), we say that (K,R, F ) is asplitting p-modular system of (X,S) if all adjacency algebras ofsubconfigurations and quotient configurations over K and F aresplitting algebras. (In general, a finite dimensional k-algebra A iscalled a splitting algebra if A/J(A) is isomorphic to a direct sum offull matrix k-algebras.)

It is enough to consider the case that K and F are big enough.

(If you do not understand, consider (Q,Z, GF (p)), though this is nota p-modular system.)


Let (X,S) be a coherent configuration, and let (K,R, F ) is asplitting p-modular system of (X,S).

For a KS-module M , there is an RS-lattice M̃ such thatK ⊗R M̃ ∼= M . We call M̃ an R-form of M . (An R-form is notuniue.)

This means that, for every representation Φ of KS, there exists asimilar representation Φ′ such that Φ′(σs) ∈ Matd(R) (s ∈ S).

Through an R-form, we can get an FS-module M∗ = M̃/πM̃ .

M∗ is not uniquely determined, but its composition factors areunique.

So the modular character, the trace of the representation, isdetermined.

For FS-modules V and W , we write V ↔W if they have the samecomposition factors.


Question 4.2

Suppose that we know the (ordinary) character table of (X,S). Can wedetermine modular irreducible characters (modules) ?

The answer is NO, in general.

It is difficult even for finite groups. (Problem to determinedecomposition numbers.)

If a group G is solvable, then every simple FG-module has a formM∗ for some simple KG-module (Fong-Swan’s Theorem).

Problem 4.3

Consider a good definition of “solveble coherent configurations”.(Definition by French-Zieschang ?)


Decomposition matrices and Cartan matrices

Let M be a simple KS-module.

Determine dM,V by M∗ ↔⊕

V ∈IRR(FS) dM,V V .

We call dM,V the decomposition number.

Put D = (dM,V )IRR(KS)×IRR(FS) and call this matrix thedecomposition matrix.

Let V be a simple FS-module.

There is a primitive idempotent eV such that eV FS/eV J(FS) ∼= V .

In this case, eV FS is the projective cover P (V ) of V .

For V , W ∈ IRR(FS), definecV,W = dimF HomFS(eV FS, eWFS) = dimF eWFSeV .

We call cV,W the Cartan invariant.

Remark that cV,W is the number of V in P (W ) as simpleconstituents.

Put C = (cV,W )IRR(FS)×IRR(FS) and call this matrix the Cartanmatrix.


It is known thattDD = C.

Especially, the Cartan matrix C is symmetic.

Problem 4.4

Consider properties of decomposition matrices and Cartan matrices.

For group representations,

C is non-singular, andelementary divisors of C are p-power.

But they are not true for (homogeneous) coherent configurations.


Modular irreducible representations of commutativeschemes

Let (X,S) be a commutative association scheme (homogeneouscoherent configuration).

Every irreducible representation over K has degree 1 and has valuesin R.

Taking values modulo πR, we can get modular irreducible characters.

Every modular irreducible character is obtained in this way.


Example

Let (X,S) be an association scheme defined by a Fano plane.

0 1 1 1 1 1 1 2 2 2 3 3 3 31 0 1 1 1 1 1 3 3 2 2 2 3 31 1 0 1 1 1 1 2 3 3 3 2 2 31 1 1 0 1 1 1 2 3 3 2 3 3 21 1 1 1 0 1 1 3 2 3 3 2 3 21 1 1 1 1 0 1 3 3 2 3 3 2 21 1 1 1 1 1 0 3 2 3 2 3 2 32 3 2 2 3 3 3 0 1 1 1 1 1 12 3 3 3 2 3 2 1 0 1 1 1 1 12 2 3 3 3 2 3 1 1 0 1 1 1 13 2 3 2 3 3 2 1 1 1 0 1 1 13 2 2 3 2 3 3 1 1 1 1 0 1 13 3 2 3 3 2 2 1 1 1 1 1 0 13 3 3 2 2 2 3 1 1 1 1 1 1 0


The character table is

g0 g1 g2 g3 mi

χ1 1 6 3 4 1χ2 1 6 −3 −4 1

χ3 1 −1√

2 −√

2 6

χ4 1 −1 −√

2√

2 6

Let p = 7.

Then

χ∗1 = χ∗

4 and χ∗2 = χ∗

3 if we choose a prime ideal containing 3 +√

2,andχ∗1 = χ∗

3 and χ∗2 = χ∗

4 if we choose a prime ideal containing 3−√

2.

It depends on the choice of the prime ideal.


Irreducible representations of coherent configurations andtheir fibers

Irreducible representations of (non-homogeneous) a coherentconfiguration can be determined by them of its fibers.

We denote by IRR(A) the set of representatives of isomorphismclasses of simple right-A modules.

Theorem 4.5

Let K be an algebraically closed field. Let (X,S) be a coherentconfiguration with fibers Xi (i = 1, 2, · · · , r). Put A =

⊕ri=1KS

ii. Thenthere are injections Φi : IRR(KSii)→ IRR(S) (i = 1, 2, · · · , r) satisfyingthe following properties.

1 Define Φ :⋃ri=1 IRR(KSii)→ IRR(KS) by Φ(W ) = Φi(W ) if

W ∈ IRR(KSii). Then Φ is surjective.

2 For V ∈ IRR(KS), V ↓A∼=⊕

W∈Φ−1(V )W .

3 The map Φ preserves the multiplicities.


What is “multiplicity”.

It is the number of the simple module in the standard module KX assimple constituents.

Namely,

mV = dimK HomKS(P (V ),KX) = dimK KSe = rank(e),

where P (V ) is the projective cover of V and e is a primitiveidempotent corresponding to V .

If K = C, then the multiplicity is just the usual multiplicity of acharacter.


How can we get a partition of⋃ri=1 IRR(KSii) ?

Proposition 4.6

For V , W ∈ IRR(A), let e, f be primitive idempotents corresponding toV , W , respectively. Namely eA/eJ(A) ∼= V and fA/fJ(A) ∼= V . ThenΦ(V ) = Φ(W ) if and only if eKSf 6⊂ J(KS).


Examples – Coherent configurations defined byquasi-symmetric designs

Higman defined the type of coherent configurations.

Let (X,S) be a coherent configuration with fibers Xi (i = 1, · · · , r).

The type of (X,S) is an r × r matrix T = (tij) with tij = |Sij |.It is clear that T is symmetric. We omit entries below the maindiagonal.

For example, a symmetric design defines a coherent configuration of

type

(2 2

2

).

We also write the type as (2, 2; 2).


A design is said to be quasi-symmetric if any two blocks intersect ineither x of y points (x 6= y).

For example, a 2-(v, k, 1)-design is quasi-symmetric with x = 0 andy = 1.

The block graph of a quasi-symmetric design is defined as follows :

Points of the graph are blocks of the design.Two blocks are adjacent in the graph if they intersect in y points.

The block graph of a quasi-symmetric design is strongly-regular.

The incidence matrix of a quasi-symmetric design defines a coherentconfiguration of type (2, 2, ; 3).

Conversely, a coherent configuration of type (2, 2; 3) gives aquasi-symmetric design.

We will consider some 2-(v, k, 1)-designs, especially 80 isomorphismclasses of Steiner triple systems 2-(15, 3, 1).


type

(2 2

2

)←→ symmetric design

type

(2 2

3

)←→ quasi-symmetric design

type

(3 2

3

)←→ strongly regular design

strongly regular design : Higman, Klin–Reichard


Let C be an incidence matrix of a combinatorial design.

The p-ranks, the ranks of matrices in characteristic p > 0, of designswith same parameters are not constant, in general.

For 80 nonisomorphic 2-(15, 3, 1)-designs, the 2-ranks of incidencematrices are 11, 12, 13, 14, and 15.

The 3-ranks are 14, and p-ranks are 15 for p 6= 2, 3.

We will focus on the 2-(15, 3, 1)-designs and p = 2.


The parameters of a 2-(v, `, 1)-design and strongly regular graphdefined by the design are :

r =v − 1

`− 1,

b =v(v − 1)

`(`− 1),

n = b,

k = `

(v − 1

`− 1− 1

),

a =v − 1

`− 1− 2 + (`− 1)2,

c = `2.


Let (X1, X2, F ) be a 2-(v, `, 1)-design, where X1 is the set of points,X2 is the set of blocks, and F is the set of flags.

Put X = X1 ∪X2.

Define binary relations si (i = 1, · · · , 9) on X by

s1 = {(x, x) | x ∈ X1}, s2 = {(x, x) | x ∈ X2},s3 = X2

1 − s1,

s4 = {(x, y) ∈ X22 | ](x ∩ y) = 1},

s5 = {(x, y) ∈ X22 | ](x ∩ y) = 0},

s6 = F, s7 = X1 ×X2 − F,s8 = ts6 = {(y, x) | x, y ∈ s6},s9 = ts7 = {(y, x) | x, y ∈ s7}.

Put S = {s1, · · · , s9}.Then (X,S) is a coherent configuration.

Note that the subconfiguration (X2, {s2, s4, s5}) defines a stronglyregular graph.


We can compute the table of multiplications :

σ1 σ3 σ6 σ7

σ1 σ1 σ3 σ6 σ7

σ3 σ3 (v − 1)σ1 (`− 1)σ6 (v − `)σ6

+(v − 2)σ3 +`σ7 +(v − `− 1)σ7

σ8 σ8 (`− 1)σ8 `σ2

+`σ9 +σ4 (`− 1)σ4

+`σ5

σ9 (v − `)σ8 (v − `)σ2

σ9 +(v − `− 1)σ9 (`− 1)σ4 +(v − 2`+ 1)σ4

+`σ5 +(v − 2`)σ5


σ2 σ4 σ5 σ8 σ9

σ2 σ2 σ4 σ5 σ8 σ9σ4 kσ2 (r − 1)σ8 (k − r + 1)σ8

σ4 +aσ4 (k − a− 1)σ4 +`σ9 +(k − `)σ9+`2σ5 +(k − `2)σ5

σ5 (b− k − 1)σ2 (b− k − 1)σ8(k − a− 1)σ4 +(b + a− 2k)σ4 (r − `)σ9 +(b− r − k + `− 1)σ9

σ5 +(k − `2)σ5 +(b− 2k − 2 + `2)σ5

σ6 σ6 (r − 1)σ6 rσ1+`σ7 (r − `)σ7 +σ3 (r − 1)σ3

σ7 (k − r + 1)σ6 (b− k − 1)σ6 (b− r)σ1σ7 +(k − `)σ7 +(b− r − k + `− 1)σ7 (r − 1)σ3 +(b− 2r + 1)σ3

We remark that the coefficients are polynomial of v, `, k, a, r, and b.

Lemma 5.1

If ` and r = (v − 1)/(`− 1) are odd, then v, a, and b are odd and k iseven.


Theorem 5.2

Let F be a field of characteristic 2. Let A be the adjacency algebra of acoherent configuration defined by a 2-(15, 3, 1)-design over F . Supposethat ` and r = (v − 1)/(`− 1) are odd. Then the adjacency algebra of acoherent configuration defined by a 2-(v, `, 1)-design over F is isomorphicto A.

We determine the structure of the algebra A.

What should we do ?

Give informations enough to apply representation theory of finitedimensional algebras.


Morita equivalence – changing the dimensions of simple modules suchthat the module categories are equivalent.

Example : K and Matn(K) are Morita equivalent.

A finite dimensional algebra is said to be basic if every simplemodules are one-dimensional.

Every finite dimensional algebra is Morita equivalent to a basic algebra :

Put IRR(A) = {Vi | i = 1, · · · , `}. Let ei be a primitiveidempotent corresponding to Vi. Put e =

∑ri=1 ei. Then

eAe is basic and Morita equivelent to A.

For Matn(K), e11Matn(K)e11∼= K, where e11 is the

matrix unit.


Let Q = (V, P, s, t) be a quiver where V is a set of vertices, P is a setof arrows, s : P → V , and t : P → V .

A quiver is a directed graph with loops and multiple edges.

The quiver algebra KQ is a K-algebra whose basis is the set of allpaths and multiplication is composition of paths.

The quiver algebra is not necessary finite dimensional.

Let J be the ideal of KQ generated by all arrows.

An ideal I of KQ is said to be admissible if J2 ⊃ I ⊃ Jn for some n.


Let A be a basic algebra.

For V ∈ IRR(A), we denote by eV a primitive idempotentcorresponding to V .

We define the Gabriel quiver Q(A) of A.

The point set is IRR(A).

The number of arrows from V to W is dimK eV (J(A)/J2(A))eW .

Theorem 5.3 (Gabriel)

Let A be a basic algebra. Then there is an admissible ideal I of KQ(A)such that A ∼= KQ(A)/I.

We will compute the Gabriel quiver and the admissible ideal foradjacency algebra of a coherent configuration defined by a2-(15, 3, 1)-design.


Let (X,S) be a coherent configuration defined by a2-(15, 3, 1)-design.

The character table of fibers (Xi, Sii) (i = 1, 2) are

s1 s3 multiplicity

ϕ1 1 14 1ϕ2 1 −1 14

s2 s4 s5 multiplicity

ψ1 1 18 16 1ψ2 1 3 −4 14ψ3 1 −3 2 20

The possibilities of irreducible characters of (X,S) are{ϕ1 + ψ1, ϕ2, ψ2, ψ3} and {ϕ1 + ψ1, ϕ2 + ψ2, ψ3}.Degrees are {2, 1, 1, 1} and {2, 2, 1}.Since |S| = 9 is the sum of squares of degrees, we have the charactertable.

s1 s3 s2 s4 s5 multiplicity

χ1 1 14 1 18 16 1χ2 1 −1 1 3 −4 14χ3 0 0 1 −3 2 20


We consider the modular character table for p = 2.

The character table of fibers are

s1 s3 multiplicity

ϕ1 1 14 1ϕ2 1 −1 14


ψ1 1 18 16 1ψ2 1 3 −4 14ψ3 1 −3 2 20

Consider them in characteristic 2. We have

s1 s3 multiplicity

1 0 11 1 14


1 0 0 11 1 0 34

We can see that the modular character table of (X,S) is


ξ1 1 0 1 0 0 1ξ2 1 1 0 0 0 14ξ3 0 0 1 1 0 34


Bys1 s3 s2 s4 s5 multiplicity

χ1 1 14 1 18 16 1χ2 1 −1 1 3 −4 14χ3 0 0 1 −3 2 20


ξ1 1 0 1 0 0 1ξ2 1 1 0 0 0 14ξ3 0 0 1 1 0 34

we can determine the decomposition matrix and the Cartan matrix.

D =

1 0 0

0 1 10 0 1

C = tDD =

1 0 0

0 1 10 1 2


We write A, B, C for the simple modules corresponding to ξ1, ξ2,and ξ3.

By the Cartan matrix

1 0 0

0 1 10 1 2

, we have the Loewy structures of

projective covers P (A) and P (B).

P (A) = A, P (B) =

[BC

]

The projective cover P (C) is either

CBC

or

[C

B C

].

Note that the algebra has two blocks B1 and B2. The block B1

(containing A) is isomorphic to Mat2(F ). So we want to know B2.

Remark that the algebra is not basic but B2 is basic.


Now we consider the Gabriel quiver.

We can choose primitive idempotent eB = σ3 and eC = σ4 + σ5.

Since eBFSeC = Fσ7 and eCFSeB = Fσ9, we put α = σ7 andβ = σ9.

We have

eB = σ3, eBα = σ7, (e2α)β = 0

eC = σ4 + σ5, eCβ = σ9, (eCβ)α = σ5.

This means that the Gabriel quiver is

Q : eBα //

eCβ

oo

and the relation is αβ = 0.


Theorem 5.4

Let F be a field of characteristic 2. Under the above notations,

FS ∼= Mat2(F )⊕ FQ/(αβ)

where Q is the following quiver :

Q : •α //

•β

oo

There are three simple module A, B, C and the Loewy structures of theprojective covers are

P (A) = [A] , P (B) =

[BC

], P (C) =

CBC

.


Now we apply representation theory of finite dimensional algebras andconsider the standard module FX.

There are three representation types (very rough definition!):

finite : there are finitely many indecomposable modules(infinite) tame : there are infinitely many indecomposable modules butthey can be classified(infinite) wild : there are infinitely many indecomposable modules butthey can not be classified

Problem 5.5

Consider representation types of adjacency algebras of coherentconfigurations (association schemes).

Our case is very easy and we can see that the representation type isfinite.


All indecomposable B2-modules are

M1 =[C], M2 =

[CB

], M3 =

CBC

,N1 =

[B], N2 =

[BC

].

So we can write

FX ∼= [A]⊕ g1M1 ⊕ g2M2 ⊕ g3M3 ⊕ h1N1 ⊕ h2N2.

for some nonnegative integers g1, g2, g3, h1, h2.


Now we consider a 2-(v, `, 1)-design with odd ` and r. As we saw, theadjacency algebras are isomorphic to the algebra defined by a2-(15, 3, 1)-design.

The multiplicities are mA = 1, mB = v − 1, mC = b− 1.

We have

g2 + g3 + h1 + h2 = b− 1, g1 + g2 + gs3 + h2 = v − 1.

Since σ9 is the transposed matrix of σ7, their ranks are equal. Thismeans g2 = h2.

Put s = rank(σ7) and t = rank(σ5). Then we have

(g1, g2, g3, h1, h2) = (b− 2s− 1, s− t, t, v − 2s+ t− 1, s− t).

Remark that the usual 2-rank of the design is rank(σ6) andrank(σ6) = 1 + rank(σ7) = 1 + s.

The structures of standard modules “can” contain more informationthan p-ranks.


Example 5.6

For 80 nonisomorphic 2-(15, 3, 1)-designs, we have the followingparameters (by computation) :

] s = rank(σ7) t = rank(σ5) g1 g2 g3 h1 h2 rank(σ6)

1 10 6 14 4 6 0 4 111 11 8 12 3 8 0 3 125 12 10 10 2 10 0 2 1315 13 12 8 1 12 0 1 1458 14 14 6 0 14 0 0 15

Question 5.7

Is one parameter enough ?


Example 5.8

In [The CRC Handbook of Combinatorial Designs], we can find a list ofdesigns with odd ` and r :

No. v b r ` ]

14 15 35 7 3 80

29 19 57 9 3 ≥ 1.1× 109

57 45 99 11 5 ≥ 16

86 27 117 13 3 ≥ 1011

114 31 155 15 3 ≥ 6× 1016

120 61 183 15 5 ≥ 10

129 91 195 15 7 ≥ 2

I want to compute 2-ranks of them. But I do not have data.


Characteristic 3, 5, and 7 for 2-(15, 3, 1)-designers

Theorem 5.9

Let F be a field of characteristic 3. Under the above notations, the Loewyseries of the projective covers of simple FS-modules of coherentconfigurations obtained by 2-(15, 3, 1)-designs are

P (A) =

AB CA

, P (B) =

[BA

], P (C) =

CAC

and the structures of standard FS-modules are

FX ∼=

AB CA

⊕ 13

CAC

⊕ 7 [C]

for all 80 designs. (By computation)



Theorem 5.10

Let F be a field of characteristic 5. There are two simple FS-modules Aand B with dimF A = 2 and dimF B = 1 . The Loewy series of theprojective covers of simple FS-modules of coherent configurationsobtained by 2-(15, 3, 1)-designs are

P (A) =

[AA

], P (B) = [B]

and the structure of the standard FS-module is

FX ∼=[AA

]⊕ 13 [A]⊕ 20 [B] .



Theorem 5.11

Let F be a field of characteristic 7. There are three simple FS-modules A,B, and C with dimF A = dimF B = 1 and dimF C = 2. The Loewy seriesof the projective covers of simple FS-modules of coherent configurationsobtained by 2-(15, 3, 1)-designs are

P (A) =

[AB

], P (B) =

BAB

, P (C) = [C]

and the structure of the standard FS-module is

FX ∼=

BAB

⊕ 19 [B]⊕ 14 [C] .


Thank you very much !


Documents

Representations of association schemes and coherent con ...combin/ac2014/docs/hanaki.pdf · splitting p-modular systemof (X;S) if all adjacency algebras of subcon gurations and quotient