arXiv:math/0104160v1 [math.QA] 16 Apr 2001 filearXiv:math/0104160v1 [math.QA] 16 Apr 2001...

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Decomposition of the Moonshine Module

with respect to a code over Z2k

Hiroki SHIMAKURA

Graduate School of Mathematical Sciences,

University of Tokyo, Komaba 3-8-1, Tokyo 153-8914, Japan

e-mail: [email protected]

Abstract

In this paper, we give a decomposition of the moonshine module V with re-spect to an extremal Type II code over Z2k for an integer k ≥ 2. Then we obtainautomorphisms of V , some 4A and 2B elements of the Monster with respect to thedecomposition. We give examples of such a decomposition for some k and give theMcKay-Thompson series for a 4A element.

1 Introduction

The aim of this paper is to give a decomposition of the moonshine module V with respect

to an extremal Type II code over Z2k = Z/2kZ.

The notation of a vertex operator algebra (VOA) is introduced in [Bo, FLM]. One

of the most interesting examples of a vertex operator algebra is the moonshine module

V constructed in [FLM]. The automorphism group of V is the Monster, the largest

sporadic finite simple group.

V contains a subVOA isomorphic to the tensor product of 48 copies of L(1/2, 0),

called a Virasoro frame in [DGH], and the VOA structure of V was reconstructed from

a Virasoro frame in [Mi2]. By using a Virasoro frame, the McKay-Thompson series for

some elements of the Monster are explicitly computed.

Since L(1/2, 0) is isomorphic to W2 algebra at c = 1/2, we should study the first

member of the unitary series of Wn algebras. In particular, W4 algebra at c = 1 is

realized as the fixed-point subspace V +L of the lattice VOA VL corresponding to the rank

one lattice L = Zα with 〈α, α〉 = 6 with respect to the −1 automorphism of the lattice.

Its fusion rules have a nice symmetry according to [Ab] and for each embedding of V +L

into V , we get a 4A element of the Monster [Ma].

1

http://arxiv.org/abs/math/0104160v1

In this paper, we consider more general case L = Zα with 〈α, α〉 = 2k for an integer

k ≥ 2. Since the Leech lattice Λ contains many elements of squared length 2k and V

contains V +Λ as a subVOA, V contains many copies of V +

L . We consider a set of 24

mutually orthogonal pairs of opposite vectors of Λ with squared length 2k. We call such a

set a 2k-frame of Λ. We have a natural bijection between equivalence classes of 2k-frames

and extremal Type II codes of length 24 over Z2k. By using a 2k-frame, we show that V

contains a subVOA isomorphic to the tensor product of 24 copies of V +L . Then we give

the decomposition of V as a (V +L )⊗24-module, which is described in terms of the extremal

Type II code of length 24 over Z2k corresponding to the 2k-frame. By the fusion rules of

V +L , we define automorphisms of V with respect to this decomposition. When k is odd,

those are 4A elements of the Monster, and when k is even, those are 2B elements of the

Monster. We give examples of a 2k-frame for some k and the corresponding decomposition

of V . Then we compute the McKay-Thompson series for a 4A element of the Monster.

Professor Ching Hung Lam studies the subject independently.

Throughout the paper, we will work over the field C of complex numbers unless oth-

erwise stated. We denote the set of integers by Z and the ring of the integers modulo k

by Zk.

Acknowledgements. The author wishes to thank Professor Atsushi Matsuo, my re-

search supervisor, for his advice and warm encouragement. He also thanks Professor

Masaaki Kitazume and Professor Ching Hung Lam for helpful advice.

2 Preliminaries

In this section, we recall or give some definitions and facts necessary in this paper.

2.1 Vertex operator algebra V +L and its irreducible modules

In this section, we give the vertex operator algebra V +L and its irreducible modules. Note

that the description of V +L in this paper is slightly different from those given in [DN, Ab].

Let L = Zα be an even lattice of rank one with 〈α, α〉 = 2k. Set H = C ⊗Z L

and we regard L as a subgroup of H . We extend the form 〈·, ·〉 to a C-bilinear form

on H . Let L = h ∈ H|〈h, α〉 ∈ Z = L/2k be the dual lattice of L. Let L be the

trivial extension of L by the order 2 cyclic group 〈−1〉. Form the induced L-module

CL = C[L] ⊗C[±1] C, where C[·] denotes the group algebra and −1 acts on C as

multiplication by −1. Note that CL ∼= C[L] as an algebra and we denote an element

of CL corresponding to µ ∈ L by eµ. Set VL = S(H ⊗ t−1C[t−1]) ⊗ CL. For

2

λ + L ∈ L/L, we set Vλ+L = S(H ⊗ t−1C[t−1])⊗ Cλ + L, where λ + L is regarded as

a subset of L and Cλ + L is the subspace of CL spanned by eµ|µ ∈ λ + L. It iswell known that VL has a VOA structure and Vλ+L has a VL-module structure for each

λ+ L ∈ L/L (cf. [FLM]). Note that all the irreducible VL-modules are given by the set

Vλ+L|λ + L ∈ L/L (cf. [D1]). For convenience, we denote h ⊗ tn by h(n) for h ∈ H

and n ∈ Z.

Let ρ: L → C× be a group homomorphism such that ρ(2µ) = (−1)〈µ,µ〉/2 for µ ∈ L.

Let θL be the automorphism of L given by θL(eµ) = ρ(2µ)e−µ for µ ∈ L and θL(−1) = −1.

Let θVL be the unique commutative algebra automorphism of VL such that θVL(h(n)⊗1) =

−h(n)⊗ 1 and θVL(1⊗ eµ) = 1⊗ θL(eµ) for h ∈ H and µ ∈ L.

Let λ be a representative of a coset in L/L. Let θλ: Vλ+L → V−λ+L be the linear

map defined by θλ(x ⊗ eλ+µ) = ρ(2µ)θH(x) ⊗ e−λ−µ, where θH acts on S(H ⊗ t−1C[t−1])

as the unique algebra automorphism such that θH(h(n)) = −h(n) for h ∈ H . Then

θλ is an isomorphism of VL-modules. When λ ∈ L/2, the linear map Vλ+L → V−λ+L,

x 7→ ρ(−2λ)x is an isomorphism of VL-modules and we have the automorphism of Vλ+L

θλ(x ⊗ eλ+µ) = ρ(2λ + 2µ)θH(x) ⊗ e−λ−µ. By abuse of notation, we denote θλ by θVL ,

and we view θVL as an automorphism of VL . For each θVL-stable subspace M of VL , let

M± denote the ±1 eigenspaces of M with respect to θVL respectively. Note that V +L is a

subVOA of VL.

Let KL = θL(a)a−1|a ∈ L be the subgroup of L. Let T0 and T1 be irreducible

L/KL-modules over C such that eα acts by ρ(α) and eα acts by −ρ(α) respectively. Set

V TiL = S(H⊗t−1/2C[t−1])⊗Ti. Then it has an irreducible θVL-twisted VL module structure.

Moreover, V T0L and V T1

L give all the inequivalent irreducible θVL-twisted VL-modules (cf.

[D2]).

Let θH be the unique commutative algebra automorphism of S(H ⊗ t−1/2C[t−1]) such

that θH(h(n)) = −h(n) for h ∈ H and n ∈ 1/2+Z. We consider the linear automorphism

of V TiL given by θH ⊗ 1Ti , where 1Ti is the identity operator of Ti. By abuse of notation,

we denote both automorphisms of V T0L and V T1

L by θV TL . Since α = θVL(α) on Ti, θV TL is

an automorphism as a VL-module. We denote the ±1 eigenspaces of V TiL with respect to

θV TL by V Ti,±L respectively. Then V Ti,±

L becomes an irreducible V +L -module.

Note that V +L and its irreducible modules defined above are isomorphic to those given

in [DN]. By Theorem 5.13 of [DN], the set V ±L , V

±α/2+L, V

Ti,±L , Vrα/2k+L|i = 0, 1, r =

1, . . . , k − 1gives all inequivalent irreducible V +L -modules.

2.2 Moonshine module

Let us review the moonshine module V from [FLM] for what we need in this paper.

3

Let Λ be the Leech lattice with the positive-definite Z-bilinear form 〈· , · 〉. It is a

unique positive-definite even unimodular lattice of rank 24 without roots.

Let Λ be the central extension of Λ by the cyclic group 〈−1〉 ∼= Z2:

1 → 〈−1〉 → Λ → Λ → 0 (2.1)

with the commutator map c0(β, γ) = 〈β, γ〉+ 2Z for β, γ ∈ Λ.

Let θΛ be the automorphism of Λ given by θΛ(a) = a−1(−1)〈a,a〉/2 for a ∈ Λ. We denote

the center of Λ by Cent Λ. Set KΛ = θΛ(a)a−1|a ∈ Λ ⊂ Cent Λ. Then KΛ is a normal

subgroup of Λ; consider the group Λ/KΛ. Note that the center of Λ/KΛ is 〈−KΛ〉 ∼= Z2.

By Theorem 5.5.1 of [FLM], we have the following proposition.

Proposition 2.1. ([FLM]) Set G = Λ/KΛ and let χ be a character of Cent G = 〈−KΛ〉such that χ(−KΛ) = −1. Let A be a maximal abelian subgroup of G and let ψ: A →C× be a character with ψ|Cent G = χ. Then T = IndG

ACψ is the unique irreducible G-

module on which x ∈ Cent G acts by χ(x)1T , where Cψ is the one-dimensional A-module

corresponding to ψ. Moreover, T ∼= ⊕Cϕ, where ϕ ranges over the characters of A whose

restriction to Cent G is χ, and dim T = 212.

Set the induced Λ-module CΛ = C[Λ]⊗C[±1]C, where −1 acts on C as multiplication

by −1. We extend θΛ to C[Λ] linearly. Since θΛ fixes −1, we view the automorphism θΛ

as an automorphism of CΛ.Set h = C ⊗Z Λ. We regard Λ as a subgroup of h. We extend 〈·, ·〉 to a C-bilinear

form on h. Let T be the irreducible Λ/KΛ-module given in Proposition 2.1. Set VΛ =

S(h ⊗ t−1C[t−1]) ⊗ CΛ and V TΛ = S(h ⊗ t−1/2C[t−1]) ⊗ T . For convenience, we also

denote h⊗ tn by h(n) for h ∈ h and n ∈ Z/2. Let θVΛ be the unique commutative algebra

automorphism of VΛ such that θVΛ(h(n)⊗1) = −h(n)⊗1 and θVΛ(1⊗b) = 1⊗θΛ(b), whereh ∈ h, n ∈ Z<0 and b ∈ CΛ. Let θh be the unique commutative algebra automorphism

of S(h ⊗ t−1/2C[t−1]) such that θh(h(n)) = −h(n) for h ∈ h, n ∈ 1/2 + Z<0. Then

θV TΛ = θh ⊗ (−1T ) is an automorphism of V TΛ , where 1T is the identity operator on T .

Thus we have V = V +Λ ⊕ V T,+

Λ , where V +Λ is the θVΛ-fixed-point subspace of VΛ and V T,+

Λ

is the θV TΛ -fixed-point subspace of V TΛ .

2.3 2k-frames and codes over Z2k

In this subsection, we give some terminology on a code over Z2k (cf. [DHS]) and formulate

the inverse of the generalized Construction A relating lattices and codes.

4

A (linear) code C of length n over Z2k is a Z2k-submodule of Zn2k. We fix an ordered

basis of Zn2k and denote i-th element of this basis by (0, . . . , 0, 1, 0, . . . , 0), where 1 only

appears in the i-th position. An element of C is called a codeword. The Euclidean

weight is Ewt(·) on Zn2k is given by Ewt(c) =∑24

i=1m2i , where c = (m1, . . . , mn) ∈ Zn2k

and −k < mi ≤ k for i = 1, . . . , n. We define the inner product of x and y in Zn2k by

〈x, y〉 =∑n

i=1 xiyi (mod 2k), where x = (x1, . . . , xn) and y = (y1, . . . , yn). The dual code

C⊥ of C is defined as C⊥ = x ∈ Zn2k|〈x, y〉 = 0for all y ∈ C. C is self-orthogonal if

C ⊂ C⊥ and C is self-dual if C = C⊥. We define a Type II code over Z2k to be a self-dual

code with all codewords having Euclidean weight divisible by 4k.

The symmetric group Sn acts on Zn2k by the permutation of the coordinate positions.

Let εi be the operator which changes the signs of the i-th position of Zn2k. Set Π = 〈εi|i =1, . . . , n〉, and Sn = 〈Sn,Π〉 = Sn ⋉ Π, since (i j)εi(i j) = εj for i, j ∈ 1, . . . , n. Two

codes C0 and C1 over Z2k are called equivalent if they both have length n and if there

exists σ ∈ Sn such that C0 = σ(C1).

Definition 2.2. A 2k-frame of a lattice of rank n is a set of n mutually orthogonal pairs

of opposite vectors of squared length 2k.

We denote the group of isometries of a lattice Λ by Aut(Λ). Since Aut(Λ) preserves

the inner product, Aut(Λ) acts on the set of 2k-frames of Λ. We say that 2k-frames S0,

S1 of Λ are equivalent if there exists σ ∈ Aut(Λ) such that S0 = σ(S1).

It is well known that one of the constructions of a lattice from a code over Z2k is the

generalized Construction A and such a lattice contains a 2k-frame (cf. [CS, BDHO]). Let

us consider the inverse map of the correspondence.

First, we recall the generalized Construction A. Let π: Zn → Zn2k be the canonical

map. Let C be a code of length n over Z2k. By the generalized Construction A, a positive-

definite lattice of rank n is given by A2k(C) = π−1(C)/√2k. Let (e1, . . . , e24) be the

standard orthogonal basis of Z24. Then Λ has a 2k-frame FC = ±√2ke1, . . . ,±

√2ke24.

Let C0, C1 be equivalent codes over Z2k and let σ be an element of Sn such that C0 =

σ(C1). Then we have A2k(C0) = σ(A2k(C1)) and FC0 = σ(FC1), where we regard σ as an

element of O(24). Therefore the equivalent codes correspond to the same lattice (up to

isomorphism), and the corresponding 2k-frames of such a lattice are equivalent. Hence

we obtain the map from the equivalence class of codes over Z2k to the equivalence class

of 2k-frames of the lattice.

We define an extremal Type II code of length 24 over Z2k to be a Type II code with

minimum Euclidean weight 8k. Note that for an extremal Type II code C of length 24

over Z2k, A2k(C) is an even unimodular lattice with no roots, namely, A2k(C) is the Leech

lattice.

5

Let Λ be the Leech lattice and suppose k ≥ 2. Let S be a 2k-frame of Λ and let N be

the sublattice of Λ generated by S. We choose an ordered basis (x1, . . . , x24) of N from

S. Since N ⊂ Λ ⊂ N = N/2k, we have the isomorphism

N/N → Z242k,

24∑

i=1

cixi/2k +N 7→ (c1, . . . , c24).

By the above isomorphism, we view Λ/N ⊂ N/N as a code of length 24 over Z2k. We

denote the codeword corresponding to v ∈ Λ by cv. For v, w ∈ Λ, we have Ewt(cv) =

2k ×min〈u, u〉|u ∈ v +N and 〈cv, cw〉 = 2k〈v, w〉 (mod 2k). Thus Λ/N is an extremal

Type II code of length 24 over Z2k. Note that the code Λ/N depends on the choice of the

ordered basis. However, it is easy to see that its equivalence class is uniquely determined

by the 2k-frame S. Therefore we denote the equivalence class of Λ/N by ϕ(S). For any

σ ∈ Aut(Λ), we have Λ/N = σ(Λ)/σ(N) = Λ/σ(N). Hence the equivalence class of codes

is uniquely determined by the equivalence class of the 2k-frame. Then ϕ is a map from

the set of the equivalence classes of extremal Type II codes over Z2k to the set of the

equivalence classes of 2k-frames of the Leech lattice.

Proposition 2.3. The map ϕ is bijective. Moreover, ϕ is the inverse map of the map

given by the generalized Construction A.

Proof. Let C be an extremal Type II code of length 24 over Z2k and let FC be the

2k-frame of Λ given by the generalized Construction A. It is easy to see that C is an

element of the equivalence class ϕ(FC). Therefore ϕ is surjective. Next, we will show

that ϕ is injective. Let F = ±x1, . . . ,±x24 be a 2k-frame of Λ and let C be a code

of ϕ(F ). Let FC be the 2k-frame corresponding to a code C. It is suffices to show that

F and FC are equivalent. Note that Λ = ∑ cixi/2k|(c1, . . . , c24) ∈ C and A2k(C) =

∑

ciei/√2k|(c1, . . . , c24) ∈ C. Then we have the isomorphism of the lattice

A2k(C) → Λ,∑

ciei/√2k 7→

∑

cixi/2k,

and this map sends FC to F . Therefore FC is equivalence to F .

Proposition 2.4. Let S be a 2k-frame of Λ and let N be the sublattice of Λ generated by

S. Let C = Λ/N be a code over Z2k and let C2 = (Λ ∩ (N/2))/N be a binary code. Set

m = dimC2 over Z2.

(1) |Λ/N | = (2k)12.

(2) m ≥ 12.

6

(3) |N/(2Λ ∩N)| is an elementary abelian 2-group with order 224−m.

(4) If k is odd, then C2 is a Type II code.

Proof. For any lattice R, we denote the dual lattice of R by R. Consider the inclusion

N ⊂ Λ = Λ ⊂ N = N/2k. It is well known that |Λ/N | = |N/Λ|. Since N/N ∼= Z242k,

we have |Λ/N | = |N/Λ| = (2k)12. Thus we obtain (1). Since for each element of C,

its order is divisible by 2k, we obtain (2). Consider the inclusion 2N ⊂ 2Λ ∩ N ⊂ N

and we obtain |N/(2Λ ∩ N)||(2Λ ∩ N)/2N | = |N/2N | = 224. Since (Λ ∩ (N/2))/N ∼=(2Λ∩N)/2N ∼= Zm2 , we have |N/(2Λ∩N)| = 224−m and we obtain (3). Suppose k is odd.

Let v =∑

cixi/2, w =∑

dixi/2 be elements of Λ∩ (N/2) and let cv = (c1, . . . , c24), cw =

(d1, . . . , d24) be corresponding codewords of C2. Then we have 〈v, w〉 = (k/2)∑

cidi ∈ Z,

namely 〈cv, cw〉 =∑

cidi ∈ 2Z. Thus C2 is self-orthogonal. Since C2 is a Syolw 2-group,

the dimension of C2 is 12, and C2 is self-dual. Since 〈v, v〉 ∈ 2Z, we have 〈cv, cv〉 ∈ 4Z.

Hence we obtain (4).

3 Decomposition of V and Z2k codes

In this section, for an integer k ≥ 2, we give the decomposition of V as a (V +L )⊗24-module

associated with an extremal Type II code over Z2k.

For convenience, we use the following notation. For c = (c1, . . . , c24) ∈ Z242 , we set

M(c)+ =⊕

ei∈±∏

ei=+

24⊗

i=1

V eiciα/2+L

,

MT (c)− =⊕

ei∈±∏

ei=−

24⊗

i=1

VTci ,ei,

L ,

where we regard ci ∈ Z2 as an element of Z.

Remark 3.1. Let c = (c1, . . . , c24) be a codeword of Z242 .

(1) M(c)+ is the fixed-point subspace of ⊗Vciα/2+L with respect to ⊗θVL .

(2) MT (c)− is the −1 eigenspace of ⊗V TciL with respect to ⊗θV TL .

For c = (c1, . . . , c24) ∈ Z242k, we set

M(c) =

24⊗

i=1

Vciα/2k+L.

Note that M(c) ∼= M(σ(c)) as a (V +L )⊗24-module for c ∈ Z24

2k and σ ∈ Π, where Π is the

set of the operators which change signs of some positions.

7

3.1 Main result

The following is our main theorem.

Theorem 3.2. Let C be an extremal Type II code of length 24 over Z2k and set C2 =

(c1, . . . , c24) ∈ C|ci = 0 (mod k) for all i, which is binary code. Set m = dimC2 over

Z2.

(1) There is an embedding

V ⊃ V +Λ ⊃ (V +

L )⊗24

of subVOAs.

(2) V +Λ is decomposed as a (V +

L )⊗24-module as follows:

V +Λ

∼=⊕

c∈C2

M(c)+ ⊕ 1

2

⊕

c∈C\C2

M(c).

(3) V T,+Λ is decomposed as a (V +

L )⊗24-module as follows:

V T,+Λ

∼=⊕

c∈C⊥2

2m−12MT (c)−.

Remark 3.3. This decomposition is uniquely determined by the extremal Type II code

of length 24 over Z2k, up to the permutation of the 24 tensor products for each components

of V .

Remark 3.4. [Ch, GH] For any integer k ≥ 2, there exists an extremal Type II code of

length 24 over Z2k. However, the classification of extremal Type II codes of length 24

over Z2k is still missing.

The rest of this section is devoted to the proof of the theorem.

3.2 Existence of (V +L )⊗24 in V

In this subsection, we will show Theorem 3.2 (1). In order to do it, we have to construct

an appropriate 2-cocycle of Λ.

Let S be a 2k-frame corresponding to C and let N be the sublattice of Λ generated

by S.

8

Lemma 3.5. There exists a set I of representatives of Λ/N satisfying the following con-

ditions:

(H1) (Λ ∩ (N/2)) ∩ I = ∑m

i=1 dihi|di ∈ 0, 1, where hi ∈ Λ ∩ (N/2).

(H2) For any λ ∈ I \ ((Λ ∩N/2) ∩ I), −λ ∈ I.

Proof. Let h1 + N, · · · , hm + N be a set of generators of (Λ ∩ (N/2))/N . Then

we choose ∑m

i=1 dihi|di ∈ 0, 1 as the set of representatives of (Λ ∩ (N/2))/N . For

λ ∈ b + N ∈ Λ/N with b /∈ N/2, we choose the representative of −b +N ∈ Λ/N as −λ.Therefore we obtain the set of representatives satisfying (H1) and (H2).

Lemma 3.6. Let ρ: 2Λ ∩ N → 〈−1〉 be a group homomorphism such that ρ(2γ) =

(−1)〈γ,γ〉 for γ ∈ N . There exist a set-theoretic section Λ → Λ, β 7→ eβ and a set I of

representatives of Λ/N satisfying

eβeγ = eβ+γ , (3.1)

θΛ(eλ+β) =

ρ(2λ+ 2β)e−λ−β if λ ∈ I ∩N/2,ρ(2β)e−λ−β if λ ∈ I ∩ (Λ \ N/2), (3.2)

for any β ∈ N , γ ∈ Λ.

Proof. Let (γ1, . . . , γ24) be an ordered basis of Λ. Consider the Z-bilinear form ε0

Λ× Λ → Z2 defined by

ε0(γi, γj) =

〈γi, γj〉 if i > j,〈γi, γi〉/2 if i = j,

0 if i < j,

for i, j ∈ Ω. It is easy to see that ε0 satisfies

ε0(β, β) = 〈β, β〉/2, (3.3)

ε0(β, γ)− ε0(γ, β) = 〈β, γ〉, (3.4)

for β, γ ∈ Λ. In other words, ε0 is a 2-cocycle corresponding to (2.1).

Let s be a set-theoretic section Λ → Λ corresponding to ε0. Then s(β)s(γ) = s(β +

γ)(−1)ε0(β,γ) for β, γ ∈ Λ. By (3.3), we have θΛ(s(β)) = s(−β)(−1)ε0(β,β)(−1)〈β,β〉/2 =

s(−β) for β ∈ Λ. Since 〈N,N〉 ⊂ 2Z, there exists a 2-coboundary η: N → Z2 such that

ε0(β, γ) = η(β + γ)− η(β)− η(γ) (3.5)

for β, γ ∈ N .

Let I be a set of representatives of Λ/N satisfying (H1) and (H2) of Lemma 3.5. Set

ρ(β) = (−1)ρ0(β) for β ∈ 2Λ ∩N . Let us show that we can choose η so that it satisfies

η(2λ) = ρ0(2λ) (3.6)

9

for any λ ∈ I ∩ (Λ ∩N/2).Set Ω0 = 1, . . . , m. Let hi|i ∈ Ω0 be a subset of I satisfying (H1), namely,

∑m

i=1 dihi|di ∈ 0, 1 = I ∩ (Λ ∩ (N/2)). Note that the set hi +N |i ∈ Ω0 is a set of

generators of (Λ ∩ (N/2))/N , namely, 2hi + 2N is a set of generators of (2Λ ∩N)/2N .

Let δ: (2Λ ∩ N)/2N → Z2 be the linear map defined by δ(2hi + 2N) = η(2hi) + ρ0(2hi)

for i ∈ Ω0. Since η + ρ0 is linear on 2Λ ∩ N , we have δ(2h + 2N) = η(2h) + ρ0(2h) for

h ∈ I ∩ (Λ∩ (N/2)). Note that η+ ρ0 is not zero on 2N , and so δ depend on a choice of a

set satisfying (H1). By Proposition 2.4, (2Λ∩N)/2N ∼= Zm2 is a subgroup of N/2N ∼= Z

242 .

Thus we can extend the linear map δ to N/2N . Therefore there exists a linear map δ:

N → Z2 such that δ(2h) = η(2h) + ρ(2h) for h ∈ I ∩ (Λ∩ (N/2)). Hence we may assume

that η satisfies (3.6), since η + δ satisfies (3.5) and (3.6).

Since ε0 is Z-bilinear, we can extend η to Λ so that

ε0(β, γ) = η(β + γ)− η(β)− η(γ) (3.7)

for β ∈ N , γ ∈ Λ. In fact, for λ ∈ I, we choose a value of η(λ), and we define η(β + λ) =

ε0(β, λ) + η(β) + η(λ) for β ∈ N . It is easy to see that η satisfies (3.7).

Now, let us modify the section s. Set s0(β) = (−1)η(β)s(β) for β ∈ Λ. Then s0 satisfies

(3.1), and we have

θΛ(s0(λ+ β)) = s0(−λ− β)(−1)η(λ+β)+η(−λ−β) = s0(−λ− β)(−1)η(λ)+η(−λ)+〈β,β〉/2 ,

for λ ∈ I and β ∈ N . By (H2) of Lemma 3.5, we can choose a value of η(λ) satisfying

η(λ) + η(−λ) = 0, for λ ∈ I \ (I ∩ N/2). Then s0 satisfies (3.2) on Λ \ (Λ ∩ N/2). Note

that η(β) + η(−β) = η(2β) for β ∈ Λ ∩N/2. By (3.6), we have the results.

Let ι: C[Λ] → CΛ be the canonical homomorphism. For a subset M of Λ, let CMdenote the subspace of CΛ generated by ι(a)|a ∈M. Let s: Λ → Λ be a section and

let I be a set of representatives of Λ/N satisfying (3.1). Consider the linear isomorphism

CΛ ∼= C[Λ], ι(s(β)) 7→ eβ for β ∈ Λ. By (3.1), the restriction to CN is an algebra

isomorphism and the restriction to Cλ+N is an isomorphism of those algebra modules

with respect to the left multiplication.

Let (x1, . . . , x24) be the ordered basis of N consisting of the 2k-frame S. For a repre-

sentative λ + N of Λ/N , we set Vλ+N = S(h ⊗ t−1C[t−1]) ⊗ Cλ + N ⊂ VΛ, where we

regard λ+N as a subset of Λ. Note that VN is a subVOA of VΛ. Let Li be the rank one

sublattice of N generated by xi. It is well known that VN ∼=⊗24

i=1 VLi∼= V ⊗24

L . Let ρi:

Li → C be the group homomorphism such that ρi(2µ) = (−1)〈µ,µ〉/2 corresponding to VLiand set ρ = ⊕24

i=1ρi. For x ∈ 2Λ ∩N , we have ρ(2x) = 1, namely, ρ(x) ∈ 〈−1〉. By (3.2),

we obtain θVΛ = ⊗θVL on VN . Hence we have V +Λ ⊃ V +

N ⊃ (V +L )⊗24. It is obvious that

(V +L )⊗24 is a subVOA of V +

Λ . Thus we have obtained Theorem 3.2 (1).

10

Remark 3.7. Let V +L,R and V

Rbe the real forms of V +

L and V as constructed in [FLM].

By [FLM], those have the positive-definite symmetric bilinear forms. By Theorem 3.2 (1),

we have the embedding V +L,R ⊂ V

R. In this embedding, the positive-definite symmetric

bilinear form is invariant.

3.3 Decomposition of the untwisted space V +Λ

In this subsection, we give the decomposition of V +Λ and show Theorem 3.2 (2).

From [D1], it follows that VΛ is decomposed as

VΛ =⊕

λ+N∈Λ/N

Vλ+N (3.8)

as a VN -module. Let (x1, . . . , x24) be the ordered basis of N given in Section 3.2. Let

λ =∑24

i=1 cixi/2k be an element of I. It is well known that

Vλ+N ∼=24⊗

i=1

Vciα/2k+L =M(c) (3.9)

as a (V +L )⊗24-module, where c = (c1, . . . , c24) ∈ C = Λ/N . By (3.2), we have θVΛ = ⊗θVL

on VΛ.

Next, we consider the θVΛ-fixed-point subspace of VΛ. We obtain the following lemma.

Lemma 3.8. Let c be a codeword of C = Λ/N .

(1) If c ∈ C2 = (Λ ∩N/2)/N , then M(c) is θVΛ-invariant, and M(c)+ is the θVΛ-fixed-

point subspace.

(2) If c ∈ C \C2, then M(c)⊕M(−c) is θVΛ-invariant, and the θVΛ-fixed-point subspace

(M(c)⊕M(−c))+ ∼=M(c) as a (V +L )⊗24-module.

Proof. By Remark 3.1, we obtain (1). Suppose c ∈ C\C2. Note that θVΛ is an involution

and maps M(c) toM(−c). Then M(c)⊕M(−c) is θVΛ-invariant and (M(c)⊕M(−c))+ is

generated by x+ θVΛ(x) for x ∈M(c). Since θVΛ commutes with the action of (V +L )⊗24 on

M(c) ⊕M(−c), we get an isomorphism of (V +L )⊗24-modules (M(c) ⊕M(−c))+ → M(c)

by x+ θVΛ(x) 7→ x. Hence we get (2).

By Lemma 3.8, (3.8) and (3.9), we obtain Theorem 3.2 (2).

11

3.4 Decomposition of the twisted space V T,+Λ

In this subsection, we give the decomposition of V T,+Λ and show Theorem 3.2 (3).

Set K(N) = KΛ ∩ N and K(2N) = KΛ ∩ ˆ2N . Since T is a Λ/KΛ-module, T is

a NKΛ/KΛ-module. Note that −1 ∈ Λ acts on T as multiplication by −1. Let M

be a maximal abelian subgroup of Λ/KΛ such that M ⊃ NKΛ/KΛ. Note that |M | =213. By Proposition 2.1, T decomposes into the direct sum of all the irreducible M-

modules on which −KΛ acts by −1. Therefore T decomposes into the direct sum of

irreducible NKΛ/KΛ-modules on which −KΛ acts by −1. By Proposition 2.4 (3), we

have |NKΛ/KΛ| = 2|N/(N ∩ 2Λ)| = 225−m, and any multiplicities of the irreducible

NKΛ/KΛ-module is 2m−12. Since NKΛ/KΛ∼= N/K(N), T is a N/K(N)-module. Thus

we obtain

T ∼=⊕

ψ∈Irr(N/K(N))ψ(−K(N))=−1

2m−12Tψ, (3.10)

where Irr(N/K(N)) is the set of the characters for N/K(N) and Tψ is the one-dimensional

module C with a character ψ.

By using the canonical map π: N/K(2N) → N/K(N), we view T as a N/K(2N)-

module. Note that the kernel of π is K(N)/K(2N). Therefore, as a N/K(2N)-module,

T is decomposed

T ∼=⊕

χ∈Irr(N/K(2N))χ(−K(2N))=−1

mχTχ, (3.11)

where mχ is the multiplicity. By (3.10), we have mχ ∈ 0, 2m−12. By (3.11), V TΛ has a

decomposition,

V TΛ

∼=⊕

χ∈Irr(N/K(2N))χ(−K(2N))=−1

mχVTχN , (3.12)

where VTχN is the θVΛ-twisted VN -module corresponding to N/K(2N)-module Tχ.

As Section 3.2, we fix the ordered basis of N as (x1, . . . , x24). Let Li = Zxi be a

sublattice of N and set K(Li) = KΛ ∩ Li. For χ ∈ Irr(N/K(2N)), we set the character

of Li/K(Li) χi(exi) = χ(exi). For a character χ = ⊗24

i=1χi such that χ(−K(2N)) =

−1, we set P (χ) = (c1, . . . , c24) ∈ Z242 , where χi(e

xi) = ρi(xi)(−1)ci . For exK(2N) =

(∏

edixi)K(2N) ∈ N/K(2N), we set Q(x) = (d1, . . . , d24) ∈ Z242 . Then we have χ(x) =

ρ(∑

dixi)(−1)∑

cidi = ρ(x)(−1)〈P (χ),Q(x)〉.

Now, let us determine the multiplicities mχ. Let χ be an element of Irr(N/K(2N))

such that χ(−K(2N)) = −1. Set n = 2m−12. Then mχ = n if and only if there exists χ ∈

12

Irr(N/K(N)) such that χ = χ π. If χ(x) = 1 for any x ∈ Kerπ = K(N)/K(2N), then

χ is regarded as a character of N/K(N) and mχ = n. Note that K(N) = θΛ(a)a−1|a ∈Λ ∩ N/2 = ρ(−β)eβ|β ∈ 2Λ ∩ N. Therefore mχ = n if and only if χ(x) = 1 for

any x ∈ K(N)/K(2N), namely, 〈P (χ), Q(x)〉 = 0 for any x ∈ K(N)/K(2N). Note that

K(N)/K(2N) ∼= (N ∩ 2Λ)/2N ∼= (Λ ∩ (N/2))/N = C2. Hence mχ = n if and only if

P (χ) ∈ C⊥2 . Then we have an isomorphism

V TΛ

∼= 2m−12⊕

χ∈Irr(N/K(2N))χ(−K(2N))=−1

P (χ)∈C⊥2

VTχN . (3.13)

Next, we fix a character χ and consider the space VTχN . If P (χ) = (c1, . . . , c24), then

Tχ = ⊗Tci. Thus, we get an isomorphism VTχN

∼= ⊗V TciL . Note that we have θV TΛ =

−(⊗θV TL ), because θV TΛ acts by −1 on Tχ and θV TL acts by 1 on Tci. Therefore, for χ ∈Irr(N/K(2N)) with χ(−K(2N)) = −1, we have an isomorphism of (V +

L )⊗24-modules

VTχ,+N

∼=MT (P (χ))−. (3.14)

By (3.13) and (3.14), we get Theorem 3.2 (3).

4 Character and automorphisms of the moonshine

module

In this section, we give the character of V and give some automorphisms of V .

4.1 Character of the moonshine module

In order to give the character of V , we consider the symmetrized weight enumerator of

a code over Z2k. The symmetrized weight enumerator of a code C over Z2k is defined as:

sweC(p0, . . . , pk) =∑

c∈C

pn0(c)0 p

n1(c)1 . . . p

nk(c)k ,

where ni(c) denotes the number of j such that cj = ±i. Note that the symmetrized weight

enumerators of equivalence codes are same. For 0 ≤ i ≤ k, we set

ai = chViα/2k+L =1

φ(q)

∑

j∈Z

qk(j+i/2k)2

,

b = (chV +L − chV −

L ) =1

φ(q)

∑

j∈Z

(−1)jqj2

,

where φ(q) =∏

n≥1(1− qn).

By Theorem 3.2, we obtain the following corollary.

13

Corollary 4.1. Let C be an extremal Type II code of length 24 over Z2k. Then we have

ch(V ) = q(J(q)− 744) =1

2sweC(a0, a1, . . . , ak) +

1

2b24 + 211q3/2

( φ(q)24

φ(q1/2)24− φ(q2)24φ(q1/2)24

φ(q)48)

.

Remark 4.2. We have sweC(a0, a1, . . . , ak) = ΘΛ(q)/(φ(q))24, where ΘΛ(q) is the theta

function associated with Λ. Since b = φ(q)/φ(q2), we obtain

ch(V ) =1

2

(ΘΛ(q)

φ(q)24+

φ(q)24

φ(q2)24)

+ 211q3/2( φ(q)24

φ(q1/2)24− φ(q2)24φ(q1/2)24

φ(q)48)

.

This equation is given in Remark 10.5.8 of [FLM].

4.2 Automorphisms of the moonshine module

By the fusion rules of V +L determined in [Ab], we have the following proposition (cf. [Ma]).

Proposition 4.3. (1) If k is odd then the linear map τ defined by setting

1 on V ±L , Viα/k+L|1 ≤ i ≤ (k − 1)/2,

−1 on V ±α/2+L, V(2i−1)α/2k+L|1 ≤ i ≤ (k − 1)/2,

√−1 on V T0,±

L ,−√−1 on V T1,±

L

is an automorphism of the fusion algebra of V +L .

(2) If k is even then the linear map τ defined by setting

1 on V ±L , V

±α/2+L, Viα/2k+L|1 ≤ i ≤ (k − 1),

−1 on V T0,±L , V T1,±

L

is an automorphism of the fusion algebra of V +L .

Let Γ be the set of inequivalent irreducible V +L -modules. Suppose given a decomposi-

tion

V =⊕

(W1,...,W24)∈Γ24

mW1,...W24W1 ⊗ · · · ⊗W24

of the moonshine module as a (V +L )⊗24-module, where mW1,...,W24 is the multiplicity. Con-

sider the map µ : Γ → C× defined by

τ(W ) = µ(W )W

14

for W ∈ Γ. For each i ∈ Ω, we define a linear automorphism σi of V by

σi(x) = µ(Wi)x

for x ∈⊗24

j=1Wj . Then we have the following proposition (cf. [Mi1]).

Proposition 4.4. σi is a VOA automorphism of V .

Proof. By the definition, σi fixes any element of (V +L )⊗24. In particular, σi fixes the

Virasoro element and the vacuum vector. Note that the fusion rules of tensor products

of modules are the tensor products of the fusion rules of those modules. Let Γ24 be the

set of inequivalent irreducible modules for (V +L )⊗24. Let µi: Γ24 → C× be a map such

that µi(W ) = µ(Wi) for W = ⊗Wj ∈ Γ24. Let W1, W 2 be elements of Γ24. For v ∈ W 1,

w ∈ W 2, we have Y (v, z)w ∈ (W 1 ×W 2)[z, z−1], where Y (·, z) is the vertex operator.

Therefore we have Y (σi(v), z)σi(w) = µi(W1)µi(W

2)Y (v, z)w = µi(W1×W 2)Y (v, z)w =

σi(Y (v, z)w).

Remark 4.5. If k = 3, then σi is a 4A element of the Monster (cf.[Ma]) and the McKay-

Thompson series T4A is given by

T4A(q) = q−1∑

(W1,...,W24)∈Γ24

mW1,...,W24 µ(Wi)

24∏

i=1

chWi.

Proposition 4.6. Suppose k is odd. In the decomposition of V given by Theorem 3.2, σi

is a 4A element of the Monster. More precisely, σi ∈ 21+24+ ⊂ 21+24

+ .Conway1 ⊂ Monster ,

where 21+24+ .Conway1 is a non-split extension of Conway1 (largest simple Conway group)

by the extra-special group 21+24+ .

Proof. By Theorem 3.2 and Proposition 4.3, σ2i acts by 1 on V +

Λ and acts by −1 on

V T,+Λ . By [FLM], the centralizer of σ2

i in the Monster simple group is 21+24+ .Conway1 .

Then we have σi ∈ 21+24+ .Conway1 . Since σi preserves the space Ceβ for any β ∈ Λ, we

have σi ∈ 21+24+ . It is well known that there are the four types of Monster elements 1A,

2A, 2B and 4A contained in 21+24+ . Therefore σi is a 4A element of the Monster.

Remark 4.7. Suppose k is even. In the decomposition of V given by Theorem 3.2, σi

is a 2B element of the Monster. More precisely, σi acts by 1 on VΛ and acts by −1 on

V TΛ , namely, σi = z given in (10.4.48) of [FLM].

5 Examples of a decomposition

In this section, we give examples of a 2k-frame for some k and the corresponding de-

composition of V . Accordingly, we give an expression of the McKay-Thompson series

T4A.

15

5.1 Examples of a 2k-frame

In this subsection, we recall the Golay code and the Leech lattice. Then we give examples

of a 2k-frame for some k.

Let G24 ⊂ P(Ω) (the power set of Ω) be the binary Golay code. Then G24 is a vector

space over F2 of dimension 12. Let αi|i ∈ Ω be a basis of R24 such that 〈αi, αj〉 = 2δi,j,

where 〈·, ·〉 is the symmetric bilinear form on R24. For B ⊂ Ω, we set αB =∑

i∈B αi.

Proposition 5.1. ([CS]) The Leech lattice Λ can be realized as follows:

Λ =∑

B∈G24

Z1

2αB +

∑

i∈Ω

Z(1

4αΩ − αi).

Proof. Since G24 is a Type II code, the lattice Λ1 =∑

B∈G24Z(1/2)αB +

∑

i∈Ω Zαi is

even unimodular. Λ is unimodular if and only if Λ1 is, since the lattice Λ0 = Λ1 ∩ Λ =∑

B∈G24Z(1/2)αB +

∑

k,l∈Ω Z(αk +αl) has index 2 in both Λ1 and Λ. One can check that

Λ is even and has no roots.

Let p be a prime such that p = 3 (mod 4). In order to construct a 2k-frame, let us

consider the quadratic residues in Fp. Let Q be the set of quadratic residues in Fp. Note

that |Q| = (p − 1)/2 For i ∈ Fp, we set the subset Qi = i + j ∈ Fp|j ∈ Q = x ∈Fp|x− i ∈ Q ⊂ Fp.

Proposition 5.2. (1) |Qi ∩Qj | = (p− 3)/4, for any i, j ∈ Fp with i 6= j.

(2) For any i, j ∈ Fp such that i 6= j, we have exactly one of the following:

(I) i ∈ Qj, (II) j ∈ Qi.

Proof. Since |F×p | is not divisible by 4, Q does not contain β = −1. We view Q as a

subgroup of F×p , and let γ be a generator of Q. Note that β, γ are generators of F×

p .

By the definition, we have Qi ∩ Qj = x ∈ Fp|x − j, x − i ∈ Q and Q ∩ Qi−j = x ∈Fp|x, x− (i − j) ∈ Q for i, j ∈ Fp. Therefore we have |Qi ∩ Qj| = |Qi−j ∩ Q|. We have

to show that |Qi ∩Q| = |Qj ∩ Q| for i, j ∈ F×p . Suppose j = βsγti. Let x be an element

of Qi ∩ Q. If s = 0, then γtx, γtx − j ∈ Q and if s = 1, then γtx, γtx + j ∈ Q, where

s ∈ 0, 1 such that s = s (mod 2). Then we have the map ϕi,j: Qi ∩ Q → Qj ∩ Q,

x 7→ γtx + δs,1j. Note that ϕi,j is injective. Let us show that ϕj,lϕi,j = ϕi,l. Suppose

j = βs1γt1i and l = βs2γt2j. Then l = βs1+s2γt1+t2i. Let x be an element of Qi ∩Q. Thenϕi,jϕj,l(x) = γt1+t2x+ δs1,1γ

t2j + δs2,1l = γt1+t2x+ (δs1,1βs2 + δs2,1)l. It is easy to see that

δs1,1βs2 + δs2,1 = δs1+s2,1.

16

On the other hand, ϕi,l(x) = γt1+t2x + δs1+s2,1l. Therefore we have ϕj,lϕi,j = ϕi,l. In

particular, we have ϕj,iϕi,j = ϕi,i, namely, ϕi,j is bijective. Hence, we have |Qi ∩ Q| =|Qj ∩ Q|. Let us show that |Qi ∩ Q| = (p − 3)/4. Set λ = |Qi ∩ Q|. For i ∈ F×

p ,

we consider the injective map ψi: Qi ∩ Q → Q × Q, x 7→ (x, x − i). Since a − b = i for

(a, b) ∈ ψi(Qi∩Q), we have ψ(Qi∩Q)∩ψ(Qj∩Q) = φ. Thus we obtain⋃

i∈F×pψi(Qi∩Q) =

(a, b) ∈ Q × Q|a 6= b. Therefore we have λ(p − 1) = (p − 1)/2 × (p − 3)/2, namely,

λ = (p − 3)/4. Hence we obtain (1). By the definition, i ∈ Qj if and only if i − j ∈ Q.

Similarly, j ∈ Qi if and only if j − i ∈ Q. Q contains only one of i − j, j − i, since−1 /∈ Q. Thus we obtain (2).

First, we consider the case when p = 23. Consider the following generators of G24 (cf.

[CS]):

A1 = Ω,

Ai = 1 ∪ 2 + j|j ∈ Qi−2, (i = 2, · · · , 24),

where we regard 2 + j|j ∈ Qi−2 as a subset of Ω. For i, j ≥ 2, j ∈ Ai if and only if

j − 2 ∈ Qp,i−2. By Proposition 5.2 (1), |Qi ∩Qj| = 5 if i 6= j. Thus we have the following

remark.

Remark 5.3. (1) |Ai ∩ Aj| = 6 for i 6= j ≥ 2.

(2) For i, j ≥ 2, we have exactly one of the following:

(I) i ∈ Aj , (II) j ∈ Ai.

Proposition 5.4. For l ∈ Z, the set ±x1, . . . ,±x24 defined by

x1 =1

4αΩ + (2l − 1)α1,

xi = −1

2αAi +

1

4αΩ + (2l − 1)αi, (i = 2, · · · , 24)

is a 2(4l2 − 3l + 2)-frame.

Proof. Set m = 2(4l2 − 3l + 2). By Proposition 5.1, Λ contains ±xi. We have to

check that 〈xi, xj〉 = mδi,j . By direct calculation, we have 〈x1, x1〉 = m and 〈x1, xi〉 = 0

for i ∈ Ω\1. Since i /∈ Qi, we have i /∈ Ai. Thus, for i ∈ Ω\1, we get 〈xi, xi〉 = m. By

Remark 5.3, we obtain 〈αAi, αj〉+〈αAj , αi〉 = 2 and |Ai∩Aj | = 6 , namely, 〈αAi, αAj〉 = 12

for i, j ∈ Ω\1 with i 6= j. Therefore we have 〈xi, xj〉 = 0 for i, j ∈ Ω\1 with i 6= j.

Next, we consider the case when p = 11. Let D be an element of G24 with |D| = 12

and set D = d1, . . . , d12. By Proposition 5.2 (1), |Qi ∩ Qj| = 2 if i 6= j. We consider

17

the bijictive map F11 → D \ d1, i 7→ di+2, and we regard Qi as a subset of D. Then we

set the following subsets of D

D1 = D,

Di+2 = d1 ∪Qi. (i = 0, 1, . . . , 10)

Set Ω \D = d13, . . . , d24. Consider the bijective map F11 → D \ d13, i 7→ di+14, and

we regard Qi as a subset of Ω \D. Then we set the following subsets of Ω \D

D13 = Ω \D,Di+14 = d13 ∪Qi. (i = 0, 1, . . . , 10)

Note that |Di| = 6 for i ∈ Ω \ 1, 13. By direct calculation, we have the following

proposition.

Proposition 5.5. For l ∈ Z, the set ±x1, . . . ,±x24 defined by

x1 =1

2αD1 + 2lαd1 ,

xi =1

2αD1 − αDi + 2lαdi , (i = 2, 3, . . . , 12)

x13 =1

2αD13 + 2lαd13 ,

xi =1

2αD13 − αDi + 2lαdi, (i = 14, 15, . . . , 24)

is a 2(4l2 + 2l + 3)-frame.

Remark 5.6. As above case, we can consider the case when p = 7. Then we obtain a

4(2l2 + l + 1)-frame.

For example, we consider the case l = 1 in Proposition 5.4 and we get the 6-frameand the corresponding extremal Type II code C of length 24 over Z6. Then we obtain its

18

symmetrized weight enumerator:

sweC(p, q, r, s) = s24 + 48r24 + 36432q2r16s6 + 97152q3r12s9

+ 36432q4r8s12 + 510048q5r16s3 + 3497472q6r12s6 + 1603008q7r8s9

+ 36432q8r16 + 4048q9s15 + 6800640q9r12s3 + 8962272q10r8s6

+ 61824q12s12 + 123648q12r12 + 5100480q13r8s3 + 242880q15s9

+ 36432q16r8 + 198352q18s6 + 24288q21s3 + 48q24

+ 13248p1q1r11s11 + 971520p1q3r15s5 + 2914560p1q4r11s8 + 582912p1q5r7s11





+ 57076800p2q12r6s4 + 4080384p2q15r6s1 + 4048p3r9s12 + 24288p3r21


+ 97637760p3q6r9s6 + 11075328p3q7r5s9 + 5100480p3q8r13 + 183941120p3q9r9s3

+ 63391680p3q10r5s6 + 6800640p3q12r9 + 34974720p3q13r5s3 + 510048p3q16r5

+ 36432p4r12s8 + 36432p4q1r8s11 + 14427072p4q3r12s5 + 14281344p4q4r8s8


+ 1238688004q10r8s2 + 42989760p4q11r4s5 + 9472320p4q14r4s2 + 582912p5q1r11s7



+ 14427072p5q12r3s4 + 971520p5q15r3s1 + 198352p6r18 + 3934656p6q2r10s6


+ 8962272p6q8r10 + 97637760p6q9r6s3 + 3934656p6q10r2s6 + 3497472p6q12r6

+ 2185920p6q13r2s3 + 36432p6q16r2 + 24288p7r9s8 + 11075328p7q3r9s5


+ 24773760p7q10r5s2 + 582912p7q11r1s5 + 145728p7q14r1s2 + 759p8s16


+ 14281344p8q8r4s4 + 24288p8q9s7 + 2914560p8q11r4s1 + 36432p8q12s4

+ 242880p9r15 + 728640p9q2r7s6 + 11075328p9q5r7s3 + 1603008p9q6r3s6

+ 1603008p9q8r7 + 2963136p9q9r3s3 + 97152p9q12r3 + 1068672p10q3r6s5


+ 582912p11q7r5s1 + 36432p11q8r1s4 + 13248p11q11r1s1 + 2576p12s12

+ 61824p12r12 + 218592p12q5r4s3 + 36432p12q8r4 + 4048p12q9s3

+ 48576p13q6r3s2 + 4048p15r9 + 759p16s8 + p24.

Remark 5.7. This symmetrized weight enumerator is equal to that of the lifted Golay

codes G246 given in [BDHO].

19

5.2 McKay-Thompson series for a 4A element of the Monster

Let us explicitly compute the decomposition with respect to the 6-frame obtained in the

previous subsection.

We set

ai = chViα/2k+L =1

φ(q)

∑

j∈Z

q3(j+i/6)2

(0 ≤ i ≤ 3),

b = (chV +L − chV −

L ) =1

φ(q)

∑

j∈Z

(−1)jqj2

.

By Remark 4.5, the McKay-Thompson series T4A is given by

T4A(q) =1

2a240 +

1

2b24 − 1

2a243 + 24a242 − 24a241

+253

2a160 a83 −

253

2a80a

163 + 2024a150 a92 − 2024a91a

153

+ 4048a70a92a

83 − 4048a80a

91a

73 + 6072a40a

122 a83 − 6072a41a

82a

123

+ 6072a120 a81a42 − 6072a80a

121 a43 + 6072a21a

162 a63 − 6072a60a

161 a22

+ 6072a81a162 − 6072a161 a82 + 8096a130 a61a

32a

23 − 8096a20a

31a

62a

133

+ 12144a30a212 − 12144a211 a33 + 24288a20a

11a

142 a73 − 24288a70a

141 a12a

23

+ 30912a120 a122 − 30912a121 a123 + 36432a120 a51a42a

33 − 36432a30a

41a

52a

123

+ 42504a80a11a

82a

73 − 42504a70a

81a

12a

83 + 85008a51a

162 a33 − 85008a30a

161 a52

+ 97152a50a11a

112 a73 − 97152a10a

51a

72a

113 + 97152a110 a71a

52a

13 − 97152a70a

111 a12a

53

+ 99176a60a182 − 99176a181 a63 + 121440a110 a41a

52a

43 − 121440a40a

51a

42a

113

+ 121440a90a21a

72a

63 − 121440a60a

71a

22a

93 + 121440a90a

152 − 121440a151 a93

+ 161920a10a31a

152 a53 − 161920a50a

151 a32a

13 + 178112a100 a31a

62a

53 − 178112a50a

61a

32a

103

+ 267168a90a81a

72 − 267168a71a

82a

93 + 364320a30a

21a

132 a63 − 364320a60a

131 a22a

33

+ 655776a100 a61a62a

23 − 655776a20a

61a

62a

103 + 655776a60a

21a

102 a63 − 655776a60a

101 a22a

63

+ 680064a10a61a

152 a23 − 680064a20a

151 a62a

13 + 850080a30a

81a

132 − 850080a131 a82a

33

+ 1457280a20a71a

142 a13 − 1457280a10a

141 a72a

23 + 1493712a60a

81a

102 − 1493712a101 a82a

63

+ 1578720a20a41a

142 a43 − 1578720a40a

141 a42a

23 + 1845888a90a

51a

72a

33 − 1845888a30a

71a

52a

93

+ 1845888a70a31a

92a

53 − 1845888a50a

91a

32a

73 + 2404512a40a

31a

122 a53 − 2404512a50a

121 a32a

43

+ 2428800a80a71a

82a

13 − 2428800a10a

81a

72a

83 + 2610960a80a

41a

82a

43 − 2610960a40a

81a

42a

83

+ 5829120a30a51a

132 a33 − 5829120a30a

131 a52a

33 + 6509184a50a

71a

112 a13 − 6509184a10a

111 a72a

53

+ 7164960a50a41a

112 a43 − 7164960a40a

111 a42a

53 + 7504992a70a

61a

92a

23 − 7504992a20a

91a

62a

73

+ 9512800a40a61a

122 a23 − 9512800a20a

121 a62a

43 + 10565280a60a

51a

102 a33 − 10565280a30a

101 a52a

63.

20

References

[Ab] T. Abe, Fusion Rules for the Charge Conjugation Orbifold, Preprint.

[BDHO] E. Bannai, S.T. Dougherty, M. Harada, and M. Oura, Type II Codes, Even

Unimodular Lattices, and Invariant Rings, IEEE Trans. Inform. Theory, 45,

(1999), 1194-1205.

[Bo] R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc.

Nat’l. Acad. Sci. U.S.A, 83, (1986), 3068-3071.

[Ch] R. Chapman, Double circulant constructions of the Leech lattice, J. Aust. Math.

Soc. Ser. A, 69, (2000), 287-297.

[CS] J.H. Conway, N.J.A. Sloane, Sphere Packing, Lattices and Groups, 3rd Edition,

Springer, New York, 1999.

[D1] C. Dong, Vertex algebras associated with even lattices, J. Algebra, 161, (1993),

245-265.

[D2] C. Dong, Twisted modules for vertex algebras associated with even lattice, J.

Algebra, 165, (1994), 91-112.

[DGH] C. Dong, R. L. Griess Jr., and G. Hohn, Framed vertex operator algebras, codes

and moonshine module, Comm. Math. Phys., 193, (1998), 407-448.

[DHS] S. T. Dougherty, M. Harada and P. Sole, Self-dual codes over rings and the

Chinese remainder theorem. J. Math. Hokkaido Univ. 28 (1999), 253-283.

[DN] C. Dong and K. Nagatomo, Representations of Vertex operator algebra V +L for

rank one lattice L, Comm. Math. Phys., 202, (1999), 169-195.

[FLM] I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the

Monster, Pure and Appl. Math., Vol.134, Academic Press, Boston, (1988).

[GH] T. A. Gulliver, M. Harada, Orthogonal Frames in the Leech Lattice and a Type

II code over Z22, J. Combin. Theory. Ser. A, (to appear).

[Ma] A. Matsuo, Norton’s Trace Formulae for the Griess Algebra of a Vertex Operator

Algebra with Larger Symmetry. Preprint.

[Mi1] M. Miyamoto, Griess Algebras and Conformal Vectors in Vertex Operator Al-

gebras, J. Algebra, 179, (1996), 523-548.

21

[Mi2] M. Miyamoto, A new construction of the Moonshine vertex operator algebra

over real number field. Preprint.

[MN] A. Matsuo and K. Nagatomo, Axioms for a Vertex Algebra and the Locality of

Quantum Fields, MSJ-Memoirs 4, Mathematical Society of Japan, (1999).

22

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