New York Journal of MathematicsNew York J. Math. 25 (2019) 71–106.

Combinatorial bases of principalsubspaces of modules for twisted affine Lie

algebras of type A(2)2l−1, D

(2)l , E

(2)6 and D

(3)4

Marijana Butorac and Christopher Sadowski

Abstract. We construct combinatorial bases of principal subspaces ofstandard modules of level k ≥ 1 with highest weight kΛ0 for the twisted

affine Lie algebras of type A(2)2l−1, D

(2)l , E

(2)6 and D

(3)4 . Using these bases

we directly calculate characters of principal subspaces.

Contents

1. Introduction 72

2. Preliminaries 74

2.1. Dynkin diagram automorphisms 75

2.2. The lattice vertex operator VL and its twisted module V TL 77

2.3. Twisted affine Lie algebras 80

2.4. Gradings 81

2.5. Higher levels 82

3. Principal subspaces 83

3.1. Properties of W TLk

83

3.2. Twisted quasi-particles 84

4. Combinatorial bases 85

4.1. Relations among twisted quasi-particles 85

5. Proof of linear independence 92

5.1. The projection πR 92

5.2. The maps ∆T (λ,−z) 93

5.3. The maps eαi 95

5.4. A proof of linear independence 97

6. Characters of principal subspaces 99

Acknowledgement 103

References 103

Received March 15, 2018.2010 Mathematics Subject Classification. Primary 17B67; Secondary 17B69, 05A19.Key words and phrases. twisted affine Lie algebras, vertex operator algebras, principal

subspaces, twisted quasi-particle bases.

ISSN 1076-9803/2019

71

72 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

1. Introduction

The study of principal subspaces of standard modules for untwisted affineLie algebras was initiated in [FS] by Feigin and Stoyanovsky and was laterextended by Georgiev in [G]. Motivated by the work of J. Lepowsky andM. Primc in [LP] and extending the earlier work of Feigin and Stoyanovsky,Georgiev constructed bases of principal subspaces of certain standard mod-

ules of the affine Lie algebra of type A(1)l . These bases were described by

using certain coefficients of vertex operators, which are called quasi-particles.From quasi-particle bases, they directly obtained the characters (i.e. multi-graded dimensions) of principal subspaces. The work of Feigin and Stoy-anovsky and Georgiev has since been extended in many ways by other au-thors (cf. [Ar], [Ba], [BPT], [Bu1]–[Bu3], [FFJMM], [J1]–[J2], [JP], [Kan],[Kaw1]–[Kaw2], [Ko1]–[Ko3], [MPe], [P], [T1]–[T3], and many others).

The study of principal subspaces of basic modules for twisted affine Liealgebras was initiated in [CalLM4], where a general setting was given and

the principal subspace of the basic A(2)2 -module was studied. This work was

later extended in [CalMPe] and [PS1]–[PS2] to study the principal subspacesof the basic modules for all the twisted affine Lie algebras, and in [PSW] to acertain lattice setting. In each of these works the authors, using certain ideasfrom the untwisted affine Lie algebra setting found in [CapLM1]–[CapLM2]and [CalLM1]–[CalLM3] (see also [Cal1]–[Cal2], [S1]–[S2]), showed that theprincipal subspaces under consideration had certain presentations (i.e. couldbe defined in terms of certain generators and relations). Using these pre-sentations, the authors constructed exact sequences among the principalsubspaces and in this way obtained recursions satisfied by the charactersof principal subspaces. Solving these recursions yields the characters of theprincipal subspaces of the basic modules for the twisted affine Lie algebrasin each work.

Let ν be a Dynkin diagram automorphism of order v of a Lie algebra oftype X where X is A2l−1 with l ≥ 2, Dl with l ≥ 4, or E6. Denote byν the lifting of ν to the basic X(1)-module VL ' L(Λ0) constructed as avertex operator algebra (cf [LL]) and the twisted VL-module V T

L , which is

isomorphic to the basic vacuum module, which we denote Lν(Λ0), of the

twisted affine Lie algebra X(v) (cf. [L1], [CalLM4]). The aim of this work isto determine the characters of the principal subspaces of the level k ≥ 1 vac-uum X(v)-module Lν(kΛ0), which we denote by W T

Lk, for the twisted affine

Lie algebras of type A(2)2l−1, for l ≥ 2, D

(2)l , for l ≥ 4, E

(2)6 and D

(3)4 , extend-

ing certain results found in [PS1]–[PS2]. The approach we use, however, isdifferent than the approach found in [PS1]–[PS2]. By using vertex operatortechniques we construct combinatorial bases of principal subspaces whichare twisted analogues to those found in [G], and from which we obtain thecharacters of principal subspaces. In our proofs, we use level k analoguesof certain maps originally developed and used in [CalLM4], [CalMPe], and

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 73

[PS1]–[PS2] analogously to how they were used in [G]. We note importantlythat, in the untwisted setting, certain modes of intertwining operators playan important role in the proofs of linear independence of these bases. In ourtwisted affine Lie algebra setting, we instead use other maps developed forthe twisted setting in [CalLM4].

More specifically, in this paper, following [G] and using certain results in[Li], we construct bases using the coefficients

xνrαi(m) = Resz{zm+r−1xνrαi(z)},

of the twisted vertex operators

xνrαi(z) = Y ν(xαi(−1)r1, z),

which we, in complete analogy with the untwisted case, call twisted quasi-particles of color i, charge r and energy −m. Similar to the untwisted case,(see [Bu1]–[Bu3], [JP]), first we prove certain relations for twisted quasi-particles of the form xνrαi(m)xνr′αi(m

′) for r ≤ r′ and xνrαi(m)xνr′αj (m′),

where 1 ≤ r, r′ ≤ k, which we call relations among twisted quasi-particles.With these relations, along with the relations xν(k+1)αi

(z) = 0, we build

twisted quasi-particle spanning sets of the principal subspaces W TLk

. The re-sulting bases are analogous to the quasi-particle bases of principal subspacesin the case of untwisted affine Lie algebras of type ADE in the sense thatenergies of twisted quasi-particles in the twisted quasi-particle spanning setssatisfy similar difference conditions, which are generalizations of differencetwo conditions found in [FS] and [G]. As in the untwisted case found in[G], in the proof of linear independence of our spanning sets we consider theprincipal subspace as a subspace of tensor product of k principal subspacesof basic modules. This enables us to use the above-mentioned maps ob-tained from the construction of level one twisted modules for lattice vertexoperator algebras from [CalLM4] and [PS1]–[PS2]. Finally, we note that inthis paper we do not consider the case of the principal subspaces W T

Lkfor

the twisted affine Lie algebra of type A(2)2l , which will be considered in future

work.Our main result in this work is as follows: denote by ch W T

Lkthe character

of the principal subspace W TLk

. The characters of the principal subspacesare:

Theorem 1.1. We have for A(2)2l−1:

ch W TLk

=∑

r(1)1 ≥···≥r

(k)1 ≥0

···r(1)l−1≥···≥r

(k)l−1≥0

q12

∑l−1i=1

∑ks=1 r

(s)2

i − 12

∑l−1i=2

∑ks=1 r

(s)i−1r

(s)i∏l−1

i=1(q12 ; q

12 )r(1)i −r

(2)i

· · · (q12 ; q

12 )r(k)i

l−1∏i=1

yr(1)i +···+r(k)ii

74 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

·∑

r(1)l ≥···≥r

(k)l ≥0

q∑ks=1 r

(s)2

l −∑ks=1 r

(s)l−1r

(s)l

(q)r(1)l −r

(2)l

· · · (q)r(k)l

yr(1)l +···+r(k)ll

for D(2)l :

ch W TLk

=∑

r(1)1 ≥···≥r

(k)1 ≥0

···r(1)l−2≥···≥r

(k)l−2≥0

q∑l−2i=1

∑ks=1 r

(s)2

i −∑l−2i=2

∑ks=1 r

(s)i−1r

(s)i∏l−2

i=1(q)r(1)i −r

(2)i

· · · (q)r(k)i

l−2∏i=1

yr(1)i +···+r(k)ii

·∑

r(1)l−1≥···≥r

(k)l−1≥0

q12

∑ks=1 r

(s)2

l−1 −∑ks=1 r

(s)l−2r

(s)l−1

(q12 ; q

12 )r(1)l−1−r

(2)l−1

· · · (q12 ; q

12 )r(k)l−1

yr(1)l−1+···+r(k)l−1

l−1

for E(2)6 :

ch W TLk

=

=∑

r(1)1 ≥···≥r

(k)1 ≥0

r(1)2 ≥···≥r

(k)2 ≥0

q12

∑2i=1

∑ks=1 r

(s)2

i −∑ks=1 r

(s)1 r

(s)2∏

i=1,2(q12 ; q

12 )r(1)i −r

(2)i

· · · (q12 ; q

12 )r(k)i

∏i=1,2

yr(1)i +···+r(k)ii

·∑

r(1)3 ≥···≥r

(k)3 ≥0

r(1)4 ≥···≥r

(k)4 ≥0

q∑4i=3

∑ks=1 r

(s)2

i −∑ks=1(r

(s)2 r

(s)3 +r

(s)3 r

(s)4 )∏

i=3,4(q)r(1)i −r

(2)i

· · · (q)r(k)i

∏i=3,4

yr(1)i +···+r(k)ii

and for D(3)4 :

ch W TLk

=∑

r(1)1 ≥···≥r

(k)1 ≥0

q13

∑ks=1 r

(s)2

1

(q13 ; q

13 )r(1)1 −r

(2)1

· · · (q13 ; q

13 )r(k)1

yr(1)1 +···+r(k)1

1

·∑

r(1)2 ≥···≥r

(k)2 ≥0

q∑ks=1 r

(s)2

2 −∑ks=1 r

(s)1 r

(s)2

(q)r(1)2 −r

(2)2

· · · (q)r(k)2

yr(1)2 +···+r(k)2

2 .

2. Preliminaries

In this section, we very closely follow the setting developed in [PS1]–[PS2](cf. also [L1] and [CalLM4]), and recall many details from these works.

Let g be a finite dimensional simple Lie algebra of type A2l−1, Dl, or E6,with root lattice

L = Zα1 ⊕ · · · ⊕ ZαD,

where D is the rank of g, with its standard nondegenerate symmetric bilinearform 〈·, ·〉. Also, let

h = L⊗Z C.

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 75

We take the following labelings of the Dynkin diagrams of our Lie algebras:Type A2l−1:

α1 α2

. . .αl−1 αl αl+1

. . .α2l−2 α2l−1

Type Dl:

α1 α2

. . .αl−2 αl−1

αl

Type E6:

α1 α2 α3 α5 α6

α4

In the case of D4, we use the labeling:

α1 α2 α3

α4

Remark 2.1. We note here that in [PS2], the labeling used for E6 was:

α1 α2 α3 α4 α5

α6

and in [PS1], the labeling used for D4 was:

α2 α1 α3

α4

and we change the labeling in this work for notational simplicity. Later inthe work, we will see that the roles of operators corresponding to α4 andα6 in the case of E6 and the roles of operators corresponding to α1 and α2

in the case of D4 will be swapped compared to their counterparts found in[PS2] and [PS1].

2.1. Dynkin diagram automorphisms. Let ν be a Dynkin diagram au-tomorphism of g of order v, extended to all of h. In the case that v = 2, welet η = −1 be a primitive second root of unity and set η0 = η, and in thecase that v = 3 we let η be a cube root of unity and set η0 = −η. Following[CalLM4] and [PS1]–[PS2], we consider two central extensions of L by the

group 〈η0〉, denoted by L and Lν , with commutator maps C0 and C andassociated normalized 2-cocycles εC0 and εC , respectively:

1 −→ 〈η0〉 −→ L −→ L −→ 1

76 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

and

1 −→ 〈η0〉 −→ Lν −→ L −→ 1

We define commutator maps C0 and C by

C0 : L× L→ C×

(α, β) 7→ (−1)〈α,β〉

and

C(α, β) =v−1∏j=0

(−ηj)〈νjα,β〉.

Following [L1] and [CalLM4], we let

e : L→ L

α 7→ eα

be a normalized section of L so that

e0 = 1

and

eα = α for all α ∈ L,

satisfying

eαeβ = εC0(α, β)eα+β for all α, β ∈ L.

We choose our 2-cocycle to be

εC0(αi, αj) =

{1 if i ≤ j(−1)〈αi,αj〉 if i > j

The 2-cocycles εC and εC0 are related by (see Equation 2.21 of [CalLM4])

εC0(α, β) =∏

− v2<j<0

(−η−j

)〈ν−jα,β〉εC(α, β).

We now lift the isometry ν of L to an automorphism ν of L such that

νa = νa for a ∈ L.

and choose ν so that

νa = a if νa = a,

and thus ν2 = 1 if ν has order 2 and ν3 = 1 if ν has order 3. Indeed, set

νeα = ψ(α)eνα

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 77

where ψ : L→ 〈η〉 is defined by

ψ(α) =

εC0(α, α) if L is type A2l−1

1 if L is type Dl and the order of ν is 2

(−1)r3r4εC0(α, α) if L is type E6 and α =∑6

i=1 riαi

(−1)r2r3εC0(α, α) if L is of type D4, α =∑4

i=1 riαi

and the order of ν is 3.

From [PS1]–[PS2], we have that:

εC0(να, νβ) =

εC0(β, α) if L is type A2l−1

εC0(α, β) if L is type Dl

(−1)r4s3+r3s4εC0(β, α) if L is type E6, α =∑6

i=1 riαi

and β =∑6

i=1 siαi

(−1)r2s3+r3s2εC0(β, α) if L is of type D4, α =∑3

i=1 riαi

and β =∑3

i=1 siαi.

As in [PS1]–[PS2], we have that

ν(eαi) = eναi

for each simple root αi.

2.2. The lattice vertex operator VL and its twisted module V TL .

We assume that the reader is familiar with the construction of the latticevertex operator algebra VL (cf. [FLM1] and [LL]), and recall some importantdetails of this construction. In particular, we follow Section 2 of [CalLM4].

We view h as an abelian Lie algebra, and let

h = h⊗ C[t, t−1]⊕ Cc

with the usual bracket, and let

h− = h⊗ t−1C[t−1].

We have that

VL ∼= S(h−)⊗ C[L]

linearly. We extend ν to an automorphism of VL, which we also call ν, byν = ν ⊗ ν.

Let

h(m) = {x ∈ h | ν(x) = ηmx}.We have that

h =∐

m∈Z/vZ

h(m).

We form the twisted affine Lie algebra

h[ν] =∐m∈Z

h(m) ⊗ tm/v

78 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

where

[α⊗ tm, β ⊗ tn] = 〈α, β〉mδm+n,0c

for m,n ∈ 1vZ and α ∈ h(vm) and β ∈ h(vn) and c is central. The Lie algebra

h[ν] is 1vZ-graded by weights:

wt(α⊗ tm) = −m and wt(c) = 0.

Define the Heisenberg subalgebra

h[ν] 1v

Z =∏

m∈ 1v

Zm 6=0

h(vm) ⊗ tm ⊕ Cc

of h[ν], the subalgebras

h[ν]± =∏

m∈ 1v

Z±m>0

h(vm) ⊗ tm

of h[ν] 1v

Z, and the induced module

S[ν] = U(h[ν]

)⊗∏

m≥0 h(vm)⊗tm⊕Cc C ∼= S(h[ν]−

),

which is Q-graded such that

wt(1) =1

4v2

v−1∑j=1

j(v − j)dimh(j).

Following [L1] and [CalLM4], we set

N = (1− P0)h ∩ L,where P0 is the projection of h onto h(0). In particular, we define

α(0) = P0α =1

v

v∑i=0

νiα

In the case that v = 2, we have that

N =

D∐i=1

Z(αi − ναi)

and when v = 3 we have that:

N = {r1α1 + r3α3 + r4α4 ∈ L | r1 + r3 + r4 = 0}.

Using Proposition 6.2 of [L1], let Cτ denote the one dimensional N -moduleC with character τ and write

T = Cτ .

Consider the induced Lν-module

UT = C[Lν ]⊗C[N ] T∼= C[L/N ],

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 79

which is graded by weights and on which Lν , h(0), and zh for h ∈ h(0) allnaturally act. Set

V TL = S[ν]⊗ UT ∼= S

(h[ν]−

)⊗ C[L/N ],

which is naturally acted upon by Lν , h 1v

Z, h(0), and zh for h ∈ h.

For each α ∈ h and m ∈ 1vZ define the operators on V T

L

α(vm) ⊗ tm 7→ αν(m),

where α(vm) is the projection of α onto h(vm), and set

αν(z) =∑m∈ 1

vZ

αν(m)z−m−1.

Of most importance will be the ν-twisted vertex operators acting on V TL for

each eα ∈ L

Y ν(ι(eα), z) = v−〈α,α〉

2 σ(α)E−(−α, z)E+(−α, z)eαzα(0)+〈α(0),α(0)〉

2− 〈α,α〉

2 ,

as defined in [L1], where

E±(−α, z) = exp

∑m∈± 1

vZ+

−α(vm)(m)

mz−m

,

and

σ(α) = 1 when v = 2

and

σ(α) = (1− η2)〈να,α〉 when v = 3

for α ∈ h. For m ∈ 1v and α ∈ L define the component operators xνα (m) by

Y ν(ι(eα), z) =∑m∈ 1

vZ

xνα (m) z−m−〈α,α〉

2 = xνα (z) .

We note here that V TL is a ν-twisted module for VL, and in particular it

satisfies the twisted Jacobi identity:

x−10 δ

(x1 − x2

x0

)Y ν(u, x1)Y ν(w, x2)− x−1

0 δ

(x2 − x1

−x0

)Y ν(w, x2)Y ν(u, x1)

= x−12

1

v

∑j∈Z/vZ

δ

(ηj

(x1 − x0)1/v

x1/v2

)Y ν(Y (νju, x0)w, x2)

for u,w ∈ VL.

80 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

2.3. Twisted affine Lie algebras. We now construct the twisted affine

Lie algebras of type A(2)2l−1, D

(2)l , E

(2)6 , and D

(3)4 , and give them an action on

V TL . Define the vector space

g = h⊕∐α∈∆

Cxα,

where {xα} is a set of symbols, and ∆ is the set of roots corresponding toL.

We give the vector space g the structure of a Lie algebra via the bracketdefined

[h, xα] = 〈h, α〉xα, [h, h] = 0

where h ∈ h and α ∈ ∆ and

[xα, xβ] =

εC0(α,−α)α if α+ β = 0εC0(α, β)xα+β if α+ β ∈ ∆0 otherwise.

We note that g is a Lie algebra isomorphic to one of type A2l−1, Dl, or E6

depending on the choice of L (cf. [FLM2]). We also extend the bilinear form〈·, ·〉 to g by

〈h, xα〉 = 〈xα, h〉 = 0

and

〈xα, xβ〉 =

{εC0(α,−α) if α+ β = 00 if α+ β 6= 0

Following [L1], [CalLM4], and [PS1]–[PS2], we use our extension of ν :

L→ L to ν : L→ L to lift the automorphism ν : h→ h to an automorphismν : g→ g by setting

νxα = ψ(α)xνα

for all α ∈ ∆. Here, we are using our particular choices of ν (extended toC{L}) and section e.

For m ∈ Z set

g(m) = {x ∈ g | ν(x) = ηmx}.Form the ν-twisted affine Lie algebra associated to g and ν:

g[ν] =∐

m∈ 1v

Z

g(vm) ⊗ tm ⊕ Cc

with

[x⊗ tm, y ⊗ tn] = [x, y]⊗ tm+n + 〈x, y〉mδm+n,0c

and

[c, g[ν]] = 0,

for m,n ∈ 1vZ, x ∈ g(vm), and y ∈ g(vn). Adjoining the degree operator d to

g[ν], we define

g[ν] = g[ν]⊕ Cd,

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 81

where

[d, x⊗ tn] = nx⊗ tn,for x ∈ g(vn), n ∈ 1

vZ and [d, c] = 0. The Lie algebra g[ν] is isomorphic

to A(2)2l−1, D

(2)l , E

(2)6 , or D

(3)4 depending on the choice of L and ν, and is

1vZ-graded. We give V T

L the structure of a g[ν]-module by:

Theorem 2.1. (Theorem 3.1[CalLM4],Theorem 9.1 [L1],Theorem 3 [FLM1])

The representation of h[ν] on V TL extends uniquely to a Lie algebra repre-

sentation of g[ν] on V TL such that

(xα)(vm) ⊗ tm 7→ xνα (m)

for all m ∈ 1vZ and α ∈ L. Moreover V T

L is irreducible as a g[ν]-module.

2.4. Gradings. As in Section 2 of [CalLM4] (also Section 6 of [L1]) wehave a tensor product grading on V T

L given by the action of Lν(0), where

Y ν(ω, z) =∑m∈Z

Lν(m)z−m−2,

which we call the weight grading. In particular, we have

wt(1) =l − 1

16, for A2l−1

wt(1) =1

16, for Dl when v = 2

wt(1) =1

8, for E6,

wt(1) =1

9, for D4 when v = 3.

From [PS1]–[PS2], we recall that

wt(xνα (m)) = −m− 1 +1

2〈α, α〉

for m ∈ 1vZ and α ∈ L.

We endow V TL with charge gradings (see [PS1]–[PS2]). In particular, in

the case of A2l−1, we have

ch(xνα (m)) =⟨2⟨α, (λ1)(0)

⟩, . . . , 2

⟨α, (λl−1)(0)

⟩,⟨α, (λl)(0)

⟩). (2.1)

In the case of Dl when v = 2 we have

ch(xνα (m)) =⟨⟨α, (λ1)(0)

⟩, . . . ,

⟨α, (λl−2)(0)

⟩, 2⟨α, (λl−1)(0)

⟩). (2.2)

In the E6 case we have

ch(xνα (m)) =(2⟨α, (λ1)(0)

⟩, 2⟨α, (λ2)(0)

⟩,⟨α, (λ3)(0)

⟩,⟨α, (λ4)(0)

⟩),

(2.3)and finally in the case of D4 when v = 3 we have

ch(xνα (m)) =(3⟨α, (λ1)(0)

⟩,⟨α, (λ2)(0)

⟩). (2.4)

82 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

We note that the λi here are the fundamental weights of the underlyingfinite dimensional Lie algebra g, dual to the roots:

〈λi, αj〉 = δi,j

with 1 ≤ i, j ≤ rank(g).

2.5. Higher levels. The primary focus of this section so far has been theconstruction of V T

L , the basic module for the twisted affine Lie algebra g[ν].In the remainder of this work we will consider the standard g[ν]-moduleLν(kΛ0) of level k ≥ 1 with highest weight kΛ0 such that 〈kΛ0, c〉 = k and〈Λ0, h(0)〉 = 0 = 〈Λ0, d〉. We note that when k = 1, we have Lν(Λ0) ∼= V T

L .

Since Lν(kΛ0) is a faithful ν-twisted L(kΛ0)-module (see [Li]), whereL(kΛ0) denotes level k standard module of untwisted affine Lie algebra gwhich has a structure of a vertex operator algebra, from Proposition 2.10 in[Li] follows that for r ∈ N and xα ∈ g

Y ν(xα(−1)r1, z) = Y ν(xα(−1)1, z)r

is a twisted vertex operator.We will also use the commutator formula for twisted vertex operators[

xνα(z1), xνn′β(z2)]

= (2.5)

=∑j≥0

(−1)j

j!

(d

dz1

)jz−1

2

1

v

∑q∈Z/vZ

δ

ηq z 1v1

z1v2

Y ν(νqxα(j)xβ(−1)n′1, z2).

In this work, we realize Lν(kΛ0) as a submodule of the tensor product ofk copies of the basic module V T

L∼= Lν(Λ0) as follows:

Lν(kΛ0) ∼= U(g[ν]) · vL ⊂ V TL ⊗ · · · ⊗ V T

L = V TL⊗k,

where vL = 1T ⊗ 1T ⊗ · · · ⊗ 1T is a highest weight vector of Lν(kΛ0) andwhere 1T is a highest weight vector of V T

L (cf. [Kac]).

It is known that V ⊗kL = VL ⊗ · · · ⊗ VL has a structure of vertex operatoralgebra. If we denote by ν the automorphism ν⊗· · ·⊗ν of the vertex operatoralgebra V ⊗kL , then we have νv = 1 and one can also define vertex operators

corresponding to elements v1⊗· · ·⊗vk ∈ V TL⊗k

as tensor products of twisted

vertex operators on the appropriate tensor factors Y ν(v1, z)⊗· · ·⊗Y ν(vk, z).

In this way V TL⊗k

becomes an irreducible ν-twisted module for the vertex

operator algebra V ⊗kL (cf. [Li]), with

Y ν(xα(−1) · (1⊗ · · · ⊗ 1), z) =∑m∈ 1

vZ

xνα(m)z−m−1.

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 83

3. Principal subspaces

In this section we define the notion of principal subspace of Lν(kΛ0) andtwisted quasi-particles which we will use in the description of our bases.

First, denote by

n =∐α∈∆+

Cxα,

the ν-stable Lie subalgebra of g (the nilradical of a Borel subalgebra), itsν-twisted affinization

n[ν] =∐

m∈ 1v

Z

n(vm) ⊗ tm ⊕ Cc

and the subalgebra of n[ν]

n[ν] =∐

m∈ 1v

Z

n(vm) ⊗ tm.

Following [FS] (see also [CalLM4], [CalMPe], and [PS1]–[PS2]), we definethe principal subspace W T

Lkof Lν(kΛ0) as:

W TLk

= U (n[ν]) · vL.

3.1. Properties of W TLk

. We now recall some important properties of the

operators xνα (m) on W TLk

. We recall from [PS1]–[PS2] that

xννα(m) = xνα (m) for m ∈ Z

xννα(m) = −xνα (m) for m ∈ 1

2+ Z.

when v = 2 and that

xνα3(m) = xνα1

(m) for m ∈ Z

xνα3(m) = ηxνα1

(m) for m ∈ 1

3+ Z

xνα3(m) = η2xνα1

(m) for m ∈ 2

3+ Z

andxνα4

(m) = xνα1(m) for m ∈ Z

xνα4(m) = η2xνα1

(m) for m ∈ 1

3+ Z

xνα4(m) = ηxνα1

(m) for m ∈ 2

3+ Z

when v = 3. In particular, we need only choose one representative from theorbit of each simple root αi when working with the operators xναi(m).

For every simple root αi consider the one-dimensional subalgebra of g

nαi = Cxαi ,

84 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

and their ν-twisted affinizations

nαi [ν] =∐

m∈ 1v

Z

nαi (vm) ⊗ tm.

In the case of A(2)2l−1 we define a special subspace of n[ν]

U = U(nαl [ν])U(nαl−1[ν]) · · ·U(nα1 [ν]).

Similary, for D(2)l we define

U = U(nαl−1[ν])U(nαl−2

[ν]) · · ·U(nα1 [ν]),

in the case of E(2)6 we define

U = U(nα4 [ν])U(nα3 [ν])U(nα2 [ν])U(nα1 [ν]),

and for D(3)4 we define

U = U(nα2 [ν])U(nα1 [ν]).

The next lemma can be proved by using properties stated above and byaruing as in Lemma 3.1 of [G].

Lemma 3.1. In all of the above cases, we have that

W TLk

= U · vL.

3.2. Twisted quasi-particles. For each simple root αi, r ∈ N, and m ∈1vZ define the twisted quasi-particle of color i, charge r and energy −m by

xνrαi(m) = Resz{zm+r−1xνrαi(z)},where

xνrαi(z) =∑m∈ 1

vZ

xνrαi(m)z−m−r

is the twisted vertex operator

xνrαi(z) = Y ν(xαi(−1)r1, z).

As in the untwisted case (see [G], [Bu1]–[Bu3]), we build twisted quasi-particle monomials from twisted quasi-particles. We say that the monomial

b = bν(αl) · · · bν(α1) =

= xνnr(1)l

,lαl

(mr(1)l ,l

) · · ·xνn1,lαl(m1,l) · · ·xνn

r(1)1 ,1

α1(m

r(1)1 ,1

) · · ·xνn1,1α1(m1,1),

is of charge-type

R′ =(nr(1)l ,l

, . . . , n1,l; . . . ;nr(1)1 ,1, . . . , n1,1

),

where 0 ≤ nr(1)i ,i≤ . . . ≤ n1,i, dual-charge-type

R =(r

(1)l , . . . , r

(sl)l ; . . . ; r

(1)1 , . . . , r

(s1)1

),

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 85

where r(1)i ≥ r

(2)i ≥ . . . ≥ r

(si)i ≥ 0, and color-type

(rl, . . . , r1) ,

where

ri =

r(1)i∑p=1

np,i =

si∑t=1

r(t)i and si ∈ N,

if for every color i, 1 ≤ i ≤ l,(nr(1)i ,i

, . . . , n1,i

)and

(r

(1)i , r

(2)i , . . . , r

(s)i

)are

mutually conjugate partitions of ri (cf. [Bu1]–[Bu3], [G]).

For two monomials b and b with charge-typesR′ andR′ =(nr(1)l ,l

, ..., n1,1

)and with energies

(mr(1)l ,l

, . . . ,m1,1

)and

(mr(1)l ,l

, . . . ,m1,1

)(which we write

so that energies of twisted quasi-particles of the same color and the samecharge form an increasing sequence of integers from right to the left), re-spectively, we write b < b if one of the following conditions holds:

1. R′ < R′2. R′ = R′ and

(mr(1)l ,l

, . . . ,m1,1

)<(mr(1)l ,l

, . . . ,m1,1

),

where we write R′ < R′ if there exists u ∈ N such that n1,i = n1,i, n2,i =

n2,i, . . . , nu−1,i = nu−1,i, and either u = r(1)i + 1 or nu,i < nu,i, starting from

color i = 1. In the case that R′ = R′, we apply this definition to the energies

to similarly define(mr(1)l ,l

, . . . ,m1,1

)<(mr(1)l ,l

, . . . ,m1,1

).

4. Combinatorial bases

In this section we prove relations among twisted quasi-particles which wewill use in the construction of our combinatorial bases for W T

Lk.

4.1. Relations among twisted quasi-particles. For every color i wehave the following relations:

xν(k+1)αi(z) = 0 (4.1)

andxνrαi(z)vL ∈W

TLk

[[z]] (4.2)

when ναi = αi,

xνrαi(z)vL ∈ z− 1

2W TLk

[[z

12

]](4.3)

when ναi 6= αi and v = 2, and

xνrαi(z)vL ∈ z− 2

3W TLk

[[z

13

]](4.4)

when ναi 6= αi and v = 3, which all follow immediately from the fact that

xνrαi(m)vL = 0

whenever m ≥ 0. We will use also the following relations among quasi-particles of the same color:

86 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

Lemma 4.1. Let 1 ≤ n ≤ n′ be fixed.

a) If ναi = αi and M, j ∈ Z are fixed, the 2n monomials from the set

A = {xνnαi(j)xνn′αi(M − j), x

νnαi(j − 1)xνn′αi(M − j + 1), . . .

. . . , xνnαi(j − 2n+ 1)xνn′αi(M − j + 2n− 1)}

can be expressed as a linear combination of monomials from the set{xνnαi(m)xνn′αi(m

′) : m+m′ = M}\A

and monomials which have as a factor the quasi-particle xν(n′+1)αi(j′),

j′ ∈ Z.b) If ναi 6= αi and M, j ∈ 1

vZ are fixed, the 2n monomials from the set

B = {xνnαi(j)xνn′αi(M − j), x

νnαi(j −

1

v)xνn′αi(M − j +

1

v), . . .

. . . , xνnαi(j −2n− 1

v)xνn′αi(M − j +

2n− 1

v)}

can be expressed as a linear combination of monomials from the set{xνnαi(m)xνn′αi(m

′) : m+m′ = M}\B

and monomials which have as a factor the quasi-particle xν(n′+1)αi(j′),

j′ ∈ 1vZ.

Proof. The proof of part a) is identical to the proof of Lemma 4.4 in [JP].Part b) can be proven analogously. First, for fixed M ∈ 1

vZ in the coefficient

of z−M−n−n′−N of the formal series

1N !

(dN

dzNxνnαi(z)

)xνn′αi(z) = (4.5)

=∑

M∈ 1v

Z

(∑m,m′∈ 1

vZ

m+m′=M

(−m−nN

)xνnαi(m)xνn′αi(m

′)

)z−M−n−n

′−N

we separate the 2n monomials with m = jv ,

j−1v , . . . , j−2n−1

v for some fixedj ∈ Z (

− jv − nN

)xνnαi

(j

v

)xνn′αi

(M − j

v

)+

+

(− jv − n+ 1

v

N

)xνnαi

(j

v− 1

v

)xνn′αi

(M − j

v+

1

v

)+

+ · · ·+(− jv − n+ 2n−2

v

N

)xνnαi

(j

v− 2n− 2

v

)xνn′αi

(M − j

v+

2n− 2

v

)+

+

(− jv − n+ 2n−1

v

N

)xνnαi

(j

v− 2n− 1

v

)xνn′αi

(M − j

v+

2n− 1

v

)+

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 87

+∑

m,m′∈ 1v

Zm+m′=M

m 6= jv, j−1v,..., j−2n−1

v

(−m− nN

)xνnαi(m)xνn′αi(m

′).

Next, note that for N = 0, 1, . . . , 2n− 1, we also have

1

N !

(dN

dzNxνnαi(z)

)xνn′αi(z) = Aν1(z)xν(n′+1)αi

(z) +Aν2(z)d

dzxν(n′+1)αi

(z),

(4.6)where Aν1(z) and Aν2(z) are formal Laurent series whose coefficients are poly-nomials in xναi(m), m ∈ 1

vZ (the proof of which is the same as the proof ofLemma 4.2 in [JP]).

Now, by comparing powers of z, from (4.5) and (4.6) we obtain a systemof 2n equations whose left-hand side consists of linear combinations of the2n unknowns

xνnαi(−p− n)xνn′αi(M + p+ n), xνnαi(−p− n−1

v)xνn′αi(M + p+ n+

1

v), . . .

. . . , xνnαi(−p− n−2n− 1

v)xνn′αi(M + p+ n+

2n− 1

v),

where p = − jv − n, and whose right-hand side is equal to expressions

in terms of higher quasi-particle monomials (with respect to ordering de-fined in previous section) and other quasi-particle monomials from the set{xνnαi(m)xνn′αi(m

′) : m+m′ = M}\ B. Such an expression exists and is

unique if the coefficient matrix

(p0

) (p+ 1v

0

). . .

(p+ 2n−2v

0

) (p+ 2n−1v

0

)(p1

) (p+ 1v

1

). . .

(p+ 2n−2v

1

) (p+ 2n−1v

1

)...

......

......(

p2n−1

) ( p+ 1v

2n−1

). . .

(p+ 2n−2v

2n−1

) (p+ 2n−1v

2n−1

)

is invertible.

By using the Chu-Vandermonde identity

N∑q=0

(a

q

)(b

N − q

)=

(a+ b

N

),

we can write the coefficient matrix as a product of two invertible matrices

(p0

)0 . . . 0 0(

p1

) (p0

). . . 0 0

......

......

...(p

2n−1

) (p

2n−2

). . .

(p1

) (p0

)

·

(00

) ( 1v0

). . .

( 2n−2v0

) ( 2n−1v0

)(

01

) ( 1v1

). . .

( 2n−2v1

) ( 2n−1v1

)...

......

......(

02n−1

) ( 1v

2n−1

)...

( 2n−2v

2n−1

) ( 2n−1v

2n−1

)

.

88 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

To see that the second matrix in the product is invertible, we first, start-ing from row 2n − 1, multiply row q of the determinant of the matrix by(q−1)v−1

qv , where 2 ≤ q ≤ 2n− 1, and then add to row q + 1, which will giveus ∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 1 . . . 1 1

0 1v

2v . . . 2n−2

v2n−1v

0 0 1v2

. . . 1v2

(2n−2

2

)1v2

(2n−1

2

)...

......

......

...

0 0 1(2n−1)v

( 2v

2n−2

). . . 2n−3

(2n−1)v

( 2n−2v

2n−2

)2n−2

(2n−1)v

( 2n−1v

2n−2

)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣.

We proceed with this process of reducing the determinant of our matrix tothe determinant of an upper triangular matrix and assume that after r − 1steps we have∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 . . . 1 1 . . . 1

0 1v . . . r−1

vrv . . . 2n−1

v...

......

......

......

0 0 . . . 0 1vr . . . 1+r

vr

(2n−1r

)0 0 . . . 0 1

r(r−1)vr−1

( rv2

). . . (2n−1)···(2n−r)

vr−1(r+1)···3( r+1

v2

)...

......

......

......

0 0 . . . 0 1(2n−1)···(2n−r−1)vr−1

( rv

2n−1−r−1

). . . (2n−1)···(2n−r)

vr−1(2n−1)···(2n−1−r)( 2n−1

v2n−1−r−1

)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣.

Now, starting from row 2n−1, we multiply row q by (q−r)v−rqv , where r+1 ≤

q ≤ 2n− 1, and add to row q + 1 and obtain∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 . . . 1 1 1 . . . 1

0 1v . . . r−1

vrv

r+1v . . . 2n−1

v...

......

......

......

...

0 0 . . . 0 1vr

r+1vr . . . r+1

vr

(2n−1r

)0 0 . . . 0 0 1

vr+1 . . . 1vr+1

(2n−1r+1

)...

......

......

......

...

0 0 . . . 0 0 1(2n−1)···(2n−r−2)vr

( r+1v

2n−1−r−2

). . . (2n−1)···(2n−r−1)

vr(2n−1)···(2n−1−r)( 2n−1

v2n−1−r−2

)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣.

After 2n− 1 steps we get the determinant∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 1 . . . 1

0 1v

2v . . . 2n−1

v

0 0 1v2

. . . 1v2

(2n−1

2

)...

......

......

0 0 0 . . . 1v2n−1

∣∣∣∣∣∣∣∣∣∣∣∣∣= v−n(2n−1),

from which our claim follows. �

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 89

Remark 4.1. We note here that a more general form of this matrix ap-peared in [PSW], where is was called a “generalized Pascal matrix”. In[PSW], this matrix was shown to be invertible but its determinant was notexplicitly computed.

From Lemma 4.1 it follows that if ναi = αi, then the 2n monomial vectors

xνnαi(m)xνn′αi(m′)vL, x

νnαi(m− 1)xνn′αi(m

′ + 1)vL, . . .

. . . , xνnαi(m−2n+2)xνn′αi(m′+2n−2)vL, x

νnαi(m−2n+1)xνn′αi(m

′+2n−1)vL

such that n < n′ can be expressed as a (finite) linear combination of themonomial vectors

xνnαi(j)xνn′αi(j

′)vL such that j ≤ m− 2n, j′ ≥ m′ + 2n

and monomial vectors with a factor quasi-particle xν(n+1)αi(j1), j1 ∈ Z. If

n = n′, then the monomial vectors

xνnαi(m)xνn′αi(m′)vL such that m′ − 2n ≤ m ≤ m′

can be expressed as a linear combination of monomial vectors

xνnαi(j)xνn′αi(j

′)vL, such that j ≤ j′ − 2n

and monomial vectors with a factor quasi-particle xν(n+1)αi(j1), j1 ∈ Z.

If ναi 6= αi, then the 2n monomial vectors

xνnαi(m)xνn′αi(m′)vL, x

νnαi(m− 1)xνn′αi(m

′ + 1)vL, . . .

. . . , xνnαi(m−2n+2)xνn′αi(m′+2n−2)vL, x

νnαi(m−2n+1)xνn′αi(m

′+2n−1)vL

with n < n′ can be expressed as a (finite) linear combination of the monomialvectors

xνnαi(j)xνn′αi(j

′)vL, such that j ≤ m− 2n

v, j′ ≥ m′ + 2n

v

and monomial vectors with a factor quasi-particle xν(n+1)αi(j1), j1 ∈ 1

vZ. If

n = n′, then the monomial vectors

xνnαi(m)xνn′αi(m′)vL, such that m′ − 2n

v≤ m ≤ m′

can be expressed as a linear combination of the monomial vectors

xνnαi(j)xνn′αi(j

′)vL, such that j ≤ j′ − 2n

v

and monomial vectors with a factor quasi-particle xν(n+1)αi(j1), j1 ∈ 1

vZ.

The following lemma describes relations among quasi-particles with di-fferent colors:

Lemma 4.2. Let 1 ≤ n, n′ ≤ k be fixed. Then:

a) if v = 2 and 〈α, β〉 = −1 = 〈να, β〉, we have:

(z1 − z2)min{n,n′}xνnα(z1)xνn′β(z2) = (z1 − z2)min{n,n′}xνn′β(z2)xνnα(z1), (4.7)

90 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

b) if v = 2 and 〈α, β〉 = −1, 〈να, β〉 = 0, we have:

(z121 −z

122 )min{n,n′}xνnα(z1)xνn′β(z2) = (z

121 −z

122 )min{n,n′}xνn′β(z2)xνnα(z1), (4.8)

c) if v = 3 for α = α1, β = α2, we have:

(z1 − z2)min{n,n′}xνnα(z1)xνn′β(z2) = (z1 − z2)min{n,n′}xνn′β(z2)xνnα(z1). (4.9)

Proof. The proof follows from the commutator formula for twisted vertexoperators (2.5) and from properties of the δ-function. �

By using relations (4.1)–(4.4) among twisted quasi-particles, Lemma 4.1and relations (4.2)–(4.9), we define the following sets:

• for A(2)2l−1:

B =⋃

nr(1)1 ,1≤...≤n1,1≤k...

nr(1)l

,l≤...≤n1,l≤k

or, equivalently,⋃

r(1)1 ≥···≥r

(k)1 ≥0

···r(1)l ≥···≥r

(k)l ≥0

{b = b(αl) · · · b(α1) =

= xνnr(1)n ,l

αl(m

r(1)l ,l

) · · ·xνn1,lαl(m1,l) · · ·xνn

r(1)1 ,1

α1(m

r(1)1 ,1

) · · ·xνn1,1α1(m1,1) :

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

mp,i ∈ 12Z, 1 ≤ p ≤ r(1)

i , 1 ≤ i ≤ l − 1;

mp,i ≤ −np,i2 + 1

2

∑r(1)i−1

q=1 min {nq,i−1, np,i} −∑

p>p′>0 min{np,i, np′,i},1 ≤ p ≤ r(1)

i , 1 ≤ i ≤ l − 1;

mp+1,i ≤ mp,i − np,i if np+1,i = np,i, 1 ≤ p ≤ r(1)i − 1, 1 ≤ i ≤ l − 1;

mp,l ∈ Z, 1 ≤ p ≤ r(1)l ;

mp,l ≤ −np,l +∑r

(1)l−1

q=1 min {nq,l−1, np,l} −∑

p>p′>0 2 min{np,l, np′,l},1 ≤ p ≤ r(1)

l ;

mp+1,l ≤ mp,l − 2np,l if np,l = np+1,l, 1 ≤ p ≤ r(1)l − 1

,

• for D(2)l :

B =⋃

nr(1)1 ,1≤...≤n1,1≤k...

nr(1)l−1

,l−1≤...≤n1,l−1≤k

or, equivalently,⋃

r(1)1 ≥···≥r

(k)1 ≥0

···r(1)l−1≥···≥r

(k)l−1≥0

{b = b(αl−1) · · · b(α1) =

= xνnr(1)l−1

,l−1αl−1

(mr(1)l−1,l−1

) · · ·xνn1,l−1αl−1(m1,l−1) · · ·

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 91

· · ·xνnr(1)1 ,1

α1(m

r(1)1 ,1

) · · ·xνn1,1α1(m1,1) :∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

mp,i ∈ Z, 1 ≤ p ≤ r(1)i , 1 ≤ i ≤ l − 2;

mp,i ≤ −np,i +∑r

(1)i−1

q=1 min {nq,i−1, np,i} −∑

p>p′>0 2 min{np,i, np′,i},1 ≤ p ≤ r(1)

i , 1 ≤ i ≤ l − 2;

mp+1,i ≤ mp,i − 2np,i if np+1,i = np,i, 1 ≤ p ≤ r(1)i − 1, 1 ≤ i ≤ 2;

mp,l−1 ∈ 12Z, 1 ≤ p ≤ r(1)

l−1;

mp,l−1 ≤ −np,l−1

2 +∑r

(1)l−2

q=1 min {nq,l−2, np,l−1}−−∑

p>p′>0 min{np,l−1, np′,l−1}, 1 ≤ p ≤ r(1)l−1;

mp+1,l−1 ≤ mp,l−1 − np,l−1 if np,l−1 = np+1,l−1, 1 ≤ p ≤ r(1)l−1 − 1

,

• for E(2)6 :

B =⋃

nr(1)1 ,1≤...≤n1,1≤k...

nr(1)4 ,4≤...≤n1,4≤k

or, equivalently,⋃

r(1)1 ≥···≥r

(k)1 ≥0

...r(1)4 ≥···≥r

(k)4 ≥0

{b = b(α4) · · · b(α1) =

= xνnr(1)4 ,4

α4(m

r(1)4 ,4

) · · ·xνn1,4α4(m1,4) · · ·xνn

r(1)1 ,1

α1(m

r(1)1 ,1

) · · ·xνn1,1α1(m1,1) :

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

mp,i ∈ 12Z, 1 ≤ p ≤ r(1)

i , 1 ≤ i ≤ 2;

mp,i ≤ −np,i2 + 1

2

∑r(1)i−1

q=1 min {nq,i−1, np,i} −∑

p>p′>0 min{np,i, np′,i},1 ≤ p ≤ r(1)

i , 1 ≤ i ≤ l − 1;

mp+1,i ≤ mp,i − np,i if np+1,i = np,i, 1 ≤ p ≤ r(1)i − 1, 1 ≤ i ≤ 2;

mp,i ∈ Z, 1 ≤ p ≤ r(1)i , 3 ≤ i ≤ 4;

mp,i ≤ −np,i +∑r

(1)i−1

q=1 min {nq,i−1, np,i} −∑

p>p′>0 2 min{np,i, np′,i},1 ≤ p ≤ r(1)

i , 3 ≤ i ≤ 4;

mp+1,i ≤ mp,i − 2np,i if np,i = np+1,i, 1 ≤ p ≤ r(1)i − 1, 3 ≤ i ≤ 4

,

• for D(3)4 :

B =⋃

nr(1)1 ,1≤...≤n1,1≤k

nr(1)2 ,2≤...≤n1,2≤k

or, equivalently,⋃

r(1)1 ≥···≥r

(k)1 ≥0

r(1)2 ≥···≥r

(k)2 ≥0

{b = b(α2)b(α1) =

92 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

= xνnr(1)2 ,2

α2(m

r(1)2 ,2

) · · ·xνn1,2α2(m1,2)xνn

r(1)1 ,1

α1(m

r(1)1 ,1

) · · ·xνn1,1α1(m1,1) :

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

mp,1 ∈ 13Z, 1 ≤ p ≤ r(1)

1 ;

mp,1 ≤ −np,13 −

23

∑p>p′>0 min{np,1, np′,1}, 1 ≤ p ≤ r(1)

1 ;

mp+1,1 ≤ mp,1 − 23np,1 if np+1,1 = np,1, 1 ≤ p ≤ r(1)

1 − 1;

mp,2 ∈ Z, 1 ≤ p ≤ r(1)2 ;

mp,2 ≤ −np,2 +∑r

(1)1q=1 min {nq,1, np,2} −

∑p>p′>0 2 min{np,2, np′,2},

1 ≤ p ≤ r(1)2 ;

mp+1,2 ≤ mp,2 − 2np,2 if np,2 = np+1,2, 1 ≤ p ≤ r(1)2 − 1

,

where r(1)0 := 0.

Now, using the same proof as in [G], we have:

Proposition 4.1. The set

B = {bvL : b ∈ B} (4.10)

spans the principal subspace W TLk

.

5. Proof of linear independence

To prove that the set B is a linearly independent set, we will use certainmaps defined on our principal subspace. The first of these maps is a gener-alization of the projection map found in [G] to our twisted setting, and theremaining maps were used in [PS1] and [PS2], (see also [L1] and [CalLM4]).The proof of the linear independence of B is carried out by induction on thelinear order on the twisted quasi-particle monomials.

5.1. The projection πR. As we mentioned earlier, for k ≥ 1, we willrealize W T

Lkas a subspace of the tensor product of k principal subspaces

W TL of the basic VL-module V T

L .

W TLk⊂W T

L ⊗ · · · ⊗W TL ⊂ V T

L⊗k.

For a chosen dual-charge-type

R =(r

(1)l , . . . , r

(k)l ; . . . ; r

(1)1 , . . . , r

(k)1

),

we define a projection πR of W TLk

on

W TL(r

(k)l

,...,r(k)1 )

⊗ · · · ⊗W TL(r

(1)l

,...,r(1)1 )

,

where W TL(r

(t)l,...,r

(t)1 )

denotes the subspace of W TL consisting of the vectors of

charges r(t)l , . . . , r

(t)1 , for 1 ≤ t ≤ k (see 2.1–2.4). We will also consider a

generalization of πR, which we denote by the same symbol, to the space of

formal series with coefficients in W TL⊗k

.

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 93

Using the level 1 relation xν2αi(z) = 0, with the projection πR for fixed

1 ≤ p ≤ r(1)i , we “place” np,i, generating functions xναi(z) on first np,i tensor

factors:

xνn(k)p,iαi

(zp,i)1T ⊗ xνn(k−1)p,i αi

(zp,i)1T ⊗ · · · ⊗ xνn(2)p,iαi

(zp,i)1T ⊗ xνn(1)p,iαi

(zp,i)1T ,

where

0 ≤ n(t)p,i ≤ 1, np,i =

k∑t=1

n(t)p,i, 1 ≤ t ≤ k.

Now, it follows that the projection of generating function

xνnr(1)l

,lαl

(zr(1)l ,l

) · · ·xνn1,1α1(z1,1) vL (5.1)

of dual-charge-type R and corresponding charge-type R′ = (nr(1)l ,l

, · · · , n1,1)

is

πRxνnr(1)l

,lαl

(zr(1)l ,l

) · · ·xνn1,1α1(z1,1) vL

C xνn(k)

r(k)l

,lαl

(zr(k)l ,l

) · · ·xνn(k)1,l αl

(z1,l) · · ·xνn(k)

r(k)1 ,1

α1(zr(k)1 ,1

) · · ·xνn(k)1,1α1

(z1,1) 1T

⊗ . . .⊗

⊗xνn(1)

r(1)l

,lαl

(zr(1)l ,l

) · · ·xνn(1)1,lαl

(z1,l) · · ·xνn(1)

r(1)1 ,1

α1(zr(1)1 ,1

) · · ·xνn(1)1,1α1

(z1,1) 1T ,

where C ∈ C∗, and

0 ≤ n(t)p,i ≤ 1, 1 ≤ t ≤ k, n(1)

p,i ≥ n(2)p,i ≥ . . . ≥ n

(k−1)p,i ≥ n(k)

p,i , np,i =k∑t=1

n(t)p,i,

for every every p, 1 ≤ p ≤ r(1)i , 1 ≤ i ≤ l.

Also, from above it follows that the projection of bvL, where b ∈ B is themonomial

xνnr(1)l

,lαl

(mr(1)l ,l

) · · ·xνn1,1α1(m1,1) (5.2)

of charge-type R′ and dual-charge-type R, is a coefficient of the generatingfunction (5.1), which we will denote with πRbvL.

5.2. The maps ∆T (λ,−z). Following [CalLM4], [CalMPe], and [PS1]–[PS2] for γ ∈ h(0) and a character of the root lattice θ : L→ C, define

τγ,θ : n[ν]→ n[ν]

xνα (m) 7→ θ(α)xνα(m+

⟨α(0), γ

⟩).

We note here that in general, τγ,θ is a linear map, and for suitably chosencharacters θ, it becomes an automorphism of n[ν], which we extend to anautomorphism of U (n[ν]). In particular, we consider maps τγ,θ where γ =

94 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

γi = (λi)(0) = 12 (λi + νλi), and where the corresponding characters θ = θi

are chosen as in [PS1]–[PS2] defined by

θi(αj) = (−1)〈λ(i),αj〉,where we choose our subscripts i as follows:

• for A2l−1, let i = 1, . . . , l• for Dl when v = 2, let i = 1, . . . , l − 1• for D4 when v = 3, let i = 1, 2• E6, let i = 1, 2, 3, 4.

and define λ(i) = v(λi)(0) for v = 2, 3 (see [PS2] and [PSW] for a moregeneral definition of this symbol).

We also consider the related map ∆c(λ, x) from [CalLM4], [CalMPe], and[PS1]–[PS2].For λ ∈ {λi|1 ≤ i ≤ l}, we define the map

∆T (λ,−z) = (−1)νλzλ(0)E+(−λ, z).and its constant term ∆T

c (λ,−z). From [PS1]–[PS2], we have

∆Tc (λi,−z)(xνβ (m) 1T ) = τγi,θi(x

νβ (m))1T . (5.3)

More generally, we have linear maps

∆Tc (λi,−z) : W T

L →W TL

a · 1T 7→ τγi,θi(a) · 1T ,where a ∈ U (n[ν]).

Fix s ≤ k and consider the map

1⊗ · · · ⊗∆Tc (λi,−z)⊗ 1⊗ · · · ⊗ 1︸ ︷︷ ︸

s−1 factors

.

Let b ∈ B as in (5.2). It follows that

(1⊗ · · · ⊗ 1⊗∆Tc (λi,−z)⊗ 1⊗ · · · ⊗ 1)πRbvL

is the coefficient of

(1⊗ · · · ⊗∆Tc (λi,−z)⊗ 1⊗ · · · ⊗ 1)πRx

νnr(1)l

,lαl

(zr(1)l ,l

) · · ·xνsα1(z1,1)vL,

where, from (5.3), it follows that operator ∆Tc (λi,−z) acts only on the s-th

tensor row as:

⊗xνn(s)

r(s)l

,lαl

(zr(s)l ,l

) · · ·xνn(s)1,lαl

(z1,l) · · ·xνn(s)

r(s)i

,iαi

(zr(s)i ,i

)zr(s)i ,i· · ·

· · ·xνn(s)1,iαi

(z1,i)z1,i · · ·xνn(s)

r(s)1 ,1

α1(zr(s)1 ,1

) · · ·xνα1(z1,1)1T⊗,

for i such that ναi = αi and as

⊗xνn(s)

r(s)l

,lαl

(zr(s)l ,l

) · · ·xνn(s)1,lαl

(z1,l) · · ·xνn(s)

r(s)i

,iαi

(zr(s)i ,i

)z1v

r(s)i ,i· · ·

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 95

· · ·xνn(s)1,iαi

(z1,i)z1v1,i · · ·x

ν

n(s)

r(s)1 ,1

α1(zr(s)1 ,1

) · · ·xνα1(z1,1)1T⊗,

if i is such that ναi 6= αi, where 0 ≤ n(s)p,i ≤ 1, for 1 ≤ p ≤ r

(s)i . By taking

the corresponding coefficients, we have

(1⊗ · · · ⊗ 1⊗∆Tc (λi,−z)⊗ 1⊗ · · · ⊗ 1)πRbvL = πRb

+vL,

whereb+ = b(αl) · · · b(αi+1)b+(αi)b(αi−1) · · · b(α1),

withb+(αi) = xνn

r(1)i

,iαi(mr

(1)i ,i

+ 1) · · ·xνn1,iαi(m1,i + 1)

if ναi = αi and with

b+(αi) = xνnr(1)i

,iαi(mr

(1)i ,i

+1

v) · · ·xνn1,iαi(m1,i +

1

v)

if ναi 6= αi.

5.3. The maps eαi. Finally, we recall the maps eαi , which satisfy

eαi : V TL → V T

L

and their restriction to the principal subspace W TL ⊂ V T

L where

eαi · 1T =2

σ(αi)xναi (−1) · 1T if ναi = αi

eαi · 1T =2

σ(αi)xναi

(−1

2

)· 1T if ναi 6= αi,

when v = 2 and

eα1 · 1T =3

σ(α1)xνα1

(−1

3

)· 1T

eα2 · 1T =3

σ(α2)xνα2

(−1) · 1T

if v = 3, and commute with our operators xνβ (n) via

eαixνβ (n) = C(αi, β)xνβ

(n−

⟨β(0), αi

⟩)eαi .

Now, assume that we have a monomial

b = b(αl) · · · b(α1)xνsα1(−s

2) ∈ B

b = xνnr(1)l

,lαl

(mr(1)l ,l

) · · · · · ·xνnr(1)1 ,1

α1(m

r(1)1 ,1

) · · ·xνn2,1α1(m2,1)xνsα1

(−s2

),

of dual-charge-type

R =(r

(1)l , . . . , r

(k)l ; . . . ; r

(1)1 , . . . , r

(s)1 , 0, . . . , 0

)and a projection πRbvL, which is a coefficient of

πRxνnr(1)l

,lαl

(zr(1)l ,l

) · · ·xνn2,1α1(z1,1) 1T ⊗ · · ·

96 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

· · · ⊗ 1T ⊗ xνα1(−1

2)1T ⊗ · · · ⊗ xνα1

(−1

2)1T

= C xνn(k)

r(k)l

,lαl

(zr(k)l ,l

) · · ·xνn(k)1,l αl

(z1,l) · · ·xνn(k)

r(k)1 ,1

α1(zr(k)1 ,1

) · · ·xνn(k)2,1α1

(z1,1) 1T

⊗ . . .⊗

xνn(s+1)

r(s+1)l

,lαl

(zr(s+1)l ,l

) · · ·xνn(s+1)1,l αl

(z1,l) · · ·

· · ·xνn(s+1)

r(s+1)1 ,1

α1(zr(s+1)1 ,1

) · · ·xνn(s+1)2,1 α1

(z2,1) 1T

xνn(s)

r(s)l

,lαl

(zr(s)l ,l

) · · ·xνn(s)1,lαl

(z1,l) · · ·xνn(s)

r(s)1 ,1

α1(zr(s)1 ,1

) · · ·xνn(s)2,1α1

(z2,1) eα11T

⊗ . . .⊗

⊗xνn(1)

r(1)l

,lαl

(zr(1)l ,l

) · · ·xνn(1)1,lαl

(z1,l) · · ·xνn(1)

r(1)1 ,1

α1(zr(1)1 ,1

) · · ·xνn(1)2,1α1

(z2,1) eα11T ,

where C ∈ C∗.If we move the operator 1 ⊗ · · · ⊗ 1 ⊗ eα1 ⊗ · · · ⊗ eα1︸ ︷︷ ︸

s factors

all the way to the

left we will get a projection πR−b′vL, where

R− =(r

(1)l , . . . , r

(k)l ; . . . ; r

(1)1 , . . . , r

(s)1 − 1, 0, . . . , 0

)and

b′ = b(αl) · · · b′(α2)b′(α1),

with

b′(α1) = xνnr(1)1 ,1

α1(m

r(1)1 ,1

+ n(1)

r(1)1 ,1

+ · · ·+ n(s)

r(1)1 ,1

) · · ·

· · ·xνn2,1α1(m2,1 + n

(1)2,1 + · · ·+ n

(s)2,1)

and

b′(α2) = xνnr(1)2 ,2

α2(m

r(1)2 ,2−n

(1)

r(1)2 ,2

+ · · ·+ n(s)

r(1)2 ,2

2) · · ·

· · ·xνn1,2αj (m1,2 −n

(1)1,2 + · · ·+ n

(s)1,2

2).

Above we only considered the case when v = 2 and να1 6= α1, but note herethat the remaining cases are similar.

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 97

5.4. A proof of linear independence. Here we will prove our main the-orem:

Theorem 5.1. The setB = {bvL : b ∈ B}

is a basis of the principal subspace W TLk

.

Proof. By Proposition 4.1, the set of monomial vectors B spans W TLk

, andso it remains to show that B is linearly independent. To prove linear inde-pendence first assume that we have

bvL = 0

whereb = xνn

r(1)l

,lαl

(mr(1)l ,l

) · · ·xνn1,lαl(m1,l) · · ·

· · ·xνnr(1)1 ,1

α1(m

r(1)1 ,1

) · · ·xνn1,1α1(m1,1) ∈ B,

of charge-type

R′ =(nr(1)l ,l

, . . . , n1,l; . . . ;nr(1)1 ,1, . . . , n1,1

)and dual-charge-type

R =(r

(1)l , . . . , r

(k)l ; . . . ; r

(1)1 , . . . , r

(n1,1)1

),

which determines the projection πR, so that we have

πRbvL = 0. (5.4)

We will assume that να1 6= α1 and we will let s = n1,1. We apply 1⊗ · · · ⊗∆Tc (λ1,−z)d ⊗ 1⊗ · · · ⊗ 1︸ ︷︷ ︸

s−1 factors

to (5.4), where

∆Tc (λ1,−z)d = ∆T

c (λ1,−z) ◦ · · · ◦∆Tc (λ1,−z)︸ ︷︷ ︸

d times

and where d ∈ N is selected so that after application of this map the twistedquasi-particle of color 1 and charge s has energy − s

v . From the considera-tions in Subsection 5.2, we have

πRxνnr(1)l

,lαl

(mr(1)l ,l

) · · ·xνn1,lαl(m1,l) · · ·xνn

r(1)1 ,1

α1(m+

r(1)1 ,1

) · · · (5.5)

· · ·xνn2,1α1(m+

2,1)

(1T ⊗ · · · ⊗ 1T ⊗ xνα1

(−1

v)1T ⊗ · · · ⊗ xνα1

(−1

v)1T

)= 0,

which is a projection of a monomial vector from B. From (5.5) it followsthat

(1⊗ · · · ⊗ 1⊗ eα1 ⊗ · · · ⊗ eα1)πR−xνnr(1)l

,lαl

(mr(1)l ,l

) · · ·xνn1,lαl(m1,l) · · ·

· · ·xνnr(1)2 ,2

α2(m′

r(1)2 ,2

) · · ·xνn1,2α2(m′1,2)

98 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

xνnr(1)1 ,1

α1(m′

r(1)1 ,1

) · · ·xνn2,1α1(m′2,1)vL = 0,

whereR− =

(r

(1)l , . . . , r

(k)l ; . . . ; r

(1)1 , . . . , r

(s)1 − 1

).

By injectivity of 1⊗ · · · ⊗ 1⊗ eα1 ⊗ · · · ⊗ eα1 , we have

πR−xνnr(1)l

,lαl

(mr(1)l ,l

) · · ·xνn1,lαl(m1,l) · · ·xνn

r(1)2 ,2

α2(m′

r(1)2 ,2

) · · · (5.6)

· · ·xνn1,2α2(m′1,2)xνn

r(1)1 ,1

α1(m′

r(1)1 ,1

) · · ·xνn2,1α1(m′2,1)vL = 0.

If v = 2, for every 2 ≤ p ≤ r(1)1 such that np,1 = s′ ≤ s we have

m′p,1 = m+p,1 + s′ ≤ −np,1

2−

∑p>p′>0

min{np,1, np′,1}+ s′;

m′p+1,1 = m+p,1 + s′ ≤ m+

p,1 − np,1 + s′ = m′p,1 − np,1 if np+1,1 = np,1;

m′p,2 = m+p,2−

s′

2≤ −np,2

2+

1

2

r(1)1∑q=1

min {nq,1, np,2}−∑

p>p′>0

min{np,1, np′,1}−s′

2;

m′p+1,2 = m+p,2 −

s′

2≤ m+

p,2 − np,2 −s′

2= m′p,2 − np,2 if np+1,2 = np,2.

If v = 3, for every 2 ≤ p ≤ r(1)1 and for np,1 = s′ ≤ s we have

m′p,1 = m+p,1 +

2s′

3≤ −np,1

3− 2

3

∑p>p′>0

min{np,1, np′,1}+2s′

3;

m′p+1,1 = m+p,1 +

2s′

3≤ m+

p,1 −2

3np,1 +

2s′

3= m′p,1 −

2

3np,1 if np+1,1 = np,1;

m′p,2 = m+p,2−s

′ ≤ −np,2 +

r(1)1∑q=1

min {nq,1, np,2}−∑

p>p′>0

2 min{np,1, np′,1}−s′;

m′p+1,2 = m+p,2 − s

′ ≤ m+p,2 − np,2 − s

′ = m′p,2 − np,2 if np+1,2 = np,2.

This shows that in (5.6) we have the projection of a monomial vector fromthe set B. In the case when να1 = α1, with the above procedure will endwith a monomial vector as in (5.6), which is also from the set B, since for

every 2 ≤ p ≤ r(1)1 and for np,1 = s′ ≤ s, we have

m′p,1 = m+p,1 + 2s′ ≤ −np,1 −

∑p>p′>0

2 min{np,1, np′,1}+ 2s′;

m′p+1,1 = m+p,1 + 2s′ ≤ m+

p,1 − 2np,1 + 2s′ = m′p,1 − 2np,1 if np+1,1 = np,1;

m′p,2 = m+p,2−s

′ ≤ −np,2 +

r(1)1∑q=1

min {nq,1, np,2}−∑

p>p′>0

2 min{np,1, np′,1}−s′;

m′p+1,2 = m+p,2 − s

′ ≤ m+p,2 − 2np,2 − s′ = m′p,2 − 2np,2 if np+1,2 = np,2.

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 99

If we continue in this way, “removing” one by one twisted quasi-particlesfrom the monomial b and by checking in each step that monomial vectorsare in the set B, after finitely many steps we arrive at vL = 0, which is acontradiction.

Now, consider a linear combination of elements from B satisfying∑a∈A

cabavL = 0, (5.7)

where ba ∈ B are monomials of the same color-type and ca ∈ C. We willassume that the monomial b of charge-type R′ and dual-charge-type R isthe smallest monomial in (5.7) with respect to our linear ordering. Recallthat the dual-charge-type R determines the projection πR and note thatfrom the definition of the projection it follows that all monomial vectorsbavL in (5.7), with charge-type R′a such that R′a > R′ (see (3.2)), will beannihilated. So, after applying πR to (5.7), we will have∑

a∈AcaπRbavL = 0, (5.8)

where all monomial vectors are of the same color-charge-type. Now we applythe above-described procedure of using the maps 1 ⊗ · · · ⊗ ∆T

c (λ1,−z)d ⊗1⊗· · ·⊗ 1 and 1⊗· · ·⊗ 1⊗ eα1 ⊗· · ·⊗ eα1 to the smallest element b. Duringthis procedure, all monomial vectors ba such that ba > b (see (3.2)) willbe annihilated. From (5.8) now we have πRcabvL = 0, which then impliesca = 0. Repeating this procedure, after finitely many steps we will get thatall coefficients ca of (5.7) are zero, which proves the theorem. �

6. Characters of principal subspaces

We define character of principal subspace W TL by

chW TLk

=∑

m,r1,...,rl≥0

dim W TLk (m,r1,...,rl)

qmyr11 · · · yrll ,

where W TLk (m,r1,...,rl)

is a weight subspace spanned by monomial vectors of

weight −m and color-type (r1, . . . , rl).We write the characters of principal subspaces in terms of dual-charge-

type parts r(s)i . Therefore, to obtain characters first we rewrite the con-

ditions on the energies of twisted quasi-particles of a basis B in terms of

the dual-charge-type. In the case of A(2)2l−1 for fixed color type (rl, . . . , r1),

charge-type R′ = (nr(1)l

, . . . , n1,l; . . . ;nr(1)1

, . . . , n1,1) and dual-charge-type

R = (r(1)l , . . . , r

(k)l ; . . . ; r

(1)1 , . . . , r

(k)1 ) we have:

r(1)i∑p=1

(∑

p>p′>0

min{np,i, np′,i}+1

2np,i) =

1

2

k∑s=1

r(s)2

i , 2 ≤ i ≤ l − 1, (6.1)

100 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

r(1)l∑p=1

(∑

p>p′>0

2min{np,l, np′,l}+ np,l) =

k∑s=1

r(s)2

l , (6.2)

r(1)i∑p=1

r(1)i−1∑q=1

1

2min{np,i, nq,i−1} =

1

2

k∑s=1

r(s)i−1r

(s)i , 1 ≤ i ≤ l − 1, (6.3)

andr(1)l∑p=1

r(1)l−1∑q=1

min{np,l, nq,l−1} =k∑s=1

r(s)l−1r

(s)l . (6.4)

Expressions (6.1)–(6.4) are proved by using induction on the level k ∈ N ofthe standard module. We would like to note that obtained expressions aresimilar to the expressions (5.9) and (5.12) in [G].

For r ∈ N set (q1a ; q

1a )r = (1− q

1a ) · · · (1− q

ra ), for a = 1, a = 2, or a = 3.

We have that1

(q1a ; q

1a )r

=∑j≥0

pr(j)qja , (6.5)

where pr(j) denotes the number of partitions of j with most r parts (cf.[An]).

Now, from the definition of the set B by using (6.1)–(6.6), for A(2)2l−1 we

have:

ch W TLk

=

∑r(1)1 ≥...≥r

(k)1 ≥0

q12r(1)1

2+...+ 1

2r(k)1

2

(q12 ; q

12 )r(1)1 −r

(2)1

. . . (q12 ; q

12 )r(k−1)1 −r(k)1

(q12 ; q

12 )r(k)1

yr(1)1 +···+r(k)1

1

·∑

r(1)2 ≥...≥r

(k)2 ≥0

q12r(1)2

2+...+ 1

2r(k)2

2− 1

2

(r(1)2 r

(1)1 −...−r

(k)2 r

(k)1

)(q

12 ; q

12 )r(1)2 −r

(2)2

. . . (q12 ; q

12 )r(k−1)2 −r(k)2

(q12 ; q

12 )r(k)2

yr(1)2 +···+r(k)2

2

· · · · · · · · ·

·∑

r(1)l−1≥...≥r

(k)l−1≥0

q12r(1)l−1

2+...+ 1

2r(k)l−1

2− 1

2

(r(1)l−1r

(1)l−2−...−r

(k)l−1r

(k)l−2

)(q

12 ; q

12 )r(1)l−1−r

(2)l−1

. . . (q12 ; q

12 )r(k−1)l−1 −r(k)l−1

(q12 ; q

12 )r(k)l−1

yr(1)l−1+···+r(k)l−1

l−1

·∑

r(1)l ≥...≥r

(k)l ≥0

qr(1)l

2+...+r

(k)l

2−r(1)l r

(1)l−1−...−r

(k)l r

(k)l−1

(q)r(1)n −r

(2)l

. . . (q)r(k−1)l −r(k)l

(q)r(k)l

yr(1)l +···+r(k)ll .

We calculate characters similarly for the other cases, therefore we willgive only expressions for energies of basis twisted quasi-particles which weused.

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 101

In the case of D(2)l for fixed color type (rl−1, . . . , r1), charge-type R′ =

(nr(1)l−1

, . . . , n1,l−1; . . . ;nr(1)1

, . . . , n1,1) and dual-charge-type

R = (r(1)l−1, ..., r

(k)l−1; ...; r

(1)1 , ..., r

(k)1 )

we have:

r(1)i∑p=1

(∑

p>p′>0

2min{np,i, np′,i}+ np,i) =

k∑s=1

r(s)2

i , 1 ≤ i ≤ l − 2, (6.6)

r(1)l−1∑p=1

(∑

p>p′>0

2min{np,l−1, np′,l−1}+1

2np,l−1) =

1

2

k∑s=1

r(s)2

l−1 , (6.7)

r(1)i∑p=1

r(1)i−1∑q=1

min{np,i, nq,i−1} =k∑s=1

r(s)i−1r

(s)i , 2 ≤ i ≤ l − 1. (6.8)

In the case of E(2)6 for fixed color type (r4, . . . , r1), charge-type R′ =

(nr(1)4

, . . . , n1,4; . . . ;nr(1)1

, . . . , n1,1) and dual-charge-type R = (r(1)4 , . . . , r

(k)4 ;

. . . ; r(1)1 , . . . , r

(k)1 ) we have:

r(1)i∑p=1

(∑

p>p′>0

min{np,i, np′,i}+1

2np,i) =

1

2

k∑s=1

r(s)2

i , 1 ≤ i ≤ 2, (6.9)

r(1)i∑p=1

(∑

p>p′>0

2min{np,i, np′,i}+ np,i) =

k∑s=1

r(s)2

i , 3 ≤ i ≤ 4 (6.10)

r(1)i∑p=1

r(1)i−1∑q=1

1

2min{np,i, nq,i−1} =

1

2

k∑s=1

r(s)i−1r

(s)i , 1 ≤ i ≤ 2, (6.11)

andr(1)i∑p=1

r(1)i−1∑q=1

min{np,i, nq,i−1} =

k∑s=1

r(s)i−1r

(s)i , 3 ≤ i ≤ 4. (6.12)

In the case ofD(3)4 for fixed color type (r2, r1), charge-typeR′ = (n

r(1)2

, . . . ,

n1,2;nr(1)1 ,1

, . . . , n1,1) and dual-charge-type R = (r(1)2 , . . . , r

(k)2 ; r

(1)1 , . . . , r

(k)1 )

we have:

r(1)1∑p=1

(2

3

∑p>p′>0

min{np,1, np′,1}+1

3np,1) =

1

3

k∑s=1

r(s)2

1 . (6.13)

102 MARIJANA BUTORAC AND CHRISTOPHER SADOWSKI

r(1)2∑p=1

(∑

p>p′>0

2min{np,2, np′,2}+ np,2) =

k∑s=1

r(s)2

2 , (6.14)

andr(1)2∑p=1

r(1)1∑q=1

min{np,2, nq,1} =k∑s=1

r(s)1 r

(s)2 . (6.15)

From the above equations, we now have:

Theorem 6.1. For each of the affine Lie algebras A(2)2l−1, D

(2)l , E

(2)6 , and

D(3)4 , the principal subspace W T

Lkof Lν(kΛ0) has multigraded dimension

given by:

• for A(2)2l−1:

ch W TLk

=∑

r(1)1 ≥···≥r

(k)1 ≥0

···r(1)l−1≥···≥r

(k)l−1≥0

q12

∑l−1i=1

∑ks=1 r

(s)2

i − 12

∑l−1i=2

∑ks=1 r

(s)i−1r

(s)i∏l−1

i=1(q12 ; q

12 )r(1)i −r

(2)i

· · · (q12 ; q

12 )r(k)i

l−1∏i=1

yr(1)i +···+r(k)ii

·∑

r(1)l ≥···≥r

(k)l ≥0

q∑ks=1 r

(s)2

l −∑ks=1 r

(s)l−1r

(s)l

(q)r(1)l −r

(2)l

· · · (q)r(k)l

yr(1)l +···+r(k)ll

• for D(2)l :

ch W TLk

=∑

r(1)1 ≥···≥r

(k)1 ≥0

···r(1)l−2≥···≥r

(k)l−2≥0

q∑l−2i=1

∑ks=1 r

(s)2

i −∑l−2i=2

∑ks=1 r

(s)i−1r

(s)i∏l−2

i=1(q)r(1)i −r

(2)i

· · · (q)r(k)i

l−2∏i=1

yr(1)i +···+r(k)ii

·∑

r(1)l−1≥···≥r

(k)l−1≥0

q12

∑ks=1 r

(s)2

l−1 −∑ks=1 r

(s)l−2r

(s)l−1

(q12 ; q

12 )r(1)l−1−r

(2)l−1

· · · (q12 ; q

12 )r(k)l−1

yr(1)l−1+···+r(k)l−1

l−1

• for E(2)6 :

ch W TLk

=∑r(1)1 ≥···≥r

(k)1 ≥0

r(1)2 ≥···≥r

(k)2 ≥0

q12

∑2i=1

∑ks=1 r

(s)2

i −∑ks=1 r

(s)1 r

(s)2∏

i=1,2(q12 ; q

12 )r(1)i −r

(2)i

· · · (q12 ; q

12 )r(k)i

∏i=1,2

yr(1)i +···+r(k)ii

·∑

r(1)3 ≥···≥r

(k)3 ≥0

r(1)4 ≥···≥r

(k)4 ≥0

q∑4i=3

∑ks=1 r

(s)2

i −∑ks=1(r

(s)2 r

(s)3 +r

(s)3 r

(s)4 )∏

i=3,4(q)r(1)i −r

(2)i

· · · (q)r(k)i

∏i=3,4

yr(1)i +···+r(k)ii

COMBINATORIAL BASES OF PRINCIPAL SUBSPACES OF MODULES FOR... 103

• for D(3)4 :

ch W TLk

=∑

r(1)1 ≥···≥r

(k)1 ≥0

q13

∑ks=1 r

(s)2

1

(q13 ; q

13 )r(1)1 −r

(2)1

· · · (q13 ; q

13 )r(k)1

yr(1)1 +···+r(k)1

1

·∑

r(1)2 ≥···≥r

(k)2 ≥0

q∑ks=1 r

(s)2

2 −∑ks=1 r

(s)1 r

(s)2

(q)r(1)2 −r

(2)2

· · · (q)r(k)2

yr(1)2 +···+r(k)2

2 .

Acknowledgement

We are grateful to Mirko Primc for his useful comments and support. Thefirst named author would also like to thank Drazen Adamovic. The firstauthor is partially supported by the Croatian Science Foundation underthe project 2634 and by the QuantiXLie Centre of Excellence, a projectcofinanced by the Croatian Government and European Union through theEuropean Regional Development Fund - the Competitiveness and CohesionOperational Programme (Grant KK.01.1.1.01.0004).

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(Marijana Butorac) Department of Mathematics, University of Rijeka, RadmileMatejcic 2, 51 000 Rijeka, Croatia.mbutorac@math.uniri.hr

(Christopher Sadowski) Department of Mathematics and Computer Science, Ursi-nus College, Collegeville, PA 19426, USA.csadowski@ursinus.edu

This paper is available via http://nyjm.albany.edu/j/2019/25-3.html.