Virasoro frames and their stabilizers for the E_{8} lattice type vertex operator algebra

Robert L. Griess, Jr.^{∗} and Gerald Höhn^{†}

3 January, 2001

Abstract

The concept of a framed vertex operator algebra (FVOA) is new (cf. [DGH]). This article contributes to this theory with a full analysis of all Virasoro frame stabilizers in V, the important example of the E_{8} level 1 affine Kac-Moody VOA, which is isomorphic to the lattice VOA for the root lattice of E_{8}(ℂ). We analyze the frame stabilizers, both as abstract groups and as subgroups of Aut(V) ≅ E_{8}(ℂ). Each frame stabilizer is a finite group, contained in the normalizer of a 2B-pure elementary abelian 2-group in Aut(V), but is not usually a maximal finite subgroup of this normalizer. In particular, we prove that there are exactly five orbits for the action of Aut(V) on the set of Virasoro frames, thus settling an open question about V in Section 5 of [DGH]. The results about the group structure of the frame stabilizers can be stated purely in terms of modular braided tensor categories, so this article contributes also to this theory.

There are two main viewpoints in our analysis. The first is the theory of codes, lattices, markings and the resulting groups of automorphisms. The second is the theory of finite subgroups of Lie groups. We expect our methods to be applicable to the study of other FVOAs and their frame stabilizers. Appendices present aspects of the theory of automorphism groups of VOAs. In particular, there is a general result of independent interest, on embedding lattices into unimodular lattices so as to respect automorphism groups and definiteness.

∗Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109 USA.

E-mail: rlgmath.lsa.umich.edu.

†Mathematisches Institut, Universität Freiburg, Eckerstraße 1, 79104 Germany. E-mail:

gerald@mathematik.uni-freiburg.de.

The first author acknowledges financial support from the University of Michigan Department of Mathematics and NSA grant USDOD-MDA904-00-1-0011.

1991 Mathematics Subject Classification. Primary 17B69. Secondary 22E40, 20B25

Contents

Abstract 1

Notation and terminology 3

1 Introduction 4

2 Stabilizers for framed lattice VOAs 7 2.1 General integral even lattices . . . . . . . . . . . . . . . . . . . . 8 2.2 Lattices from marked binary codes . . . . . . . . . . . . . . . . . 14

3 General results about Virasoro frames in V_{E_{8}} 15

4 The five classes of frames in V_{E_{8}} 21 4.1 The case k = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 The case k = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3 The case k = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.4 The case k = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.5 The case k = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Appendix 30 5.1 Equivariant unimodularizations of even lattices . . . . . . . . . . 30 5.2 Lifting Aut(L) to the automorphism group of V_{L} . . . . . . . . . 33 5.3 Nonsplit Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.4 Orbits of parabolic subgroups of orthogonal groups . . . . . . . . 38

Notation and terminology

Aut(V)  The automorphism group of the VOA V.
C = C(F)  The binary code determined by the T_{r}-module structure of V^{0}.
D = D(F)  The binary code of the I ⊆ {1, …, r} with V^{I} ≠ 0.
Δ ≅ L/M  A ℤ_{4}-code associated to a lattice L with fixed frame sublattice M.
D_{X}  The normal subgroup of W_{X} which stabilizes each subset {±x} of X.
E_{8}  The root lattice of E_{8}(ℂ).
E_{8}(ℂ)  The Lie group of type E_{8} over the field of complex numbers.
η(h) = exp(2πih_{0}) For h ∈ V_{1}, η(h) is an automorphism of the VOA V.
F = {ω_{1}, …, ω_{r}} A Virasoro frame.
FVOA  Abbreviation for framed vertex operator algebra.
G = G(F)  The subgroup of Aut(V) fixing the VF F of V.
G_{C} = G_{C}(F)  The normal subgroup of G(F) acting trivially on T_{r}.
G_{D} = G_{D}(F)  The normal subgroup of G(F) acting trivially on V^{0}.
H_{8}, H_{16},  The Hamming codes of length 8 resp. 16.
𝔥  A Cartan subalgebra in V_{L}
k  The dimension of D.
L, L^{∗}  An integral lattice of rank n, often self-dual and even, and its dual.
L_{C}  The even lattice constructed from a doubly-even code C.
M  A frame sublattice of L. This is a sublattice isomorphic to D_{1}^{n}
M(h_{1}, …, h_{r})  The irreducible T_{r}-module of highest weight (h_{1}, …, h_{r}) ∈ \{0, \frac{1}{2}, \frac{1}{16}\}^{r}.
N  The normalizer of T in Aut(V_{L}).
r  The number of elements in a VF.
t  The Miyamoto map F ⟶ G_{D}(F).
T  A toral subgroup of Aut(V_{L}) for integral even lattice L

with a frame sublattice M.

T_{r} = M(0)^{⊗r}  The tensor product of r simple Virasoro VOAs of rank \frac{1}{2}.
V  An arbitrary VOA, or the VOA V_{E_{8}}.
V_{L}  The VOA constructed from an even lattice L.
VF  Abbreviation for Virasoro frame.
VOA  Abbreviation for vertex operator algebra.
V_{𝔥}  The canonical irreducible module for the Heisenberg algebra

based on the finite dimensional vector space 𝔥.

V^{I}  The sum of irreducible T_{r}-submodules of V isomorphic to

M(h_{1}, …, h_{r}) with h_{i} = \frac{1}{16} if and only if i ∈ I.

V^{0} = V^{∅}  This is V^{I}, for I = ∅.
W_{X}  The stabilizer of a lattice frame X of a lattice L in Aut(L).
Y ( . , z)  A vertex operator.
x(−n)  Abbreviation for the element t^{−n} ⊗ x in V_{L}.
X  A lattice frame of L. These are the vectors of norm 4 in a frame sublattice

1 Introduction

In this article, we determine, up to automorphisms, the Virasoro frames and their stabilizers for V_{E_{8}}, the lattice type vertex operator algebra based on the E_{8}-lattice.

In [DGH], the basic theory of framed vertex operator algebras (FVOAs) was established. It included some general structure theory of frame stabilizers, the subgroup of the automorphism group fixing the frame setwise. It is a finite group with a normal 2-subgroup of class at most 2 and quotient group which embeds in the common automorphism group of a pair of binary codes. There was no procedure for computing the exact structure. To develop our understanding of FVOA theory, we decided to settle the frame stabilizers definitively for the familiar example V_{E_{8}}. This result, with further general theory for analyzing frame stabilizers in lattice type FVOAs, is presented in this article. Even simple questions such as whether the C-group (defined below) can be nonabelian or of exponent greater than 2 did not seem answerable with the techniques in [DGH] (the C-groups for V_{E_{8}} turn out to be nonabelian for four of the five orbits and elementary abelian for the last orbit). Furthermore, in V_{E_{8}}, we also show that there are just five orbits on frames, a point which was left unsettled in [DGH].

The study of FVOAs is a special case of the general extension problem of nice rational VOAs. The problem can be formulated completely in terms of the associated modular braided tensor category or 3d-TQFT (cf. [H] and the introduction of [DGH]). There has been recent progress in this direction [B, M99], proving also conjectures from [FSS], but a general theory for such extensions is unknown, even for FVOAs. The analysis of Virasoro frame stabilizers contributes to this problem by computing the automorphisms of such extensions. Furthermore, our classification result for the five VFs in V_{E_{8}} can be used to show the uniqueness of the unitary self-dual VOA of central charge 24 with Kac-Moody subVOA V_{A_{1,2}}^{⊗16} (cf. [DGH], Remark 5.4) since up to roots of unity the associated modular braided tensor category is equivalent to the one for the Virasoro subVOA L_{1/2}(0)^{⊗16} (cf. [MS]). This seems to be the first uniqueness result for one of the 71 unitary self-dual VOA candidates of central charge 24 given by Schellekens [Sc] which is not the lattice VOA of a Niemeier lattice.

Before stating our main results, we review some material about Virasoro frames from [DGH].

A subset F = {ω_{1}, …, ω_{r}} of a simple vertex operator algebra (VOA) V is called a Virasoro frame (VF) if the ω_{i} for i = 1, …, r generate mutually commuting simple Virasoro vertex operator algebras of central charge 1/2 and ω_{1} + ⋯ + ω_{r} is the Virasoro element of V. Such a VOA V is called a framed vertex operator algebra (FVOA).

We use the notation of [DGH] throughout. In particular, we shall use G for the stabilizer of the Virasoro frame F in the group Aut(V). There are two binary codes, C, D and we use k for the integer dim(D). There will be some obvious modifications of the [DGH] notation, such as D(F) to indicate dependence of the code D on the Virasoro frame F, G(F), G_{D}(F), G_{C}(F), etc. We call the group G_{D} the D-group of the frame and we call G_{C} the C-group of the frame (see [DGH], Def. 2.7).

Denote for an abelian group A with Â = Hom(A, ℂ^{×}) the dual group. Throughout this paper, we use standard group theoretic notation [Go, H]. For instance, if J is a group and S a subset, C(S) or C_{J}(S) denotes the centralizer of S in J, N(S) or N_{J}(S) denotes the normalizer of S in J and Z(J) denotes the center of J.

We summarize the basic properties of G.

Proposition 1.1 (i) G_{D} ≤ G_{C} and G_{D} and G_{C}/G_{D} are elementary abelian 2-groups;

(ii) G_{D} ≤ Z(G_{C});

(iii) G_{D} ≅ D̂ and G_{C}/G_{D} embeds in Ĉ;

(iv) G is finite, and the action of G on the frame embeds G/G_{C} in Sym_{r}.

Proof. [DGH], Th. 2.8. The assertion G_{D} ≤ Z(G_{C}) is easy to check from the definitions, but unfortunately was not made explicit in [DGH].

We have that G_{C}/G_{D} embeds in Ĉ, but general theory has not yet given a definitive description of the image.

We summarize our main results below. See Section 2 for certain definitions. Note that Main Theorem I (ii) just refers to the text for methods.

Main Theorem I

(i) In the case of a lattice type VOA based on a lattice L and a VF which is associated to a lattice frame, X, we have a description of G ∩ N, where N is the normalizer of a natural torus T (see Th. 2.8). It is an extension of the form (G ∩ T).W_{X}, where W is the automorphism group of the lattice and W_{X} is the stabilizer in W of X. Let D_{X} be the subgroup of W_{X} which stabilizes each set {±x}, for x ∈ X. Let n = rank(L) and suppose that L is obtained from the sublattice spanned by X by adjoining “glue vectors” forming the ℤ_{4}-code Δ ≅ 2^{ℓ} × 4^{k}. We have G_{C} ≤ N, G_{C} ∩ T ≅ 2^{ℓ} × 4^{k}, G_{C}/(G_{C} ∩ T) ≅ D_{X} and G ∩ T ≅ 2^{n−ℓ−k} × 4^{ℓ} × 8^{k}.

(ii) Assume that in the situation (i) the lattice comes from a marking of a binary code. Then a triality automorphism σ is defined (cf. [DGH], after Theorem 4.10) and one has G ≥ ⟨G ∩ N, σ⟩ > G ∩ N. In particular the group of permutations induced on the VF by ⟨G ∩ N, σ⟩ strictly contains the group induced by G ∩ N. We give conditions for identifying these permutation groups. In the case of V_{E_{8}}, the cases dim(D) = 1, 2 and 3 come from a marking and we prove that ⟨G ∩ N, σ⟩ = G.

Main Theorem II Let V be the lattice VOA based on the E_{8}-lattice.

(i) There are exactly five orbits for the action of Aut(V) ≅ E_{8}(ℂ) on the set of VFs in V.

(ii) These five orbits are distinguished by the parameter k, the dimension of the code D, and in these respective cases G = G(F), the stabilizer of the Virasoro frame F, has the following structure:

k  G
1  2^{1+14}Sym_{16}
2  2^{2+12}[Sym_{8} ≀ 2]
3  [2^{3+9} ≅ 2^{4+8}]2^{8}[Sym_{3} ≀ Sym_{4}] ≅
[2^{3+9} ≅ 2^{4+8}][Sym_{4} ≀ Sym_{4}] ≅
2^{4+16}[Sym_{3} ≀ Sym_{4}]
4  2^{4+5}[2 ≀ AGL(3, 2)] ≅ [2^{4} × 8^{4}]2.AGL(3, 2)
5  2^{5}AGL(4, 2) ≅ 4^{4}[2·GL(4, 2)]

(iii) In these five cases, the frame stabilizers G are determined up to conjugacy as subgroups of E_{8}(ℂ) by group theoretic conditions. Sets of conditions which determine them are found in Section 4 and listed below for each k.

k = 1: G is the normalizer of the unique up to conjugacy subgroup isomorphic to 2_{+}^{1+14}; equivalently, the unique up to conjugacy subgroup isomorphic to 2_{+}^{1+14}Sym_{16}.

k = 2: G satisfies the hypotheses of this conjugacy result:

In E_{8}(ℂ), there is one conjugacy class of subgroups which are a semidirect product X⟨t⟩, where t has order 2, X = X_{1}X_{2} is a central product of groups of the form [2 × 2^{1+6}]Sym_{8} such that X_{1} ∩ X_{2} = Z(X_{1}) = Z(X_{2}) and conjugation by t interchanges X_{1} and X_{2}.

k = 3: G is a subgroup of E_{8}(ℂ) characterized up to conjugacy as a subgroup X satisfying the following conditions:

(a) X has the form [2^{3+9} = 2^{4+8}]Sym_{4}^{4}.Sym_{4} ≅ [2^{4+16}]Sym_{3} ≀ Sym_{4};

(b) X has a normal subgroup E ≅ 2^{3} which is 2B-pure. k = 4: G has the form [2^{4} × 8^{4}]2.AGL(3, 2) ≅ 2^{4+5+8}AGL(3, 2) and is

characterized up to conjugacy by this property: it is contained in a subgroup G_{1} of the form [2^{4} × 8^{4}][2·GL(4, 2)], which is uniquely determined in E_{8}(ℂ) up to conjugacy in the normalizer of a GL(4, 2)- signalizer (defined in Section 4.4); in particular, G is determined uniquely up to conjugacy in G_{1} as the stabilizer of a subgroup isomorphic to 2^{3} in the GL(4, 2)-signalizer.

k = 5: G is conjugate to a subgroup of the Alekseevski group (see [A, G76] and Prop. 3.5) of the form 2^{5}AGL(4, 2) ≅ 4^{4}[2·GL(4, 2)] and the set of all such subgroups of the Alekseevski group form a conjugacy class in the Alekseevski group.

Remark 1.2 We stress that there are two main viewpoints to the analysis in this article. One is group structures coming from binary codes and lattices via markings and frames as in [DGH]; and the other is the theory of finite subgroups of E_{8}(ℂ) (for a recent survey, see [GR99]).

Appendix 5.1 contains a proof that, given an even lattice L of signature (p, q), there is an integer m ≤ 8 so that L embeds as a direct summand of a unimodular even lattice M of signature (mp, mq) and so that Aut(M) contains a subgroup which stabilizes L and acts faithfully on L as Aut(L). This is similar in spirit to results of James and Nikulin [J, N] (which display such embeddings into indefinite lattices) and gives a useful containment of VOAs V_{L} ≤ V_{M}.

Appendix 5.2 is a construction of a group extension W̃ of the automorphism group W of a lattice L by an elementary abelian 2-group. This extension plays a natural role in the automorphism group of V_{L}. While this construction is not new, it it useful to make things explicit for certain proofs in this article. Also, there are some historical remarks.

Appendix 5.3 discusses the group extension aspect of the frame stabilizers. At first, it looks like the groups G/G_{D} might split over G_{C}/G_{D}, but some do not.

Appendix 5.4 is a technical result about permutation representations for a classical group.

2 Stabilizers for framed lattice VOAs

In [DGH], we used the following concept.

Definition 2.1 A lattice frame in a rank n lattice L ≤ ℝ^{n} is a set, X, of 2n lattice vectors of squared length 4 in L such that two elements are equal, opposite or orthogonal. Every lattice frame spans a lattice M ≅ D_{1}^{n}, called the frame sublattice.

Clearly, in a given lattice, there is a bijection between lattice frames and frame sublattices (the frame defining the frame sublattice is the set of minimal vectors in that sublattice). Note that in [DGH] the term lattice frame means sublattice.

In Chapter 3 of [DGH], we constructed, for every integral lattice containing a frame sublattice, a Virasoro frame for the associated rank n lattice VOA and determined the decomposition into modules for the Virasoro subVOA T_{2n} belonging to this Virasoro frame. In Section 2.1, we will determine the subgroup of the Virasoro frame stabilizer which is visible from this construction. As in [DGH], we will use the language of ℤ_{4}-codes. We also prove a result about the centralizer of G_{C} for some framed lattices. In Section 2.2, we look at lattices with frame sublattices constructed from marked binary codes as in Chapter 4 of [DGH]. Here, a triality automorphism is defined; see Theorem 2.19.

2.1 General integral even lattices

Let V_{L} be the lattice VOA, based on the integral even lattice L. For every frame sublattice M of L there is the associated VF F = {ω_{1}, …, ω_{2n}} inside V_{M} ⊂ V_{L} (cf. [DGH], Def. 3.2). If X is the lattice frame contained in M, the associated VF F is the set of all \frac{1}{16}x(−1)^{2} ± \frac{1}{4}(e^{x} + e^{−x}), x ∈ X. We use x(−n) for the element t^{−n} ⊗ x in V_{L}.

Using the notation of [DGH], we can describe some structure of the Cartan subalgebra 𝔥 = t^{−1} ⊗_{ℂ} (L ⊗_{ℤ} ℂ) ⊂ V_{L} associated to a frame sublattice of L:

Proposition 2.2 (Cartan subalgebra) Let M be a frame sublattice spanned by a lattice frame inside an integral even lattice L of rank n and let T_{2n} be the subVOA of V_{M} ≤ V_{L} generated by the associated Virasoro frame of the lattice VOA V_{L}. Then

(i) 𝔥 = (V_{M})_{1} is an abelian Lie algebra of rank n.

(ii) It is the n-dimensional highest weight space for the T_{2n}-submodule of V_{L} isomorphic to the direct sum

M(\frac{1}{2}, \frac{1}{2}, 0, …, 0) ⊕ M(0, 0, \frac{1}{2}, \frac{1}{2}, 0, …, 0) ⊕ ⋯ ⊕ M(0, …, 0, \frac{1}{2}, \frac{1}{2}).

The summands are spanned by vectors of the form x(−1), where x is in the lattice frame.

Proof. For the first statement, recall that as a graded vector space V_{M} = V_{𝔥} ⊗ ℂ[M]. Since the minimal nonzero squared length of a vector x in the lattice M ≅ D_{1}^{n} is 4, i.e. e^{x} ∈ ℂ[M] has conformal weight 2, the weight one part of V_{M} is just the the weight one part of the Heisenberg VOA V_{𝔥}, i.e., in the usual notation, 𝔥 = t^{−1} ⊗_{ℂ} (M ⊗_{ℤ} ℂ). It inherits a toral Lie algebra structure from the Lie algebra V_{1}.

For the second statement use Corollary 3.3. (1) of [DGH]. Since M(h_{1}, …, h_{2n}) has minimal conformal weight h_{1} + ⋯ + h_{2n} (this is the smallest i so that M(h_{1}, …, h_{2n}) has an L(0)-eigenvector for the eigenvalue i) and the weight one part of M(0, …, 0) is zero, the assertion follows.

Throughout this article, when we work with a VOA based on a lattice with lattice frame, we write 𝔥 for the above Cartan subalgebra (V_{M})_{1} of (V_{L})_{1}.

Corollary 2.3 In the situation where the VF comes from a lattice frame, G_{C} normalizes the Cartan subalgebra 𝔥 of (V_{L})_{1}.

Proof. We use the proof of Prop. 2.2 and its notation. For every (h_{1}, …, h_{2n}), the group G_{C} leaves the submodule associated to M(h_{1}, …, h_{2n}) in the Virasoro module decomposition invariant, so it normalizes the Lie algebra 𝔥 ≤ (V_{L})_{1}.

Definition 2.4 Elements of (V_{L})_{1} act as locally finite derivations under the 0^{th} binary composition on V_{L}. Such endomorphisms may be exponentiated to elements of Aut(V_{L}). For h ∈ 𝔥, we define η(h) := exp(2πih_{0}), so that η is a homomorphism from 𝔥 to Aut(V_{L}). Let T := η(𝔥) be the associated torus of automorphisms. The scale factor 2πi gives us the exact sequence

0 ⟶ L^{∗} ⟶ 𝔥 ⟶^{η} T ⟶ 1.

Let N := N(T) be the normalizer of the torus T in Aut(V_{L}) and denote by W̃ the lift of W := Aut(L) to a subgroup of Aut(V_{L}), as described in Appendix 5.2. Finally, we need the subgroup K := ⟨exp(2πix_{0}) | x ∈ (V_{L})_{1}⟩.

Proposition 2.5 For any lattice VOA we have

(i) N = T W̃ and N/T ≅ W;

(ii) Aut(V_{L}) = KN and K is a normal subgroup.

Proof. Part (i) follows from [DN] and the construction of W̃ given in Appendix 5.2; part (ii) is due to [DN].

Definition 2.6 For a subset X of L, let W_{X} be its setwise stabilizer. We can identify X as a subset X of V_{L} via the embedding L ⊂ 𝔥. Let W̃_{X} indicate the setwise stabilizer of X in W̃.

When X is a lattice frame, D_{X} denotes the subgroup of W_{X} which stabilizes each subset {±x} of X (so D_{X} acts “diagonally” with respect to the double basis X). Always, −1 ∈ D_{X}. Let D̃_{X} be the preimage in W̃.

Given a lattice frame X ⊂ L we will describe the intersection G ∩ N of the frame group G for the associated VF F with N. By using Prop. 2.2 we will show how to get G_{C} in the course of studying G ∩ T and G ∩ N.

Definition 2.7 (The code Δ and integers k, ℓ) Recall n = rank(L). Let X be the lattice frame and M the associated sublattice. We observe that M ≤ L ≤ L^{∗} ≤ M^{∗} = \frac{1}{4}M and M determines a ℤ_{4}-code Δ ≤ ℤ_{4}^{n} which corresponds to L/M ≤ M^{∗}/M by the identification M^{∗}/M ≅ ℤ_{4}^{n} extending some \{±1\}-equivariant bijection of \frac{1}{4}X with the set of vectors (0, …, 0, ±1, 0, …, 0) (cf. [DGH], p. 462).

There are integers ℓ and k such that the code Δ is, as an abelian group, isomorphic to 2^{ℓ} × 4^{k}.

By Th. 4.7 of [DGH], one has k = dim(D). We have ℓ + 2k ≤ n since L is integral and ℓ + 2k = n if and only if L is self-dual. Note that since L contains a frame sublattice, its determinant must be an even power of 2. In terms of the ℤ_{4}-code Δ we get for its automorphism group

Aut(Δ) ≅ W_{X} ≤ Mon(n, ℤ_{4}) ≅ 2^{n}:Sym_{n}

and D_{X} is a normal subgroup of sign changes in W_{X}.

Theorem 2.8 (The intersection G ∩ N) For the frame stabilizer G and the normal subgroups G_{C} and G_{D} we have:

(i) G_{D} ≤ T, G_{D} = η(\frac{1}{2}M + L^{∗}) ≅ (\frac{1}{2}M + L^{∗})/L^{∗} ≅ 2^{k};

(ii) G_{C} ≤ N, G_{C} ∩ T ≅ Δ ≅ 2^{ℓ} × 4^{k}, G_{C}/(G_{C} ∩ T) ≅ D_{X};

(iii) G∩T ≅ 2^{n−ℓ−k}×4^{ℓ}×8^{k}, G∩N = (G∩T)W̃_{X}, (G∩N)/G_{C} ≅ 2 ≀ (W_{X}/D_{X}), where the wreath product is taken with respect to the action of W_{X}/D_{X} on the n-set of pairs {±x}, x ∈ X.

Proof. First, we describe G ∩ T. The group T acts on the VF, consisting of elements of the form \frac{1}{16}x(−1)^{2} ± \frac{1}{4}(e^{x} + e^{−x}), x ∈ X. A transformation η(h) ∈ η(𝔥) = T will fix x(−1)^{2} and send e^{x} + e^{−x} to a e^{x} + a^{−1}e^{−x}, with a = e^{2πi(h,x)}.

Using [DGH], Th. 4.7, we see that η(h) ∈ T is in G_{D} if (h, L ∩ \frac{1}{2}M) ≤ ℤ, i.e., if h ∈ 2M^{∗} + L^{∗} = \frac{1}{2}M + L^{∗}. Since η(\frac{1}{2}M + L^{∗}) ≅ (\frac{1}{2}M + L^{∗})/L^{∗} ≅ 2^{k} has the same order as D̂ ≅ G_{D}, part (i) is proven.

For an element η(h) of T to centralize the frame, the requirement is all of the a above equal 1, which is equivalent to (h, M) ≤ ℤ. This defines the set M^{∗} = \frac{1}{4}M, and its image is η(M^{∗}) ≅ (M^{∗} + L^{∗})/L^{∗} = M^{∗}/L^{∗} ≅ L/M ≅ Δ ≅ 2^{ℓ} × 4^{k}.

For η(h) ∈ T to stabilize the frame, the requirement is that all a ∈ {±1}, which is equivalent to (h, M) ≤ \frac{1}{2}ℤ, i.e., h ∈ \frac{1}{2}M^{∗} = \frac{1}{8}M. The image of \frac{1}{8}M under η is isomorphic to (\frac{1}{8}M + L^{∗})/L^{∗} ≅ 2^{n−ℓ−k} × 4^{ℓ} × 8^{k}. The first part of (iii) follows.

For the rest of (ii), one has G_{C} ≤ N by Corollary 2.3. Recall notations from Def. 2.6. It is clear that G_{C} ≤ (G∩T)D̃_{X}. The action of G∩T on the VF implies that G_{C} meets every coset of G∩T in (G∩T)D̃_{X}, i.e., (G∩T)D̃_{X} = (G∩T)G_{C}, whence G_{C}/(G_{C} ∩ T) ≅ D_{X}.

The last two statements of (iii) follow from (ii), the structure of G ∩ T and Proposition 5.10 (ii). In more detail, G∩N acts on X and on F. In fact, there is the G ∩ N-equivariant map F ⟶ X/\{±1\} by \frac{1}{16}x(−1)^{2} ± \frac{1}{4}(e^{x} + e^{−x}) ↦ \{±x\}, for x ∈ X. Now, on X/{±1}, G ∩ N induces W_{X}/D_{X}. The kernel modulo G_{C} is at most a group of order 2^{n}, where n = |X/{±1}|. On the other hand, G ∩ T induces a group of this order on X and it is in the kernel of this action. Consider the partition P of X into the n pairs {x, −x}, x ∈ X. The stabilizer of P in Sym_{X} is a wreath product 2 ≀ Sym_{n}, where the normal 2^{n} subgroup is the kernel of the action on the n-set P. By the Dedekind lemma, any subgroup H of Stab(P) which contains this normal subgroup is a semidirect product, and this remark applies to the action of our group G ∩ N on X.

From the above proof, we get these observations.

Corollary 2.9 The exponent of the group G ∩ T is 8 if k ≥ 1. The order of G_{C} is 2^{ℓ+2k+e}, where 2^{e} = |D_{X}|. Since −1 ∈ D_{X}, e ≥ 1. See Tables 2 and 3 in Section 3 for certain values of k and e when V_{L} = V_{E_{8}}.

In fact, one can consider G_{C} as a group extension in several ways. The above discussion shows that G_{C} is an extension of the abelian group G_{C}∩T and a group isomorphic to D_{X}. On the other hand, G_{C} has the central elementary abelian subgroup G_{D} ≅ 2^{k} giving elementary abelian quotient 2^{ℓ+k+e} (cf. Prop. 1.1).

We next describe the centralizer of G_{C}; to this end, we need some background from the theory of Chevalley groups.

Recall that, by definition, a Chevalley group over a field is contained in the automorphism group of a Lie algebra and has trivial center. The associated Steinberg group is a central extension of the Chevalley group and has the property that the preimage of a Cartan subgroup is abelian. See [Ca].

In the next result, we formally extend the “normal form” (or “canonical form”) concept for Steinberg groups to a wider class of groups which includes automorphism groups of lattice type VOAs. The notation of [Ca] is fairly standard: G, H, N, B, U stand for a Steinberg group, a Cartan subgroup, a subgroup of the normalizer of the Cartan subgroup, a Borel subgroup containing H, the normal subgroup of B generated by root groups. These groups participate in the so-called (B, N) pair and normal form structures for G.

In this article, the symbols G and N have already been assigned meanings, so we shall deviate from traditional Chevalley theory notations slightly for the next two results. We shall use notations R, H_{R}, N_{R}, B_{R}, U_{R} for a group R with subgroups H_{R}, N_{R}, B_{R}, U_{R} of R which will have properties like G, H, N, B, U. In an algebraic group R, H_{R} stands for a maximal torus. If R is a torus, H_{R} = N_{R} = R and U_{R} = 1. If R is an algebraic group and the connected component of the identity R^{0} is a torus, then H_{R} = R^{0}, N_{R} = R, U_{R} = 1. If R is a reductive algebraic group, H_{R} is a maximal torus and N_{R} is its normalizer. In Steinberg groups, the group N_{R} is not necessarily the full normalizer of H_{R} and H_{R} = 1 is possible (for small fields).

The automorphism groups of lattice type VOAs are reductive algebraic groups over the complex numbers, and so are an instance of the group R of Lemma 2.10 (iii) with R^{0} the connected component of the identity and N_{R} the normalizer of a maximal torus, usually denoted by the symbol T instead of H (these subgroups lie in R^{0}).

Lemma 2.10 (i) In a Steinberg group R over a field, every element may be written uniquely in the “normal form” uhn_{w}u′, where, in usual notation,

• h ∈ H_{R}, a Cartan subgroup;

• N_{R} is a subgroup containing H_{R} as a normal subgroup and for each w in N_{R}/H_{R}, there is a choice of preimage n_{w} in N_{R};

• u ∈ U_{R}, the subgroup generated by root groups for the set of positive roots;

• u′ ∈ U_{R,w}, the subgroup of U_{R} generated by root groups associated to positive roots r such that w(r) is negative.

(ii) We have a normal form as in (i) for direct products of Steinberg groups and tori, and even for central products of universal Steinberg groups and tori.

Suppose that R = R_{1} ⋯ R_{n} is a central product where R_{i} is a torus or a Steinberg group, all over a common field. To each index i is associated a sequence R_{i}, H_{R_{i}}, N_{R_{i}}, B_{R_{i}}, U_{R_{i}} for which we have a normal form as in (i). Then we have a normal form for R given by the sequence R, H_{R}, N_{R}, B_{R}, U_{R}, where R = R_{1} ⋯ R_{n} and H_{R} is the product of the H_{R_{i}} and similarly for N_{R}, B_{R}, U_{R}.

(iii) We have a normal form as in (i) for groups R which contain a normal subgroup R_{0} which is a direct product as in (ii) such that there exists a subgroup N_{R} of R satisfying N_{R}∩R_{0} = N_{R_{0}}. It suffices to take H_{R} = H_{R_{0}}, U_{R} = U_{R_{0}}, B_{R} = B_{R_{0}}.

Proof. For (i), see [Ca]. Parts (ii) and (iii) are formal.

Corollary 2.11 Suppose that R is a group as in Lemma 2.10 (iii). Let S be a subset of the Cartan subgroup H_{R}. Then C_{R}(S) = EC_{N_{R}}(S), where E is generated by all the root groups centralized by S.

Proof. Let g ∈ R and s ∈ S. Consider g = uhn_{w}u′ in normal form and study the conjugate ^{s}g := sgs^{−1} = ^{s}u^{s}h^{s}n_{w}^{s}u′ = ^{s}uh^{s}n_{w}^{s}u′. Observe that the Cartan subgroup normalizes each root group, hence also U_{R} and U_{R,w}, as in Lemma 2.10. If we write ^{s}n_{w} = h_{1}n_{w}, for an element h_{1} ∈ H, then we get the normal form ^{s}u(hh_{1})n_{w} · ^{s}u′ for sgs^{−1}. Therefore, g commutes with s if and only if u, u′ and n_{w} commute with s.

Note that the intersection hypothesis of Lemma 2.10 (iii) and R = R^{0}N_{R} is satisfied by complex reductive algebraic groups, in particular by Aut(V_{L}). (R^{0} is the connected component of the identity and N_{R} is the normalizer in R of a maximal torus.)

Corollary 2.12 Let S be a subset of T. Then C_{Aut(V_{L})}(S) = EC_{N}(S), where E is generated by all the root groups in K centralized by S.

Definition 2.13 In a torus T, let T_{(m)} = {u ∈ T | u^{m} = 1} ≅ ℤ_{m}^{rank(T)}, for an integer m > 0.

Corollary 2.14 (i) C_{T} (G_{C}) ≤ T_{(2)}.

(ii) G′_{C} ≠ 1.

(iii) C_{Aut(V_{L})}(G_{C}) ≤ C_{Aut(V_{L})}(G_{C} ∩ T) ≤ N.

(iv) T C_{Aut(V_{L})}(G_{C})/T corresponds to a subgroup of W_{X} under the identification of N/T with W.

Proof. Since G_{C} contains an element u corresponding to −1 ∈ D_{X}, we have C_{T} (G_{C}) ≤ C_{T} (u) = T_{(2)}, proving (i).

For the proof of (ii), note that G_{C} ∩ T has exponent 4.

For (iii), recall that G_{C} ∩ T = η(M^{∗}). Since η(M^{∗}) ≤ T, a maximal torus, we use Lemma 2.12 to deduce that the centralizer of η(M^{∗}) in Aut(V_{L}) is EC_{N}(η(M^{∗})), where E is generated by all the root groups with respect to T which are centralized by η(M^{∗}). There are no such root groups since (M^{∗})^{∗} = M, which contains no roots. So, E = 1.

For (iv), just observe that C_{Aut(V_{L})}(G_{C}) is a subgroup of N, and leaves invariant η(M^{∗}). So, in its action on 𝔥, it preserves M^{∗} and M, hence also X.

Theorem 2.15 Suppose that L^{∗} contains no nonzero elements of \frac{1}{4}X + \frac{1}{4}X (e.g., this holds if L^{∗} has no vectors of squared length \frac{1}{2} and 1). Then C_{Aut(V_{L})}(G_{C}) ≤ T_{(2)}.

Proof. Let C := C_{Aut(V_{L})}(G_{C}). Use Cor. 2.14 (iv) and suppose that c ∈ C corresponds to a nonidentity element of W_{X}. Then there is x ∈ X so that cx ≠ x. Since c is trivial on G_{C} ∩ T = η(M^{∗}) ≅ M^{∗}/L^{∗}, we get \frac{1}{4}x − \frac{1}{4}cx ∈ L^{∗} \ \{0\}, a contradiction. So, C ≤ T. From Cor. 2.14 (i), we get C = C_{T} (G_{C}) ≤ T_{(2)}.

Corollary 2.16 C_{Aut(V_{L})}(G_{C}) ≤ T_{(2)} if L is self-dual.

Proof. Since L = L^{∗}, every element of L^{∗} has even integer norm, so this is obvious.

Remark 2.17 (i) When the conclusion of Theorem 2.15 holds, the subgroup C_{Aut(V_{L})}(G_{C}) of T_{(2)} depends just on the action of D_{X} on T_{(2)} ≅ \frac{1}{2}L^{∗}/L^{∗}.

(ii) In the E_{8} lattice example, C_{E_{8}(ℂ)}(G_{C}) ≤ T_{(2)}, by Cor. 2.16. For k = 1, C_{E_{8}(ℂ)}(G_{C}) is contained in G_{C}; for k = 4, it is not contained in G_{C} ≅ 4^{4}:2 (in this case, C_{E_{8}(ℂ)}(G_{C}) = T_{(2)}). See Section 4.

2.2 Lattices from marked binary codes

In the more special situation where a VOA is constructed with the help of binary codes there exists a so-called triality automorphism σ. The triality automorphism was first defined in [G82] as an automorphism of the Griess algebra and in [FLM] it was extended to an automorphism of the Moonshine module V^{♮}. In [DGM, DGM2], it was shown that for any doubly even self-dual code C one can define σ for both of the VOAs V_{L_{C}} and ṼL̃_{C} (cf. [DGH], Sect. 4, for the notation).

For binary codes we introduced in [DGH] the notation of a marking.

Definition 2.18 Let n be even. A marking of a length n binary code C is a partition of the n coordinates into n/2 sets of size 2.

The binary code is a subspace of 𝔽_{2}^{n}, but it may be considered as a subset of ℂ^{n} by interpreting the coordinates 0 and 1 modulo 2 as the ordinary integers 0 and 1.

For a binary code there is the lattice

L_{C} = \{\frac{1}{\sqrt{2}}(c + x) | c ∈ C, x ∈ (2ℤ)^{d}\}.

The lattice L_{C} is integral and even if the code C is doubly-even. Every marking of the binary code C determines a frame sublattice inside L_{C} (cf. [DGH], p. 425).

Again, let F be the VF associated to this lattice frame.

Theorem 2.19 (cf. [DGH], p. 432) For the VF F inside the lattice VOA V_{L_{C}} constructed from a marked binary doubly-even self-dual code C, there is a triality automorphism σ inside G(F).

Its image σ̄ ∈ G/G_{C} ≤ Sym_{F} ≅ Sym_{2n} interchanges the Virasoro elements ω_{4i−2} and ω_{4i−1} for i = 1, …, n/2 and fixes the others.

Proof. The existence of σ was proven [FLM], Th. 11.2.1, and [DGM, DGM2].

The description is the following (see [FLM], (11.1.72), and [DGM]): Let A_{1} = \sqrt{2}ℤ be the root lattice of SL_{2}(ℂ). The group SL_{2}(ℂ) acts on the lattice VOA V_{A_{1}} and its module V_{A_{1}+1/\sqrt{2}}. On

(V_{A_{1}})_{1} ≅ ℂ x(−1) ⊕ ℂ e^{x} ⊕ ℂ e^{−x} ≅ 𝔰l_{2}(ℂ),

where x is a generator of the A_{1} root lattice, let μ̄ ∈ PSL_{2}(ℂ) ≅ SO_{3}(ℂ) be the linear map defined by

μ̄(x(−1)) = e^{x} +e^{−x}, μ̄(e^{x} +e^{−x}) = x(−1) and μ̄(e^{x} −e^{−x}) = −(e^{x} −e^{−x}).

Let μ be one of the two elements in SL_{2}(ℂ) which maps modulo the center to μ̄. On V_{L_{C}} ≅ ⨁_{c∈C} V_{A_{1}+c_{1}/\sqrt{2}} ⊗ ⋯ ⊗ V_{A_{1}+c_{n}/\sqrt{2}}, there is the tensor product action of the direct product of n copies of SL_{2}(ℂ). The triality automorphism is defined as the diagonal element σ = (μ, …, μ) ∈ SL_{2}(ℂ) × ⋯ × SL_{2}(ℂ) in it. For n/8 odd, the definition has to be adjusted by replacing the last component by κμ, where κ is the nontrivial central element of SL_{2}(ℂ).

Since the code C is even, the action of σ on V_{L_{C}} is independent of the choice for μ (cf. [FLM], Remark 11.2.3). One has σ^{2} = 1.

Now we describe how σ acts on the Virasoro frame F ⊂ V_{A_{1}^{n}} for the standard marking {(1, 2), (3, 4), …, (n−1, n)}: For i = 1, …, n/2, the Virasoro elements are

ω_{4i−3} = \frac{1}{16}(x_{2i−1}(−1) + x_{2i}(−1))^{2} + \frac{1}{4}(e^{x_{2i−1}+x_{2i}} + e^{−x_{2i−1}−x_{2i}}),
ω_{4i−2} = \frac{1}{16}(x_{2i−1}(−1) + x_{2i}(−1))^{2} − \frac{1}{4}(e^{x_{2i−1}+x_{2i}} + e^{−x_{2i−1}−x_{2i}}),
ω_{4i−1} = \frac{1}{16}(x_{2i−1}(−1) − x_{2i}(−1))^{2} + \frac{1}{4}(e^{x_{2i−1}−x_{2i}} + e^{−x_{2i−1}+x_{2i}}),
ω_{4i} = \frac{1}{16}(x_{2i−1}(−1) − x_{2i}(−1))^{2} − \frac{1}{4}(e^{x_{2i−1}−x_{2i}} + e^{−x_{2i−1}+x_{2i}}),

where x_{i} is the generator of the i-th component of the lattice A_{1}^{n}. The action of SL_{2}(ℂ)_{2i−1} × SL_{2}(ℂ)_{2i} on x_{2i−1}(−1)^{2} and x_{2i}(−1)^{2} is trivial and on the vector space spanned by the nine elements

x_{2i−1}(−1)x_{2i}(−1), x_{2i−1}(−1)e^{x_{2i}}, …, e^{−x_{2i−1}}e^{−x_{2i}} = e^{−x_{2i−1}−x_{2i}}

it is the tensor product action of the adjoint action of both factors. The remaining factors SL_{2}(ℂ)_{j} act trivially. It follows immediately by computation that σ fixes ω_{4i−3} and ω_{4i} and interchanges ω_{4i−2} with ω_{4i−1}.

Remark 2.20 A Virasoro frame coming from a lattice frame alone may not have a triality.

3 General results about Virasoro frames in V_{E_{8}}

From now on, let V be the E_{8} lattice VOA, so one has Aut(V) = E_{8}(ℂ) (cf. [DN]). In [DGH], we found the five Virasoro frames Γ, Σ, Ψ, Θ and Ω inside V. They were constructed with the help of frames in the root lattice E_{8} and markings of the binary Hamming code H_{8} of length 8 as follows:

There are, up to automorphisms, three markings of the Hamming code H_{8}, denoted by α, β, γ (Th. 5.1 of [DGH]). They give the three frame sublattices K_{8}, K′_{8} and L_{8} inside the E_{8} lattice. The final frame O_{8} can be obtained by a twisted construction from γ (see Th. 5.2. of [DGH]). The four VFs Γ, Σ, Ψ, and Θ come from the frames K_{8}, K′_{8}, L_{8} and O_{8}. The fifth VF Ω is obtained by a twisted construction from O_{8} (see Th. 5.3. of [DGH]). Table 1 summarizes this. In it, the arrow ↙ (resp. ↘) denotes the untwisted (resp. twisted) construction.

Table 1: The markings of H_{8}, resp. frames of E_{8} and V and their relations

object  marking/frame
H_{8}  α  β  γ
↙ ↘  ↙ ↘  ↙ ↘
E_{8}  K_{8}  K′_{8}  L_{8}  O_{8}
↙ ↘  ↙ ↘  ↙ ↘  ↙ ↘
V  Γ  Σ  Ψ  Θ  Ω

Table 2: The frame sublattices in the E_{8} lattice

orbit origin Type of Δ D_{X}  W_{X}
K_{8}  α  2^{6} × 4^{1} 2^{7}  2^{7}:Sym_{8}
K′_{8}  β, α̃  2^{4} × 4^{2} 2^{6} 2^{6}.(Sym_{4} ≀ 2)
L_{8}  γ, β̃  2^{2} × 4^{3} 2^{4} 2^{4}.(2 ≀ Sym_{4})
O_{8}  γ̃  4^{4} 2 2.AGL(3, 2)

In [DGH], Th. 5.3, we showed further that the possible values of k = dim(D) for any VF are 1, 2, 3, 4, 5 and that they occur for the VFs Γ, Σ, Ψ, Θ, and Ω, in this order. For the values k ∈ {1, 2, 3, 4}, the VF is unique up to automorphisms of V. For k = 5 this was a conjecture, now proven in Theorem 4.15.

To apply the general discussion about G∩N from the last section, we summarize in Table 2 again the necessary information about the four frame sublattices and the associated ℤ_{4}-code Δ already given in [DGH], Th. 5.2, and [CS].

This table together with Theorem 2.8 proves the next theorem.

Theorem 3.1 For k = 1, 2, 3, and 4 the structure of G_{C} and (G ∩ N)/G_{C} is given in Table 3.

The groups G_{D} are well known subgroups of Aut(V) ≅ E_{8}(ℂ), see Prop. 3.5.

Table 3: Structural Information about G ∩ N

k Structure of G_{C}  (G ∩ N)/G_{C} d = 2^{5−k}
1 2^{1+14} ≅ (2^{6} × 4).2^{7} 2 ≀ Sym_{8} 16
2 2^{2+12} ≅ (2^{4} × 4^{2}).2^{6} 2 ≀ [Sym_{4} ≀ 2]  8
3 2^{3+9} ≅ (2^{2} × 4^{4}).2^{4} 2 ≀ [2 ≀ Sym_{4}]  4
4  2^{4+5} ≅ 4^{4}.2  2 ≀ AGL(3, 2)  2

Table 4: Centralizers and Normalizers of 2B-pure elementary abelian subgroups of E_{8}(ℂ)

k Centralizer  Normalizer
1 HSpin(16, ℂ) HSpin(16, ℂ)
2  2^{2}D_{4}^{2}:2  2^{2}D_{4}^{2}:[2 × Sym_{3}]
3  2^{4}A_{1}^{8}  2^{4}A_{1}^{8}.AGL(3, 2)
4  T_{8}.2^{1+6}  T_{8}.2^{1+6}.GL(4, 2)
5  2^{5+10}  2^{5+10}GL(5, 2)

For general background on subgroups of E_{8}(ℂ), see [CG] and the recent survey [GR99].

Proposition 3.2 (Involutions in E_{8}(ℂ)) In the group E_{8}(ℂ), there are two classes of involutions, denoted 2A and 2B. Their respective centralizers are connected groups of types A_{1}E_{7}, D_{8}, and in more detail, their isomorphism types are HSpin(16, ℂ), 2A_{1}E_{7}, which is a nontrivial central product of a fundamental SL(2, ℂ) subgroup with a simply connected group of type E_{7} (the factors have common center of order 2). On the adjoint module, of dimension 248, the spectra are 1^{136}, −1^{112} and 1^{120}, −1^{128}.

Remark 3.3 Conjugacy classes and centralizers for elements of small orders are discussed in several articles, e.g. [CG, G91].

We recall some information about 2-local subgroups of E_{8}(ℂ).

Definition 3.4 A subgroup S of a group K is Y -pure if all nonidentity elements of S lie in the conjugacy class Y of K. (This definition is often used for elementary abelian p-subgroups of a larger group.)

Proposition 3.5 For each integer k = 1, 2, 3, 4, and 5, there is up to conjugacy a unique 2B-pure elementary abelian subgroup of rank k in E_{8}(ℂ). These groups are toral for k ≤ 4 and nontoral for k = 5. Their centralizers and normalizers are described in Table 4.

Proof. [CG], (3.8); [G91].

Let F be any VF in V and let D the associated binary code. We will identify G_{D} as a 2B-pure subgroup in Prop. 3.7.

Proposition 3.6 The binary code D is equivalent to a code generated by the first k = 1, 2, 3, 4, 5 codewords of the list 1^{16}, 1^{8}0^{8}, (1^{4}0^{4})^{2}, (1100)^{4}, (10)^{8}.

Proof. This result was obtained during the proof of Theorem 5.3 of [DGH] by using the decomposition polynomial.

Proposition 3.7 Every involution in G_{D} is of type 2B, so G_{D} is 2B-pure.

Remark 3.8 For a general FVOA, one may not expect the group G_{D} to be pure.

Proof. From Proposition 3.6, the code D has one codeword of weight 0, 2^{k} − 2 codewords of weight 8 and one of weight 16. For the components V^{I} of V = ⨁_{I∈D} V^{I} we have the following decomposition into T_{16}-modules (see again [DGH] proof of Th. 5.3): For I = 0^{16}: V^{0} = ⨁_{c∈C} V (c) with C = D^{⊥}. For the weight of I equal to 8: V^{I} = ⨁_{h_{i}∈\{0, \frac{1}{2} , \frac{1}{16} \}} 2^{5−k}M(h_{1}, …, h_{16}), where h_{i} = \frac{1}{16} for i ∈ I and h_{i} = \frac{1}{2} for an odd number of i ∉ I and h_{i} = 0 otherwise. For I = 1^{16}: V^{I} = 2^{8−k}M(\frac{1}{16}, …, \frac{1}{16}). This gives 8 · (2^{5−k} − 1), 8 · 2^{5−k}, and 2^{8−k} for the dimension of the weight one part V_{1}^{I} of V^{I}, respectively.

Let μ ∈ G_{D} ≅ D̂ be an involution. We investigate its action on the weight one part V_{1} of V. The 2^{k−1} codewords I ∈ D for which μ acts by −1 on V^{I}, i.e., with μ(I) ≠ 0, have weight 8 or 16. It follows from above that the −1-eigenspace of μ on V_{1} has dimension 2^{k−1}·2^{8−k} = 128 and the +1-eigenspace has dimension 248−128 = 120. Therefore, μ is an involution of type 2B described in Prop. 3.2.

In our case, the 2B-pureness of G_{D} implies that when it is toral (i.e., k ≤ 4), it lies in a maximal totally singular subspace of T_{(2)}. (The natural quadratic form on this subgroup of the torus is given in [G91].) For E_{8}(ℂ), the set of totally singular subspaces of T_{(2)} of a given dimension form an orbit under the Weyl group. For this reason, G_{D} is characterized up to conjugacy in E_{8}(ℂ) by the integer k. See [CG, G91] for a more detailed discussion.

Definition 3.9 Given a frame F, we denote by t(f) the Miyamoto involution of type 2 associated to f ∈ F (see [M96]).

The D-group G_{D}(F) equals ⟨t(F)⟩ = ⟨t(f) | f ∈ F ⟩, which is naturally identified with D̂, cf. Prop. 1.1. We have seen that for a frame F the D-group G_{D}(F) is 2B-pure of rank k and all values k ∈ {0, 1, 2, 3, 4, 5} are obtained by the frames of [DGH] and described at the beginning of this section. We shall see that the correspondence between frames and 2B-pure groups is not monic in general. Note also that there is a bijection between D-groups and subVOAs denoted V^{0} in [DGH], defined by ⟨t(F)⟩ ↦ V^{⟨t(F)⟩}.

Proposition 3.10 Let F = {ω_{1}, …, ω_{16}} be a Virasoro frame for V and let k = dim(D) ∈ {1, 2, 3, 4, 5}. One has |{t(ω_{i}) | ω_{i} ∈ F }| = 2^{k−1} and so there are 2^{k−1} distinct Miyamoto involutions coming from the frame. Their set-theoretic complement in the D-group G_{D} is a codimension one subspace.

Proof. Every Miyamoto involution t(ω_{i}) acts on V^{I}, I ∈ D, by −1 or 1 depending on whether the index i is contained in the support of I or not. Using the base of the code D given in Proposition 3.6, we see that t(ω_{i}) always acts as −1 on the first base vector and that any combination of signs on the remaining k − 1 base vectors is possible.

Corollary 3.11 The normalizer of the set of Miyamoto involutions induces AGL(k − 1, 2) on both the D-group and on the set of Miyamoto involutions in it.

Proof. Let G_{D} be the D-group. Since G_{D} is 2B-pure, N(G_{D})/C(G_{D}) ≅ GL(k, 2) for all k ∈ {1, 2, 3, 4, 5} by Prop. 3.5, whence the Corollary.

Lemma 3.12 For all k, the factor group G/G_{C} embeds in Sym_{d} ≀AGL(k−1, 2), where d = 2^{5−k}.

Proof. By Theorem 2.8 (3) of [DGH], G/G_{C} must be a subgroup of Aut(m_{h}(V)) ≤ Aut(D) ≤ Sym_{16} (see [DGH], Def. 2.7 for the notation). Using Proposition 3.6 and Theorem C.3 (ii) of [DGH] we see immediately that Aut(D) ≅ Sym_{d} ≀ AGL(k − 1, 2).

Remark 3.13 We shall see in Corollary 3.19 that G(F) induces the permutation group Sym_{2^{5−k}} ≀ AGL(k − 1, 2) on F. Unfortunately, this statement does not seem to follow in an obvious way from the previous result.

Definition 3.14 Suppose that the group J acts on the set Ω. A block is a nonempty subset B ⊆ Ω so that g(B) = B or B ∩ g(B) = ∅ for all g ∈ J. A partition of Ω into blocks is called a system of imprimitivity. If the only systems of imprimitivity are the trivial ones ({Ω}, and all 1-sets), we call the action of J on Ω primitive. Otherwise, we call the action imprimitive. Note that primitivity implies transitivity if |Ω| > 2.

Theorem 3.15 (Jordan, 1873) A primitive subgroup of Sym_{n} which contains a p-cycle, for a prime number p ≤ n − 3, is Alt_{n} or Sym_{n}.

Proof. [W], Th. 13.9.

Notation 3.16 We say that the partition of the natural number n has type or partition type p^{a}q^{b} ⋯ (for distinct natural numbers p, q, …) if it has exactly a parts of size p, b parts of size q etc., and the sum ap+bq +⋯ is n. In Sym_{n}, the stabilizer of such a partition is isomorphic to Sym_{p} ≀ Sym_{a} × Sym_{q} ≀ Sym_{b} × ⋯, and when we write such an isomorphism type for a subgroup of Sym_{n}, it is understood to be the stabilizer of a partition (unless stated otherwise).

Lemma 3.17 If p > 1, a > 1, Sym_{p} ≀ Sym_{a} (the natural subgroup of Sym_{pa} fixing a partition of type p^{a}) is a maximal subgroup of Sym_{pa}.

Proof. Let P be the partition fixed by the natural subgroup H ≅ Sym_{p} ≀ Sym_{a} of Sym_{pa} and let K be a subgroup, H ≤ K ≤ Sym_{pa}. If K acts primitively on the pa letters, it is Sym_{pa}, by the Jordan theorem 3.15. We suppose K acts imprimitively and seek a contradiction. Let B be a block from such a system of imprimitively and suppose that B ∩ A ≠ ∅ for some A ∈ P. Then, since the action of Stab_{H}(A) on A is Sym_{A}, we get A ⊆ B. Therefore, B is a union of parts of P. Since the action of H on the parts of P is Sym_{P}, we deduce that B = A since |B| < pa. Therefore, P is the system fixed by K, whence H = K, and we are done.

Lemma 3.18 Let n = pqr with p > 1, q > 1, r ≥ 1 and let H ≅ Sym_{p} ≀ Sym_{qr} be the subgroup of Sym_{n} fixing a partition P of type p^{qr}. Let Q be a partition of type (pq)^{r} of which P is a refinement and let H_{0} be the subgroup of Sym_{n} which permutes the parts of P and fixes all the parts of Q. Thus, H_{0} ≅ ∏_{1}^{r} Sym_{p}≀Sym_{q}.

Let G_{0} ≅ ∏_{1}^{r} Sym_{pq} be the subgroup of Sym_{n} fixing all parts of Q. Let g ∈ G_{0} and write g = g_{1} ⋯ g_{r} as a product of permutations for which supp(g_{i}) is contained in Q_{i}, the i^{th} part of Q. Suppose that each g_{i} is not in H_{0}. Then ⟨H_{0}, g⟩ = ⟨H_{0}, g_{1}, …, g_{r}⟩ = G_{0} ≅ ∏_{1}^{r} Sym_{pq}.

Proof. Define J = ⟨H_{0}, g⟩ and let G_{i} ≅ Sym_{pq} be the set of permutations which are the identity outside Q_{i}, the i^{th} part of Q, and let H_{i} = H∩G_{i} ≅ Sym_{p}≀Sym_{q}.

We study the action of J on Q_{i}, and use the property that H_{i} is maximal in G_{i} (Lemma 3.17). We have the natural projection maps G_{0} = G_{1} × ⋯ × G_{r} → G_{i}. Since J projects onto G_{i} (by maximality of H_{i} and the hypotheses on the g_{i}), it follows that J ∩ G_{i} is a normal subgroup of G_{i}. The only normal subgroup of G_{i} which contains H_{i} is G_{i}: Since H_{i} contains a transposition, a normal overgroup contains all transpositions, so equals G_{i}. We conclude that J contains each G_{i} and is therefore equal to G_{0}.

Corollary 3.19 For k = 1, 2, 3, 4, let d = 2^{5−k}. Then G/G_{C} ≅ Sym_{d} ≀ AGL(k − 1, 2).

Proof. Use Lemma 3.12 and Theorem 3.1. For k = 4 we are already done. For k = 1, 2, and 3 apply Lemma 3.18 with p = 2, q = 2^{4−k}, r = 2^{k−1} and use the description of the image σ̄ ∈ G/G_{C} of the triality automorphism σ given in Theorem 2.19.

Remark 3.20 The conclusion of Cor. 3.19 is also true for k = 5, but there is no Cartan subalgebra naturally associated to the frame in the nontwisted lattice construction of V, hence no obvious analogue of Theorem 2.8.

4 The five classes of frames in V_{E_{8}}

In this section, we describe and characterize the frame stabilizer as a subgroup of Aut(V) = E_{8}(ℂ). We also prove the uniqueness of the Virasoro frame for k = 5.

4.1 The case k = 1

Theorem 4.1 (Characterization of G, k = 1) The C-group G_{C} is extraspecial of plus type, i.e., G_{C} ≅ 2_{+}^{1+14}. Also, G_{C} is the unique subgroup of its isomorphism type up to conjugacy in Aut(V) and G = N_{Aut(V)}(G_{C}) ≅ 2_{+}^{1+14}.Sym_{16}

Proof. We now show that G_{C} is extraspecial. Since G_{C} is nonabelian and since G_{D} ≅ 2 and G_{C}/G_{D} is elementary abelian, it follows that G′_{C} = G_{D}.

Now let Z be the center of G_{C}. It is clear from the action of O_{2}(W̃_{X}) that Z ∩ G_{C} has order 2, corresponding to the coset e_{i} + L. Due to this action, which is faithful (note that an element corresponding to −1 inverts elements of order 4 in T ∩ G_{C}), it follows that Z has order 2. Since G_{C} has nilpotence class at most 2, it follows that Z = G′_{C} and so G_{C} is extraspecial.

From the action of a Sym_{8} subgroup of W̃_{X}, we see that its composition factors within G_{C} are irreducibles of dimensions 1 (three times) and 6 (twice), and that the section G_{C} ∩ T/Z is an indecomposable module with ascending factors of dimensions 6, 1 and that G_{C}/G_{C} ∩ T is the dual module, an indecomposable module with ascending factors of dimensions 1, 6.

By [G91], (12.2), this extraspecial group is uniquely determined up to conjugation in C_{E_{8}(ℂ)}(Z) ≅ HSpin(16, ℂ), and so its normalizer looks like 2_{+}^{1+14}.Sym_{16}. By Corollary 3.19, G = N_{Aut(V)}(G_{C})

4.2 The case k = 2

We get the structure 2^{2+12}[Sym_{8} ≀ 2] for G, Cor. 3.19, but now we want to understand G as a subgroup of E_{8}(ℂ).

Definition 4.2 A wreathing element in a wreath product G = H ≀ 2 is an element of G outside the direct product H_{1} × H_{2} of two copies of H which are used in the definition of the wreath product. The same term applies to elements of G/Z outside H_{1} × H_{2}/Z, where Z is a normal subgroup of G such that Z ∩ H_{1} = 1 = Z ∩ H_{2} (it follows that Z is central).

Theorem 4.3 (Characterization of G, k = 2) G satisfies the hypotheses of this conjugacy result: In E_{8}(ℂ), there is one conjugacy class of subgroups, each of which is a semidirect product X⟨t⟩, where t has order 2, X = X_{1}X_{2} is a central product of groups of the form [2×2^{1+6}]Sym_{8} such that X_{1}∩X_{2} = Z(X_{1}) = Z(X_{2}) and t interchanges X_{1} and X_{2}.

Proof. Let H := N(G_{D}) ≅ 2^{2}D_{4}^{2}[2 × Sym_{3}] and let H_{1}, H_{2} be the two central factors of H^{0}. Define G_{i} := G ∩ H_{1}. From the structure of H/H^{0}, it is clear that G_{0} := G ∩ H^{0} has index dividing 4 in 2^{2+12}[Sym_{8} ≀ 2], whence G_{0} ≥ O_{2}(G). For i ∈ {1, 2}, let i′ be the other index.

First we argue that G_{i} ≅ [2 × 2^{1+6}]Sym_{8}. Consider the quotient map π_{i} : H^{0} → H^{0}/G_{i} ≅ PSO(8, ℂ). Then π_{i}(O_{2}(G)) is an elementary abelian 2-group in H^{0}/G_{i}, so has rank at most 8 [G91]. Since G_{D} ≤ ker(π_{i}) = H_{i′} and rank(G_{C}/G_{D}) = 12, it follows that G_{C} ∩H_{i} > G_{D}, for all i. Since G_{C} ∩H_{i} is normal in G_{0}, it has order 2^{8} or 2^{14}. The latter is impossible since then we would get a rank 14 elementary abelian subgroup of PSO(8, ℂ), which is impossible [G91]. The normal structure of G_{0} now implies that G_{i}/G_{D} contains a copy of 2^{6}:Alt_{8} and equals this subgroup or has the shape 2^{6}:Sym_{8}. We now argue that the latter is the case, and this follows from embedding of G in C ≅ HSpin(16, ℂ), the centralizer in E_{8}(ℂ) of Z(G) ≅ 2 and using the 16-dimensional projective representation of C, noting that G/G_{C} maps isomorphically to a natural subgroup of the form PSO(8, ℂ) ≀ 2, with image [2^{6}:Sym_{8}] ≀ 2. It is clear from this representation, that G_{0} = G_{1}G_{2} and that the outer elements G \ G_{0} correspond to wreathing elements in the above PSO(8, ℂ)≀2. In H/H^{0} ≅ 2×Sym_{3} (with the second factor corresponding to simultaneous graph automorphisms on both H_{i}), such elements would correspond to an involution which projects nontrivially on the first factor. The reason is that a subgroup of PSO(8, ℂ):Sym_{3} isomorphic to 2^{6}:Sym_{8} is self-normalizing. The wreathing elements in G act nontrivially on G_{D} (see Corollary 3.11). This means that an involution of G \ G_{0} projects nontrivially on the second factor of H/H^{0} ≅ 2 × Sym_{3}. This implies that G is unique up to conjugacy in E_{8}(ℂ) and this uniqueness follows from just the isomorphism type of G and the fact that G_{D} is 2B-pure.

It follows that |G : G_{0}| = 2 and that G_{0} = G_{1}G_{2} (central product), G_{1}∩G_{2} = Z(G_{1}) = Z(G_{2}) ≅ 2^{2}. The isomorphism type of G will be uniquely determined if we show that there is an involution in G \ G_{0}. But this follows from the above representation in SO(16, ℂ) since a wreathing involution has spectrum {1^{8}, −1^{8}} and so lifts in the spin group to an involution [G91, Ch]. For i = 1, 2, define z_{i} by ⟨z_{i}⟩ := O_{2}(G_{i})′ ≅ ℤ_{2}.

Finally, we observe from the above 8-dimensional representations of G_{i} that O_{2}(G_{i})/O_{2}(G_{i})′ ≅ 2^{7} is an indecomposable module for G_{i}/O_{2}(G_{i}) with ascending socle factors of dimensions 1 and 6. Also, for both i, we may think of O_{2}(G_{i})′ ≅ 2 as the kernel of the representation of H_{i} as SO(8, ℂ) (rather than as a half spin group), since the above wreathing involution is realized in the degree 16 orthogonal projective representation of C(Z(G)). It follows that O_{2}(G_{1})′ = O_{2}(G_{2})′, i.e., z_{1} = z_{2}.

4.3 The case k = 3

Let H := N(G_{D}) ≅ 2^{4}A_{1}^{8}.AGL(3, 2); then H^{0} = C(Z(H^{0})) and C(O_{2}(H)) ≅ H^{0}.2^{3}. See [CG, G91].

We have already determined in Theorem 3.1 that G_{C} has the shape 2^{3+9} = 2^{4+8}, the latter decomposition coming from the general structure of H, above.

From earlier sections, we know G ∩ N ≅ 2^{3+9}. This group, with the triality of Theorem 2.19, will generate G, a group of the form below (see Cor. 3.19):

[2^{3+9} = 2^{4+8}]Sym_{4}^{4}.Sym_{4} ≅ [2^{4+16}]Sym_{3} ≀ Sym_{4}.

What we need is a group theoretic characterization of such a subgroup of E_{8}(ℂ).

Definition 4.4 Let the groups X_{1}, …, X_{r} be given and let X = X_{1} ⋯ X_{r} be a central product. Let J be the set of indices {1, …, r} and let S be a subset of X. Let Z be the subgroup of Z(X) generated by all X_{i} ∩ ⟨X_{j} | j ≠ i⟩; then X/Z is a direct product of the X_{i}Z/Z. We define the quasiprojection of S to X_{J} := ⟨X_{i} | i ∈ J⟩ to be X_{J} ∩ S^{∗}, where S^{∗} is the preimage in X_{J} of the projection of SZ/Z to X_{J}/Z, a direct factor of X/Z, complemented by X_{{1,...,r}\J}. (We use this concept in the case where X_{i} is quasisimple.)

Let Q_{J} be the quasiprojection of Q to H_{J}, J ⊆ {1, …, 8}.

Theorem 4.5 (Characterization of G, k = 3) Let X be a finite subgroup of E_{8}(ℂ) such that

(i) X has the form [2^{3+9} = 2^{4+8}]Sym_{4}^{4}.Sym_{4} ≅ [2^{4+16}]Sym_{3} ≀ Sym_{4};

(ii) X has a normal subgroup E ≅ 2^{3} which is 2B-pure.

Then:

(a) X is unique up to conjugacy in E_{8}(ℂ).

(b) If H := N_{G}(E) ≅ 2^{4}A_{1}^{8}.AGL(3, 2), then X ∩ H^{0} ≅ [2^{3+9} = 2^{4+8}]Sym_{4}^{4} = 2^{4+16}Sym_{3}^{4}.

(c) If H_{1}, …, H_{8} are the SL(2, ℂ)-components of H^{0}, then H_{i} ∩ X ≅ Quat_{8} and there is a partition Π of {1, …, 8} into 2-sets so that if A ∈ Π, then ⟨H_{i} | i ∈ A⟩ ∩ X ≅ [Quat_{8} × Quat_{8}].Sym_{3} (the top Sym_{3} layer sits diagonally over the two Quat_{8}-factors).

Proof. Denote by U the normal subgroup indicated by 2^{3+9} = 2^{4+8}. Let P ∈ Syl_{3}(X), P ≅ 3 ≀ 3 × 3. Then, R := P ∩ H^{0} is abelian (property of a connected group of type A_{1}^{8}), whence R ≅ 3^{4} is the unique maximal abelian subgroup of P. In H, any 2-group S satisfying S = [S, R] lies in a central product Q := Q_{1} ⋯ Q_{8} of groups Q_{i} ≅ Quat_{8}, Q_{i} ≤ H_{i}, for i ∈ {1, …, 8}. It follows that G_{C} lies in Q ≅ 2^{4+8} (see [G91]).

Now, let Z := O_{2}(H). Then, Z = Z(Q) = Q′ and Q/E ≅ 2_{+}^{1+16} is extraspecial with center Z/E. The outer automorphism group of Q/E is O^{+}(16, 2), in which the stabilizer of a maximal isotropic subspace I of the natural module M := 𝔽_{2}^{8} has the form

2^{120}:GL(8, 2).

In this stabilizer, an elementary abelian subgroup of order 3^{4} acts on I by a direct sum of four distinct 2-dimensional irreducibles, and it has equivalent (= dual) action on M/I. It follows that R decomposes into a direct product R = R_{1} × R_{2} × R_{3} × R_{4} and that the index set {1, …, 8} has a partition into 2-sets A(1), …, A(4) so that R_{i} ≅ 3 centralizes H_{j} iff j ∉ A(i) and [Q, R_{i}] = Q_{A(i)} ≅ 2^{2+4} (from [G91], if the involution z_{i} generates Z(Q_{i}) and a product of r > 0 distinct z_{i} equals 1, then r ≥ 4).

So far, we have shown that the conjugacy class of a subgroup of X of the shape 2^{4+16}:3^{4} is unique, namely the conjugacy class of the subgroup QR. Our frame stabilizer contains a group Y of the form 2^{4+16}.Sym_{3}^{4} ≅ 2^{4+8}.Sym_{4}^{4} (Cor. 3.19). In N_{G}(Q) ≅ 2^{4+16}Sym_{3}^{8}.AGL(3, 2), there is a unique group containing QR which has the form 2^{4+16}.Sym_{3}^{4} ≅ 2^{4+8}.Sym_{4}^{4} (so we now have G ∩ H^{0} up to conjugacy). Such a group is contained in H^{0} and in fact it is QRT, where T = T_{1} × ⋯ × T_{4}, where T_{i} ≤ H_{A(i)}, T_{i} ≅ 4 and R_{i}T_{i} ≅ 3:4 embeds diagonally in H_{A(i)} ≅ SL(2, ℂ) × SL(2, ℂ). Given the partition, all such choices of T_{i} are equivalent under conjugation by an element of R^{∗}, the unique group of order 3^{8} in the torus C(R)^{0}. We now deduce the uniqueness up to conjugacy of a subgroup of the form [2^{4+16}.Sym_{3}^{4} ≅ 2^{4+8}.Sym_{4}^{4}].Sym_{4} by a Frattini argument on the group T_{1}T_{2}T_{3}T_{4}Q/Q of order 2^{4} in QRT/Q ≤ Y/Q, which maps to H/H^{0} as the subgroup of the degree 8 permutation group AGL(3, 2) which stabilizes a partition [DGH], Appendix A., p. 441. Therefore, G is determined up to conjugacy.

Definition 4.6 We use the notation in the proof of the preceeding result. We say that an element of Q may be written as a product of elements in the Q_{i} and the number of such elements which are outside Z(Q_{i}) is the Q-weight. We define the Z(Q)-weight of a product of a set of n distinct z_{i} to be n.

Remark 4.7 In this sense, the products of the z_{i} which are the identity are those whose support forms a word in a Hamming code H_{8} with parameters [8, 4, 4].

Corollary 4.8 For k = 3, G′_{C} = G_{D}.

Proof. It is clear that every element of G_{C}/Z(Q) has even Q-weight [GR94]. In fact, weights 0, 2, 4 and 8 all occur. It is also clear by looking in the Q_{A(i)} that we get all even words in the generators z_{j} as commutators in G_{C}.

4.4 The case k = 4

We have G_{C} ≤ N = N(T), Cor. 2.3, and we get G_{C} ∩ T ≅ 4^{4}. Also, G_{C} ≅ 4^{4}:2 (the outer 2 corresponds to an inverting element (note that every inverting element is an involution in this case [CG, G91]); here, D_{X} = ⟨−1⟩), G_{D} = G′_{C} = Z(G_{C}), G_{C}/G_{D} ≅ 2^{5}.

Next, (G ∩ T)G_{C} ≅ [2^{4} × 8^{4}]:2 and (G ∩ T)G_{C} ◁ G ∩ N. Also, D_{X} ≅ 2, W_{X} ≅ 2.AGL(3, 2).

So, G_{D} ≤ O_{2}(G)′. Since C(G_{D})^{0} = T, we get G ≤ N(G_{D}) ≤ N(T) = N and so G ≤ N. (This is a consequence of Cor. 3.19, proved by different arguments. We remark that since G = G∩N, there is no “triality” in this case, a consequence of the earlier result Cor. 3.19).

The structure of G is given by Theorem 2.8 and so G ≅ [2^{4}×8^{4}].2.AGL(3, 2) ≅ 2^{4+5+8}.AGL(3, 2).

We now develop a characterization of G as a subgroup of Aut(V).

Definition 4.9 A subgroup A ≅ 4^{4} of E_{8}(ℂ) is called a GL(4, 2)-signalizer if N(A)/C(A) has a composition factor isomorphic to GL(4, 2) (notice that Aut(A) has the form 2^{16}.GL(4, 2). (This terminology is adapted from the signalizer concept in finite group theory.)

Lemma 4.10 There is one conjugacy class of GL(4, 2)-signalizers in E_{8}(ℂ).

Proof. Clearly, N(A) has one orbit on the involutions of E := Ω_{1}(A). Since there is no pure 2A-subgroup of rank greater than 3 by [CG], this must be a 2B-pure group of order 16. Such a group is unique up to conjugacy. Let T be its connected centralizer, a rank 8 torus. In W := W_{E_{8}}, there is no subgroup isomorphic to 2^{4} whose normalizer induces GL(4, 2) on it, so A ≤ T. Define W_{A} := Stab_{W} (A) and W_{E} := Stab_{W} (E). Then W_{E} ≅ 2^{1+6}GL(4, 2) and W_{A} is a subgroup of this containing a composition factor isomorphic to GL(4, 2). It is an exercise with the Lie ring technique that Aut(A) does not contain GL(4, 2) [G76]. We now show that W_{A} < W_{E}. If false, O_{2}(W_{E}) acts trivially on E and A/E, so acts as an elementary abelian 2-group, whereas the element of W corresponding to −1 must act nontrivially, yet is in the Frattini subgroup of O_{2}(W_{E}), contradiction. Since O_{2}(W_{E}) has just two chief factors under the action of W_{E}, the only possibility for W_{A} < W_{E} is W_{A} ≅ 2·GL(4, 2), the nonsplit extension.

Let U := T_{(2)}. Then W_{A} normalizes UA = ⟨U, A⟩ ≅ 2^{4} × 4^{4}. So does W_{E}, since U is the inverse image in T_{(2)} of E under the squaring endomorphism. The action of W_{E} on UA/E ≅ 2^{8} has kernel exactly Z(W_{E}) ≅ 2 (otherwise, we would have [A, O_{2}(W_{E})] ≤ [U, O_{2}(W_{E})] ≤ E and get a contradiction as in the previous paragraph). It follows that in UA/E, the subspace A/E has stabilizer in W_{E} equal to W_{A}. Also, UA/E is a completely reducible W_{A}-module, the direct sum of a four dimensional module and its dual.

Now let A_{1} be another GL(4, 2) signalizer in E_{8}(ℂ). By following the above procedure for A_{1} in place of A, we may arrange UA_{1} = UA. We may replace A_{1} by a conjugate with an element of W_{E} so that both A and A_{1} are stabilized by W_{A} (property of a parabolic subgroup of O^{+}(8, 2)). Since there are precisely three W_{A}-chief factors within UA, two isomorphic to the natural module for W_{A}/Z(W_{A}) ≅ GL(4, 2) and the third to the dual module, it follows that A ∩ A_{1} ≥ E and finally that A = A_{1}.

Theorem 4.11 (Characterization of G, k = 4) The group G is determined up to conjugacy, as described below, in the normalizer of A, a GL(4, 2)-signalizer, which is, in turn, unique up to conjugacy in E_{8}(ℂ).

The normalizer in E_{8}(ℂ) of A, a GL(4, 2)-signalizer is a group of the form T.[2·GL(4, 2)]. In N(A), let u be any element which acts on T by inversion. Then |u| = 2. Set B := ⟨A, u⟩. Then N(B) has the form 4^{4}.2^{8}.2·GL(4, 2) and the normal subgroup of the shape 4^{4}.2^{8} ≅ 2^{4} × 8^{4} is a characteristic subgroup of O_{2}(B). The group G is a subgroup of index 15 in N(B) stabilizing a rank three subgroup of Ω_{1}(A), and this property characterizes G up to conjugacy in N(B).

Proof. The maximal subgroup above of O_{2}(B) is the unique maximal subgroup which is abelian, so is characteristic.

4.5 The case k = 5

Let F be a frame with k = 5. The D-group G_{D} is the unique up to conjugacy 2B-subgroup of rank 5 in Aut(V) and the normalizer of G_{D} is 2^{5+10}GL(5, 2) (see Prop. 3.5). More precisely, one has:

Proposition 4.12 The extension

1 → O_{2}(N(G_{D}))/Z(O_{2}(N(G_{D}))) → N(G_{D})/Z(O_{2}(N(G_{D}))) → GL(5, 2) → 1

is split, though N(G_{D}) does not split over O_{2}(N(G_{D})). There is a subgroup X of N(G_{D}) satisfying Z(O_{2}(N(G_{D}))) ≤ X, X/Z(O_{2}(N(G_{D}))) ≅ GL(5, 2) is nonsplit over Z(O_{2}(N(G_{D}))).

Proof. [G76], Section 1.

The group X has been called the Dempwolff group, [Th, G76], and we call N(G_{D}) the Alekseevski-Thompson group. The first discussion of this group in the literature on finite subgroups of Lie groups was probably [A], but Thompson discovered it independently around early 1974 while studying the sporadic group F_{3} which embeds in E_{8}(3) [Th, G76].

It follows from Corollary 3.11 that the frame stabilizer is contained in the subgroup of N(G_{D}) which preserves the set t(F) of Miyamoto involutions, an affine hyperplane of G_{D}.

Though t restricted to F gives a bijection with a set of 16 involutions in G_{D}, it is possible that some other frame F′ ≠ F satisfies t(F) = t(F′). We shall prove now that an affine hyperplane t(F) ⊂ G_{D} ≤ E_{8}(ℂ) corresponds to a unique frame in V, i.e., all VFs with k = 5 are equivalent under Aut(V), so in particular are equivalent to the VF Ω of [DGH].

Let V^{0} = V^{G_{D}} be the subVOA fixed by the D-group. It is the vertex operator algebra studied in [G98]. It has automorphism group O^{+}(10, 2), graded dimension 1 + 0 q^{1} + 156 q^{2} + ⋯ and is isomorphic to the VOA V_{\sqrt{2}E_{8}}^{+}.

Proposition 4.13 The action of the normalizer N(G_{D}) ≅ 2^{5+10}GL(5, 2) induces the action of a parabolic subgroup P ≅ 2^{10}:GL(5, 2) of O^{+}(10, 2) on V^{0}.

Proof. The kernel of the action of G on V^{0} is G_{D}. In fact, if K is the (possibly larger) subgroup of E_{8}(ℂ) which acts trivially on V^{0}, then it acts trivially on the frame F, so is contained in G. Therefore, K = G_{D}.

Thus N(G_{D})/G_{D} acts faithfully on V^{0}, so gives a subgroup of shape 2^{10}:GL(5, 2) of O^{+}(10, 2). Such a subgroup is unique up to conjugacy. It is the stabilizer of a maximal totally singular subspace in 𝔽_{2}^{10}.

The relevant Virasoro elements of V^{0} are 496 in number and may be identified with the nonsingular points N in the space 𝔽_{2}^{10} with a maximal Witt index quadratic form (see [G98], 6.8). Under this identification, a Virasoro frame of V^{0} corresponds to a set of nonsingular vectors in 𝔽_{2}^{10} spanning a 5-dimensional subspace which is totally singular with respect to the bilinear form associated to the quadratic form (reason: all Miyamoto involutions σ of type 2 associated to a frame commute and commutativity corresponds in this case to orthogonality of the corresponding members of N). In it, the set of nonsingular vectors is the nontrivial coset of the 4-dimensional subspace of totally singular vectors and zero.

Recent work of Lam [L] shows that Aut(V^{0}) is transitive on frames within V^{0}. One should keep in mind that frames in V^{0} may have different values of k as frames in V. Next, we give a short proof of transitivity on frames.

Proposition 4.14 In V^{0}, there is one orbit of Aut(V^{0}) on frames.

Proof. By Witt’s theorem, the 5-dimensional subspace corresponding to a frame is unique up to isometry of 𝔽_{2}^{10}. Transitivity of Aut(V^{0}) on its frames follows at once.

The orbits of the parabolic subgroup P on such subspaces are studied in Appendix 5.4. We use this to show:

Theorem 4.15 For k = 5, there is one E_{8}(ℂ) orbit of such Virasoro frames in V. If F, F′ are frames with k = 5 and t(F) = t(F′), then F = F′.

For its proof we need:

Lemma 4.16 For k ≤ 4, G_{C} contains a group E^{∗} which is elementary abelian of order 2^{5} and is 2B-pure, contains G_{D} and has the property that T ∩ E^{∗} has rank 4.

Proof. Let u be any element of G_{C} which corresponds in N/T to −1. Then u is an involution (see [G91]) and it centralizes T_{(2)}. For all k ≤ 4, such u exist in G_{C} and G_{C} contains a maximal totally singular subspace of T_{(2)} containing G_{D}, say E_{1}. Take E^{∗} = E_{1}⟨u⟩. This group is 2B-pure, see [G91].

Proof of Theorem 4.15. The group G_{D} of a VF with k = 5 is up to conjugation unique in E_{8}(ℂ) and has normalizer 2^{5+10}GL(5, 2) which induces the action of 2^{10}:GL(5, 2) ≅ P on V^{0} by Prop. 4.13.

We claim that, for every k ∈ {1, 2, 3, 4, 5}, there is a frame F′ from V with k = dim(G_{D}(F′)) which is contained in V^{0}, the fixed point VOA for G_{D}. For k = 5, this is obvious. For k ≤ 4, this will follow if we show that the group G_{C}(F′) contains a conjugate of G_{D}, since if g^{−1} is an element conjugating G_{D} into G_{C}(F′) one has g(F′) ⊂ V^{gG_{C}(F′)g^{−1}} ⊂ V^{G_{D}}. By Lemma 4.16 and Prop. 3.5, we are done.

On a P -orbit, the values of k are constant. Since Proposition 5.13 shows that we have five orbits and all five values of k are represented there, P acts transitively on the set of frames of V^{0} with a fixed value of k, and in particular we have transitivity for k = 5.

We refer to Section 5.4 for the definition of the J-indicator and the collection of subspaces, Σ. We need to know something about the (k, j)-bijection {1, 2, 3, 4, 5} ↔ {0, 1, 2, 3, 4} indicated in the proof of Th. 4.15.

Definition 4.17 Define P_{j} as the stabilizer in P of some member of Σ with J-indicator j. Using G/G_{D} ≅ P, let H_{j} be the subgroup of N(G_{D}) with H_{j}/G_{D} ≅ P_{j}.

Lemma 4.18 The cases of J-indicator 0 or 4 correspond to frames F′ ⊂ V^{0} with k = 1 or 5. Thus, {1, 5} ↔ {0, 4} under the (k, j)-bijection.

Proof. From Prop. 5.14 and Cor. 5.15, H_{j} has a normal subgroup Q of order 2^{9} resp. 2^{19} and H_{j}/Q ≅ GL(4, 2). Let F′ be the associated frame. So, H_{j} ≤ G(F′) and we want to show that k = 1 or 5 for F′.

We assume k ≠ 1, 5 and derive a contradiction. According to the shapes of frame stabilizers for k ∈ {2, 3, 4}, the fact that H_{j}/Q ≅ GL(4, 2) implies k = 2. In H_{j}, the chief factors afford irreducible modules for H_{j}/Q ≅ GL(4, 2) of dimensions 1, 4 and 6. In a frame stabilizer for k = 2, only dimensions 1 and 6 occur.

Lemma 4.19 Let K be a 4-dimensional subgroup of G_{D} ≅ 2^{5}. Then, O_{2}(N(G_{D}))/K ≅ 2^{5+10}/K is isomorphic to the direct product 2^{4} with the extraspecial group 2^{1+6}.

Proof. This follows from (3) in Section 3 of [G76]. The above quotient group is nonabelian with derived group of order 2 and admits the action of N(G_{D}) ∩ N(K) ≅ 2^{5+10}AGL(4, 2). Since this subgroup embeds in a torus normalizer, the above structure is forced.

Proposition 4.20 The frame F with k = 5 corresponds to the J-indicator 0. Therefore, 5 ↔ 0 and 1 ↔ 4 under the (k, j)-bijection.

Proof. Take k = 5. We assume by Lemma 4.18 that the J-indicator is 4 and work for a contradiction. Then by Corollary 5.15, G(F) = H_{4} ≅ 2^{5+10}AGL(4, 2) and G_{C} ≅ 2^{5+10}.

The action of G_{C} on V respects the decomposition into T_{16}-modules. As in the proof of Prop. 3.7 (cf. also [M98]) one has that for an 8-set I ∈ D each irreducible T_{16}-module M(h_{1}, h_{2}, …, h_{16}) with h_{i} = \frac{1}{16} for i ∈ I and h_{i} = \frac{1}{2} for an odd number of i ∉ I has multiplicity 1. Let M be one of the eight irreducibles from above with h_{i} = \frac{1}{2} for exactly one i. In M, the weight 1 subspace M ∩ V_{1} is one dimensional.

The action of G_{D} ≅ D̂ on V^{I} ⊃ M has as kernel the 4-dimensional subgroup K of elements in D̂ whose kernel contain I. By Lemma 4.19, one has G_{C}/K ≅ 2^{1+6} × 2^{4}, where the center of the extraspecial group 2^{1+6} is G_{D}/K. Faithful irreducible modules for 2^{1+6} have dimension 8 (cf. [Go, H]), so we have a contradiction.

The Proposition follows now from Lemma 4.18.

Theorem 4.21 (Characterization of G, k = 5) The frame stabilizer for k = 5 has shape 2^{5}AGL(4, 2) ≅ 4^{4}[2 · GL(4, 2)], where the factor 2 indicates an involution inverting the normal 4^{4} subgroup. Such a subgroup is uniquely determined up to conjugacy as a subgroup of E_{8}(ℂ) by these conditions. In particular, G_{C} = G_{D} is elementary abelian.

Proof. The structure of G/G_{C} follows from Prop. 4.20 and Cor. 5.15.

For the characterization, one can modify the argument of Theorem 4.11. To do so, we must locate a suitable GL(4, 2)-signalizer within G, see Def. 4.9.

(i) We define the group A by A := [O_{2}(G), G]. It follows from the vanishing of the Ext^{1} group for modules {1, 4} (in either order) that O_{2}(G) = A⟨t(f)⟩, a semidirect product, that the four dimensional chief factors in G_{C} are isomorphic (by commutation with an element of t(f), for f ∈ F). Since these irreducibles are not self dual, O_{2}(G)′ has rank 4 and O_{2}(G) = A⟨t(f)⟩ is a semidirect product, for any f ∈ F.

We want to show that A is homocyclic of type 4^{4}. For GL(4, 2), the 6 dimensional module 4 ∧ 4 is irreducible, so A is abelian. We assume that A is elementary abelian and derive a contradiction.

Let A ≤ E, a maximal elementary abelian 2-subgroup of Aut(V) ≅ E_{8}(ℂ). The classification in [G91] shows just two conjugacy classes of such E, of ranks 8 and 9. If rank(E) = 8, we have a contradiction since GL(4, 2) is not involved in N(E). Therefore, rank(E) = 9, which means that E contains a unique subgroup E_{1} of index 2 which lies in a maximal torus, T (so E_{1} = T_{(2)}). Such a subgroup is characterized in E as the unique maximal subgroup of E whose complement in E contains only involutions in class 2B. From [G91], we may take T to be the connected centralizer of A_{1} := O_{2}(G)′, a 2B-pure rank 4 elementary abelian subgroup. In C(A_{1})/T ≅ 2^{1+6}, O_{2}(G) maps to a subgroup whose normalizer in N(A_{1}) ≅ T.2^{1+6}GL(4, 2) has a section isomorphic to GL(4, 2). This means O_{2}(G) ≤ T ⟨u⟩, where u is an involution in the torus normalizer corresponding to −1 in the Weyl group.

It follows that A = [O_{2}(G), G] ≤ [T ⟨u⟩, G] ≤ T. Therefore, A ≤ E_{1}, so A = E_{1}. We now have a final contradiction, since E_{1} is a self dual module for its normalizer, whereas A has chief factors consisting of two non self dual modules, contradiction. This proves A ≅ 4^{4}.

(ii) In G, the unique normal abelian subgroup maximal with respect to containment is A.

5 Appendix

5.1 Equivariant unimodularizations of even lattices

Sometimes it is convenient to have a lattice embedded in another lattice whose determinant avoids certain primes, and it can be useful to do this in a way which respects automorphisms.

First, we recall some basic facts concerning extensions of lattices (cf. [N]). An even lattice L defines a quadratic space (A, q), where A = L^{∗}/L, L^{∗} = {x ∈ L ⊗ ℚ | (x, y) ∈ ℤ for all y ∈ L} the dual lattice, and q : L^{∗}/L ⟶ ℚ/2ℤ is the quadratic form x (mod L) ↦ (x, x) (mod 2ℤ). Even overlattices M of L define isotropic subspaces C = M/L of (A, q) and this correspondence is one to one. An automorphism g of L extends to an automorphism of M if and only if the induced automorphism ḡ ∈ Aut(A, q) fixes the subspace C. A subgroup C of A generated by a set of elements is isotropic if the generating elements are isotropic and orthogonal to each other with respect to the ℚ/ℤ-valued bilinear form obtained by taking the values of b(x, y) = \frac{1}{2}(q(x + y) − q(x) − q(y)). The determinant det(L) of L is the order of A. If A has exponent N, then q takes values in \frac{1}{2N} (2ℤ)/2ℤ. There is an orthogonal decomposition (A, q) = ⨁_{p|det(L)}(A_{p}, q_{p}) of quadratic spaces, where A_{p} is the Sylow p-subgroup of A.

A sublattice L of a lattice M is called primitive if M/L is free. Let K = L_{M}^{⊥} the orthogonal complement of L in M. Then, L is primitive exactly if the projection of M/(L ⊕ K) to K^{∗}/K is injective.

Definition 5.1 Let M be a lattice and L a sublattice. We say that an automorphism α of L extends (weakly) to M if there is β ∈ Aut(M) so that β|_{L} = α. We say that a subgroup S ≤ Aut(L) extends (weakly) to M if every element extends and we say that it extends strongly if there is a subgroup R ≤ Aut(M) which leaves L invariant and the restriction of R to L gives an isomorphism of R onto S. In this case, call such R a strong extension of S to M.

Definition 5.2 Let L be an even lattice. An equivariant unimodularization of L is an unimodular lattice M containing L as a primitive sublattice such that Aut(L) extends strongly to M.

Theorem 5.3 (James, [J]) An equivariant unimodularization M of an even lattice L exists of rank at most 2 · rank(L) + 2.

One can take for M the orthogonal sum of rank(L) + 1 hyperbolic planes. A somewhat stronger result can be found in [N] (see Prop. 1.14.1 and Th. 1.14.2).

The above unimodularizations from [J, N] are all indefinite. The next theorem shows that one can get equivariant definite unimodularizations of a definite lattice. For its proof we need:

Lemma 5.4 (i) Let p be an odd prime and r ≥ 0. Then, −1 is the sum of two squares in ℤ/p^{r}ℤ. (ii) For all r ≥ 0, one can write −1 is a sum of four squares in ℤ/2^{r}ℤ.

Proof. (i) When r = 1, we quote the well known fact that every element in a finite field is a sum of two squares. Part (i) is now proved by induction on r: Assume that r ≥ 1 and that a, b are integers such that a^{2} + b^{2} = −1 + p^{r}m, for some integer m. Let x, y be integers and consider (a + p^{r}x)^{2} + (b + p^{r}y)^{2} = −1 + p^{r}m + 2p^{r}[ax + by] + p^{2r}e, for some integer e. Since not both a and b can be divisible by p we can solve 2[ax + by] ≡ −m (mod p) for integers x, y. Thus, −1 is a sum of two squares modulo p^{r+1}.

Part (ii) follows from a similar argument, or from Lagrange’s theorem that every nonnegative integer is a sum of four integer squares.

Theorem 5.5 Let L be an even lattice of signature (n_{1}, n_{2}). Then there exists an equivariant unimodularization of the lattice L whose rank is 8 · rank(L) and signature is (8n_{1}, 8n_{2}). If det(L) is odd, there is one whose rank is 4 · rank(L) and signature is (4n_{1}, 4n_{2}). In particular, if the lattice is definite, this unimodularization is also definite.

Proof. Assume first that det(L) is odd and let (A, q) be the finite quadratic space associated L. Let K = L ⊥ L ⊥ L ⊥ L having the associated quadratic space (B, q′) = (A, q) ⊕ (A, q) ⊕ (A, q) ⊕ (A, q). We decompose (A, q) as the orthogonal sum

(A, q) = ⨁_{p|det(L)}(A_{p}, q_{p}),

where A_{p} ≅ ℤ/p^{a_{p,1}}ℤ + ℤ/p^{a_{p,2}}ℤ + ⋯ + ℤ/p^{a_{p,np}} ℤ is an abelian p-group with a_{p,1} ≥ a_{p,2} ≥ ⋯ ≥ a_{p,n_{p}} of order p^{a_{p}}, where a_{p} := a_{p,1} + a_{p,2} + ⋯ + a_{p,n_{p}}.

Fix a prime p|det(L). Using Lemma 5.4, let r, s ∈ ℤ so that r^{2} + s^{2} ≡ −1 (mod p^{a_{p,1}}). We let

D_{p} = {(rx, sx, 0, x) | x ∈ A_{p}}  and  E_{p} = {(sx, −rx, x, 0) | x ∈ A_{p}}.

Since q_{p}(±rx) + q_{p}(±sx) + q_{p}(±x) = p^{a_{p,1}}q_{p}(x) ∈ 2ℤ/2ℤ, the groups D_{p} and E_{p} are isotropic subspaces of (A_{p}, q_{p}) ⊕ (A_{p}, q_{p}) ⊕ (A_{p}, q_{p}) ⊕ (A_{p}, q_{p}). They are orthogonal to each other, so that C_{p} = D_{p} + E_{p} is also isotropic and has order p^{2a_{p}}.

Finally, let C = ⨁_{p|det(L)} C_{p}. It is an isotropic subspace of (B, q′) with |A|^{2} elements and it is invariant under the diagonal action of Aut(L) induced on (B, q′). Since |C|^{2} = |B|, the overlattice M of K belonging to C = M/K ≤ B is a definite even unimodular lattice having an automorphism group which contains a strong extension of Aut(L) and L is also primitive.

Now, we do the case of even det(L). This time we take K = L ⊥ ⋯ ⊥ L (8 times) with associated quadratic space (B, q′) = (A, q)⊕⋯⊕(A, q) (8 times). We proceed in a similar spirit:

For p = 2, let r, s, t, u be integers such that r^{2} + s^{2} + t^{2} + u^{2} ≡ −1 (mod 2^{a_{2,1}+1}) and define

\begin{aligned}D_{2} & = & \{( rx, & sx, & tx, & ux, & x, & 0, & 0, & 0) & | x ∈ A_{2}\}, \\ E_{2} & = & \{( sx, & −rx, & ux, & −tx, & 0, & x, & 0, & 0) & | x ∈ A_{2}\}, \\ F_{2} & = & \{(−x, & 0, & 0, & 0, & rx, & sx, & tx, & ux) & | x ∈ A_{2}\} and \\ G_{2} & = & \{( 0, & −x, & 0, & 0, & sx, & −rx, & ux, & −tx) & | x ∈ A_{2}\}. \end{aligned}

Since q_{2}(±rx)+q_{2}(±sx)+q_{2}(±tx)+q_{2}(±ux)+q_{2}(±x) = 2^{a_{2,1}+1}q_{2}(x) = 2ℤ/2ℤ ∈ ℚ/2ℤ, the groups D_{2}, E_{2}, F_{2} and G_{2} are totally isotropic subspaces of (A_{2}, q_{2})⊕ ⋯ ⊕ (A_{2}, q_{2}). They are pairwise orthogonal, so that C_{2} := D_{2} + E_{2} + F_{2} + G_{2} is also isotropic and has order 2^{4a_{2}}.

For the odd primes, we let C_{p} = (D_{p} + E_{p}) ⊕ (D_{p} + E_{p}). As in the preceding cases, we see that the overlattice M of K belonging to C = ⨁_{p|det(L)} C_{p} has all the required properties.

If we try only to double the rank of L, we may not find an unimodularization in general, but we can achieve the following:

Theorem 5.6 Let L be an even lattice with signature (n_{1}, n_{2}). Then there exists an even lattice M of signature (2n_{2}, 2n_{2}) containing L as a primitive sublattice such that Aut(L) can be strongly extended to a subgroup of Aut(M) and det(M) is a power of an arbitrarily large prime.

Proof. The Dirichlet Theorem implies that there are infinitely many primes s satisfying s ≡ −1 (mod 2 det(L)). Let s be such a prime.

Let L[s] be a lattice which as a group is isomorphic to L by ψ : L ⟶ L[s] with bilinear form defined by (ψ(x), ψ(y)) = s · (x, y). Then, det(L[s]) = s^{n}det(L), where n = rank(L). Extend ψ to maps between the rational vector spaces spanned by L and L[s].

We will define M as an overlattice of K = L ⊥ L[s]. Proceeding as in the proof of the last theorem, let C = {(x, ψ(x)) | x ∈ A} ≤ (B, q′). We have q′((x, ψ(x))) = (1 + s)q(x) ∈ 2ℤ/2ℤ, i.e., C is isotropic. The determinant of the overlattice M belonging to C is det(K)/|C|^{2} = s^{n}det(L)^{2}/det(L)^{2} = s^{n}. We get an extension of Aut(L) to M by taking the diagonal subgroup of Aut(L)×Aut(L[s]), with respect to the isomorphism ψ. This diagonal subgroup preserves both L ⊥ L[s] and C hence also M.

5.2 Lifting Aut(L) to the automorphism group of V_{L}

This section is written by the first author (RLG).

We need to describe a lifting of the automorphism group W of the even lattice L to a group of automorphisms W̃ of the VOA V_{L}. This group will have a normal elementary abelian subgroup of rank equal to the rank of L and the quotient by it will be W. It will normalize a natural torus of automorphisms, whose rank is again equal to the rank of L.

For preciseness, we define the group W̃ by using the definition of L̂, the unique (in the sense of extensions) group such that there is a short exact sequence 1 → \{±1\} → L̂ → L → 1 with the property that x^{2} = (−1)^{\frac{1}{2} (x,x)} (it follows that [x, y] = (−1)^{(x,y)}) for x, y ∈ L (inner products for elements of L̂ are evaluated on their images in L).

Definition 5.7 We define W̃ as the subgroup of the automorphism group of the abstract group L̂ which preserves the given quadratic form on L via its action on the quotient by {±1}. We call W̃ the group of isometry automorphisms of L̂.

A construction will exhibit structure of W̃ and make useful actions available. It is clear from the definition that W̃ participates in a short exact sequence 1 → 2^{n} → W̃ → W → 1 where n = rank(L), so that any group constructed as a subgroup of the group of isometry automorphisms which fits into the middle of such a short exact sequence must be the group of isometry automorphisms. So, the choices made in a particular construction do not affect the isomorphism type of the group constructed.

Our construction expresses some unity between two different contexts where such extensions have occurred. In the case where L is between the root lattice and its dual, for a root system Φ whose indecomposable components have types ADE, the group W̃ will be a subgroup of G:Γ, where G is a simply connected and connected group of type Φ and Γ is the group of Dynkin diagram automorphisms, lifted to G. The basic reference for such a lift is [Ti], which relates the lift to the sign problem for the definition of a Lie algebra, given a root system. When L is the Leech lattice, we get a group W̃ which comes up in the theory of the Monster simple group [G81, G82]. Other examples of lattices and sporadic simple groups come up this way. See Remark 5.8 for more background.

We now present this author’s basic theory of W̃. It is self-contained, except for applications to VOA theory, for which we assume the standard theory of the VOA structure on the space V_{L} used in Prop. 5.9. The use of 2-regularizations here is probably new.

Remark 5.8 (Some history) Several finite dimensional representations (including projective ones) of such groups had been known for a long time to group theorists and Lie theorists. When L is the root lattice of a type ADE Lie algebra, W̃ is a subgroup of the maximal torus normalizer in the simply connected group generated by explicitly defined lifts of the fundamental reflections and the elements of order 2 in the torus [Ti]. In finite group theory, one looks at the study of 2-local subgroups in finite simple groups, especially centralizers of involutions, for occurrences of groups like W̃, up to central extension. For this author, his involvement started with [G73], in which certain linear groups of shape 2_{ε}^{1+2n}.O^{ϵ}(2n, 2) and 4.2^{2n}.Sp(2n, 2) were constructed and analyzed. (A few years later, the article [BRW] came to his attention.) See especially the ideas in [G76, G76b], which this author adapted in 1979 to an action of W̃ on the vector space V_{L} where L is the Leech lattice, and the same idea worked without change for odd determinant lattices. A version of his ideas was reported in [K], but the report seems to be flawed. See also [G81, G82, G86].

The VOA concept came later, in the mid 80s. See the basic reference [FLM] for VOA theory, which gives a treatment of L̂, W̃ and its action on certain VOAs. The action of W̃ defined earlier on the vector space V_{L} turns out to respect the VOA structure on V_{L}. One can describe W̃ as the set of group automorphisms of L̂ which preserve the bilinear form on L, a neat characterization [FLM] on which the Definition 5.7 is based. The object L̂ is not needed to construct W̃ but was used heavily in [FLM]; possibly these authors were the first to construct L̂ and show its relevance to VOA theory. The sign problem for constructing Lie algebras and VOAs had a new solution in the late 70s with the so-called epsilon function [FK, Se] and the epsilon function was later used as a cocycle for creating the group extension L̂. See [G96] and references therein for a general discussion about structure constants and group extensions.

5.2.1 The construction of W̃ when det(L) is odd

For simplicity at first, let us consider the case where L has odd determinant. This is equivalent to the nonsingularity of the 𝔽_{2}-valued bilinear form on L/2L derived from the integer valued one on L by reduction modulo 2. Then n is even and there is an extraspecial 2-group E, unique up to isomorphism, so that the squaring and commutator maps from E to E′ are essentially the 𝔽_{2}-valued quadratic form and bilinear form on L/2L.

Let M be the essentially unique faithful irreducible module for E. The action of E extends to the faithful action of a group B ≅ E.W in such a way that the action of B on E/E′ is identified with the action of W on L/2L. The group W̃ is defined by a pullback diagram:

W̃ ⟶  W
↓  ↓
B/E′ ⟶ W/{±1}

The extension 1 → E′ → E → L/2L → 0 is given by a cocycle ε : L×L → {±1} (and identification of {±1} with E′), which is bilinear as a function and so is constant on pairs of cosets of 2L in L.

The cocycle ε may be used in an obvious way to construct a group L̂, which participates in the short exact sequence 1 → Z → L̂ → L → 0, where Z := ⟨z⟩ ≅ ℤ_{2}, and which maps onto the group E. Since then L̂ also participates in a pullback diagram

L̂ ⟶ L
↓  ↓
E ⟶ L/2L,

the construction of W̃ and the above diagram makes it clear that it acts faithfully as a group of automorphisms of L̂.

5.2.2 The construction of W̃ for arbitrary nonzero values of det(L)

In general, the lattice L will not have odd determinant, so a modification of the above program is needed. One way is to choose a different (nonabelian) finite 2-group to play the role of E, but the modifications to the previous argument which one seems to need are not attractive. Instead, our idea is to embed L suitably in an odd determinant lattice and deduce what we need for L. This is achieved by any equivariant embedding into a lattice of odd determinant, see Theorems 5.3, 5.5 and 5.6. We call such a lattice a 2-regularization of L.

We now carry out the earlier construction for J, a 2-regularization of L. We have our definition of W̃_{J}, the group of isometry automorphisms of Ĵ.

Since L̂ is naturally a subgroup of Ĵ, for which the pullback diagrams are compatible, with compatible epsilon-functions, there is in W_{J}, by definition of 2-regular extension, a subgroup W_{L} which we may identify with Aut(L). Let W̃_{L,J} be the preimage of W_{L} in W̃_{J}. Finally, the group W̃ we seek for L̂ is just the image of W̃_{L,J} in Aut(L̂). The kernel of this map is the normal elementary abelian 2-subgroup of rank equal to rank(J) − rank(L) in W̃_{L,J} which acts trivially on L̂.

5.2.3 Proof that W̃ acts faithfully as a group of VOA automorphisms

We shall define the action of W̃ on V_{L} below. The definition will make it clear that W̃ acts as a group of invertible linear transformations on V_{L} which respects grading.

We have the space V := V_{L} = 𝕊 ⊗ ℂ[L], based on the rank n even integer lattice, L. The lattice L has the group of isometries, W. There is the simple way to define a linear action of W on V by w : p ⊗ e^{x} ↦ w(p) ⊗ e^{w(x)}, but this will not be an automorphism of VOA structures in general. One has to make a modification and replace an action of W with an action of a group W̃ ≅ 2^{n}.W. This construction applies to the case where L is the root lattice of a simple Lie algebra and gives the familiar subgroup of the torus normalizer lifting the Weyl group [Ti]. However, our argument is based only on properties of lattices and finite group theory, and uses no Lie theory. For this author, the ideas came from experience with centralizers of involutions in finite simple groups.

The tensor factor ℂ[L] of V should be thought of as the quotient ℂ[L̂]/(e^{1} + e^{z}) of the group algebra of L̂ by the ideal generated by e^{1} + e^{z}, which has the effect of making multiplication by e^{z} act as −1. Since W̃ acts as automorphisms of L̂, we get an action as automorphisms of its group algebra and the above quotient. The previous “naive” definition of the action of W on the space ℂ[L] would not seem to (except in the degenerate case (L, L) ≤ 2ℤ) give algebra automorphisms of ℂ[L̂]/(e^{1} + e^{z}) when this is identified with ℂ[L] by a linear mapping of the form e^{x} ↦ c_{x}ex̄, where bar indicates the natural map of L̂ onto L, and x ↦ c_{x} is a set function to the nonzero complex numbers. Sometimes, one writes V_{L} = 𝕊 ⊗ ℂ[L] or V_{L} = 𝕊 ⊗ ℂ[L]_{ϵ} to indicate that the second factor is identified linearly with the group algebra ℂ[L] (the subscript ϵ refers to a cocycle giving L̂ from L).

It follows that W̃ has a natural action on the space V_{L} via the above natural action of W on the polynomial algebra 𝕊.

Proposition 5.9 W̃ acts as automorphisms of the VOA V_{L}.

Proof. Clearly, we have a degree preserving group of invertible linear transformations. It preserves the principal Virasoro element which has the form ∑_{i} x_{i}(−1)^{2}, for an orthonormal basis {x_{i} | i = 1, …, rank(L)}, so corresponds to the natural quadratic form on the complex vector space spanned by the lattice. By linearity, the verification that W̃ is a group of automorphisms is reduced to checking preservation of products of the form a_{n}b, where a = p ⊗ e^{x} and b = q ⊗ e^{y}. For such elements, Y (a, z)b has a clear general shape. At each z^{−n−1}, we get a finite sum of monomial expressions, each evaluated with the usual annihilation and creation operations and multiplication in the algebra ℂ[L̂]. Since elements of W̃ act as automorphisms of L̂, it is easy to check that they preserve all the basic compositions involved in such monomials.

This completes our basic theory of W̃.

See Def. 2.6 and Def. 2.4 for the definitions of W_{X}, D_{X} and T, which occur in the next result.

Proposition 5.10 Let X be a lattice frame in L and let F be the associated VF, i.e., \frac{1}{16} x(−1)^{2} ± \frac{1}{4}(e^{x} + e^{−x}), x ∈ X.

(i) In the group W̃ ≅ 2^{n}.W, the stabilizer of F is just 2^{n}.D_{X}.

(ii) In N = N(T) = T W̃, the stabilizer of F has the form N_{F} = (T ∩N_{F})W̃_{X}, so as a group extension looks like (T ∩ N_{F}).W_{X}.

Proof. (i) Obvious from the form of the action of W̃.

(ii) Suppose w ∈ W̃ and the coset T w contains an element g = tw in N_{F}. Such an element takes \frac{1}{16}x(−1)^{2} ± \frac{1}{4}(e^{x} + e^{−x}) to a vector in V_{L} of the form \frac{1}{16}w(x)(−1)^{2} ± \frac{1}{4}(ae^{w(x)} + a^{−1}e^{−w(x)}), for some nonzero scalar a. For this to be in F, we need a = ±1 and the image of w in W must be in W_{X}.

5.3 Nonsplit Extensions

We discuss extension theoretic aspects of a few frame stabilizers.

First take k = 1, for which G ≅ 2^{1+14}.Sym_{16}. This group is a subgroup of H := N_{Aut(V)}(G_{D}) and in the 16-dimensional orthogonal projective representation of H, it corresponds to the determinant 1 subgroup J_{1} of a group J ≅ 2 ≀ Sym_{16} stabilizing a double orthonormal basis D. We claim that J_{1} does not contain a subgroup isomorphic to Sym_{16}, though it does obviously contain a subgroup A isomorphic to Alt_{16}. We may take A as the subgroup of J_{1} stabilizing a orthonormal basis B ⊂ D.

We claim that J_{1} does not contain a subgroup isomorphic to Sym_{16} or a central extension by a group of order 2. Suppose by way of contradiction that S is such a subgroup. The representation of J_{1} is induced and the same is true for S. Let T be a relevant index 16 subgroup of S, stabilizing the 2-set {v, −v} in D. Since the action of S is faithful, the induced representation for T must be faithful on Z(T), which has order 1 or 2. If Z(S) is nontrivial, Z(S) = Z(T) = {±1}. It follows that T splits over Z(T). This means that S splits over Z(S) and so we may assume that S ≅ Sym_{16}. So, the degree 16 representation for S is the standard degree 16 permutation module or that module tensored with the degree 1 sign representation. Neither one gives a map of S to SL(16, ℂ), a contradiction which proves the claim.

Since G/G_{D} ≅ J_{1}/Z(J_{1}) ≅ 2^{14}·Sym_{16}, the claim implies that

1 ⟶ G_{C}/G_{D} ⟶ G/G_{D} ⟶ Sym_{16} ⟶ 1

and

1 ⟶ G_{C} ⟶ G ⟶ Sym_{16} ⟶ 1

are nonsplit. However, G does contain an extension 2·Alt_{16}.

For k = 5, G does not split over G_{C}. If it did, the group denoted A in the proof of Th. 4.21 would be elementary abelian, a possibility which was disproved in that discussion.

5.4 Orbits of parabolic subgroups of orthogonal groups

Assume 𝔽 is a perfect field of characteristic 2. We set up notation to discuss certain orbits of a parabolic subgroup in O^{+}(2n, 𝔽).

Let W be a 2n dimensional vector space over the perfect field 𝔽 of characteristic 2 and suppose that W has a nonsingular quadratic form Q with maximal Witt index, i.e., there are totally singular n-dimensional subspaces J, K so that W = J ⊕ K as vector spaces. Denote by ( . , .) the associated bilinear form: Q(x + y) = Q(x) + Q(y) + (x, y), for all x, y ∈ W.

Let P be the subgroup of the isometry group Aut(Q) which stabilizes J. Thus, P has the form 𝔽^{\binom{n}{2}}:GL(n, 𝔽) and is a maximal parabolic of Ω^{+}(2n, 𝔽).

Let Σ denote the set of n-dimensional subspaces which are totally singular with respect to the bilinear form but for which the set of singular vectors and zero forms a codimension 1 subspace. This is a nonempty set, for if v ∈ W is nonsingular, 𝔽v + [J ∩ v^{⊥}] ∈ Σ. Since 𝔽 is perfect, any two members of Σ are isometric.

Remark 5.11 We observe that if A ∈ Σ and S_{1}, S_{2} are any equal dimensional subspaces of S, the codimension 1 subspace in A consisting of zero and all singular vectors, then A/S_{1} and A/S_{2} are isometric because the isometry group Aut(A) ≅ AGL(S) induces GL(S) on S.

Definition 5.12 For A ∈ Σ, we define the J-indicator j(A) = dim(A ∩ J). This is an integer from 0 to n − 1, and all these values occur.

Proposition 5.13 Two members of Σ are in the same P orbit if and only if they have the same J-indicator.

Proof. One direction is obvious, so let us assume that A, B ∈ Σ both have the same indicator d. The images of A, B in W/J are both (n − d)-dimensional, so we may assume that the images are equal since P induces the full general linear group on W/J. It follows that R := A ∩ J = B ∩ J is the radical of J + A = J + B.

Let A_{1} complement R in A and B_{1} complement R in B. Using the Remark, we know that A_{1} and B_{1} are isometric, say by an isometry ψ, which corresponds elements of A_{1} and B_{1} which are congruent modulo J: ψ is just the composite of isometries A_{1} ≅ A/R ≅ B/R ≅ B_{1}, where the middle isometry is based on congruence modulo J.

We now define an isomorphism of J + A = J + B with itself by φ : u + a ↦ u + ψ(a), for a ∈ A_{1}. We verify that this is an isometry by using a − ψ(a) ∈ J: Q(u + a) = Q(u) + Q(a) + (u, a) = Q(u) + Q(ψ(a)) + (u, ψ(a)) = Q(u + ψ(a)), where Q is our quadratic form. Observe that this map takes A = A_{1} + R to B = B_{1} + R.

Now, by Witt’s theorem, φ extends to an isometry of W. Since it obviously fixes J, this extension lies in P. Therefore, A and B lie in a single P -orbit.

Proposition 5.14 Let A ∈ Σ, let H its stabilizer inside P and let U be the unipotent radical of H (when 𝔽 is a finite field of characteristic 2, U = O_{2}(H)). With j = j(A), the J-indicator, one has H/U ≅ GL(j, 𝔽) × GL(n − j − 1, 𝔽).

Proof. This is an exercise with actions of classical groups. We consider 0 ≤ R := A ∩ J ≤ T := A + J = R^{⊥} ≤ W. Then R is the radical of T and T/R is a nonsingular space of maximal Witt index. Note that in T/R, J/R and A/R are maximal isotropic with respect to the bilinear form and give a direct sum decomposition of T/R as a vector space. Let J′ be a complement in J to R and let A′ be a complement in A to R. Then M := J′ + A′ is a nonsingular subspace whose orthogonal complement contains R.

It follows that the action of H on M is the direct sums of the actions on J′ and A′, which are dual. The action could be as large as GL(n − j, 𝔽) but is in fact just AGL(n − j − 1, 𝔽) since A′ is not totally singular with respect to the quadratic form. (More precisely, H stabilizes S = {x ∈ A | Q(x) = 0} and is trivial on A/S.)

Note that the actions of H on R and W/R are dual, and are each GL(j, 𝔽). This can be seen with action of a subgroup of H on M^{⊥}.

It is an exercise with linear algebra to work out the structure of U. For brevity, we record only the cases j = 0 and 4.

Corollary 5.15 In the notation of the previous result, take n = 5 and 𝔽 = 𝔽_{2}. Then GL(4, 2) occurs as a quotient group of H for just j = 0 and j = 4. If j = 0, |U| = 2^{4}. If j = 4, |U| = 2^{14}.

Proof. The previous result allows only j = 0 and j = 4. If j = 0, W = J ⊕ A implies that H acts faithfully on both factors. The case j = 4 is an exercise.

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