Pieces of Eight.

Semiselfdual lattices and a new foundation for the theory of Conway and Mathieu groups.

revised (version 11 July, 1997)

Robert L. Griess, Jr. Department of Mathematics

University of Michigan Ann Arbor, MI 48109 U.S.A.

Proposed running head: Pieces of Eight

Address for author’s proofs is the same as on page 1:

Robert L. Griess, Jr. Department of Mathematics University of Michigan Ann Arbor, MI 48109 U.S.A.

eaddress: rlg@math.lsa.umich.edu

A list of symbols is unnecessary since the symbols in this article (done in AMS-TeX) come from well-known TeX fonts.

Table of Contents.

1. Introduction

2. Basic Notations and Definitions.

3. The Automorphism group of the Leech lattice.

4. The Mathieu group, with the Golay code (finally!)

5. Other consequences for the Leech lattice and its automorphism group.

Appendices: Background.

Appendix A. Elementary lattice theory.

Appendix B. Orthogonal groups in characteristic 2.

Appendix C. Extraspecial p-Groups.

1. Introduction.

We give a new and easy construction of the largest groups of Mathieu and Conway, as the full automorphism groups of the Golay code and Leech lattice, respectively. In addition we get a new uniqueness proof for the Leech lattice and the Golay code. The main technique is intensive use of semiselfdual sublattices: instead of rank 1 lattices as the basis for coordinate concepts, we use scaled versions of L_{E_{8}}, the E_{8} root lattice; the semiselfdual lattices we use most of the time are isometric to \sqrt{2}L_{E_{8}}. While it has been recognized for decades that one can use copies of E_{8} to describe the Leech lattice (see [19] [25][26] and [23]), our uses of it to create the theory of Conway and Mathieu groups are new. Using these “smarter coordinates”, properties of their automorphism groups and appropriate uniqueness theorems, we get a more compact foundation of this theory (see Section 3, esp. (3.7) and (3.19)), notably in avoidance of counting arguments, the construction of explicit extra automorphisms and dependence on the usual theory of the Golay code.

The logical order of our steps differs from those of earlier versions of this theory. We first take a Leech lattice (any rank 24 even integral unimodular lattice without vectors of squared length 2; no explicit description is required) then deduce its uniqueness and properties of its automorphism group, the large Conway group. We use our uniqueness theorems to get transitivity of Λ_{2}, Λ_{3}, Λ_{4} and other configurations of vectors and sublattices without listing members of these sets or even knowing |Aut(Λ)|, in contrast with the classic approaches (some of our transitivity results may not be in the literature). Next, we deduce existence and uniqueness of the Golay code, then existence and order of the Mathieu group, M_{24}. The older versions of this theory depend on existence and uniqueness of the Golay code to prove existence and uniqueness of the Leech lattice but ours does not. Simple observations of the Golay code then give immediate results about permutation representations of M_{24}. Only minimal examination of particular codes is ever done, in contrast with older approaches [5][8][1][14]. Our logical sequence of first obtaining the Conway group, then the Mathieu groups, is in defiance of the classical theory for these groups.

The construction of the Leech lattice we give (3.3.i) is not really new; ( see [19][25][26]), but our treatment of uniqueness and of the automorphism group is different, notably in avoidance of Conway’s characterization [6] and avoidance of displaying explicit “extra automorphisms” with respect to a frame basis [5][7][14]. Furthermore, our foundation of the theory of Conway and Mathieu groups is “elementary”, if one takes the structure of the E_{8} lattice, its automorphism group and basic lattice management (2.1) for granted. We emphasize that the present article contains a complete proof of this foundation (modulo standard background material about lattices and finite groups in Section 2 and the Appendices), a fact which should be taken into account in making comparisons with other versions of this theory. The very nature of our basic theory gives a host of uniqueness and transitivity results for configurations in the Leech lattice. Such results occur relatively late in the usual presentations and can involve many special arguments.

Our characterization of the Leech lattice is a logical improvement over that of [6], which depends on the characterization of the Golay code because we deduce its characterization; so, we offer a reason to reconsider Venkov’s opinion: “Conway’s proof is very pretty and can hardly be improved.” [27], (or [8], p.438). We avoid many special counting arguments in the traditional theory of the Golay code and Leech lattice. Our break with the past is not complete since we still rely on the theta series of an E_{8} lattice and a Leech lattice and in (3.4), (3.7), (3.17), (3.18), we have to use a few elementary properties of a code associated to a frame (A.4)(A.8) (this code turns out to be the Golay code, but we do not need to quote its characterization). Posssibly, reliance on the theta series can be lessened; for instance, one would like a direct, elementary proof of the fact that in a Leech lattice, Λ, the orthogonal of a sublattice isometric to \sqrt{2}L_{E_{8}} contains a copy of \sqrt{2}L_{E_{8}} (and even \sqrt{2}L_{E_{8}} ⊥ \sqrt{2}L_{E_{8}}). Even the weaker statement that a vector x in the Leech lattice contains a copy of \sqrt{2}L_{E_{8}} in the sublattice \{ y ∈ Λ | (x, y) ∈ 2ℤ\} (the “annihilator of x mod 2”) would be useful (and seems hard to prove without using theta series).

We introduce semiselfdual involutions in (2.5), a concept with potential for wider applications. In case one wishes to follow the spirit of an earlier construction, one can choose an extra automorphism from our family of semiselfdual involutions (2.5)(3.6)(5.3).

We thank George Glauberman and Dan Frohardt for useful comments. For financial support during work on this article, we acknowledge NSF Grant DMS 9623038 and University of Michigan Faculty Recognition Grant (1993-96).

2. Basic Notations and Definitions.

Throughout this article, (·, ·) denotes a positive definite bilinear form on a lattice of finite rank or a finite dimensional rational vector space. If L is a lattice, we write ℚL for the ambient rational vector space, which may be identified with ℚ ⊗ L.

(2.1) Definition. We recall a few things about lattices and free abelian groups. The invariants or invariant factors of an integral lattice, L, are the invariant factors of the integral matrix (x_{i}, x_{j}), where x_{1}, …, x_{n} is a basis. For example, L_{E_{8}} has invariants (1^{8}) and \sqrt{2}L_{E_{8}} has invariants (2^{8}). All our lattices are positive definite, so invariants are nonzero; N.B.: we may list only the invariants greater than 1. The determinant of L (not necessarily integral) is the determinant of the matrix ((x_{i}, x_{j})). If M is a sublattice of finite index, det(M) = det(L)|L : M|^{2}. If L is a lattice in the rational vector space V (L is not necessarily integral and does not necessarily span V), the dual of L is L^{∗} := {x ∈ ℚL | (L, x) ≤ ℤ}. Then L^{∗} is a lattice, called the dual lattice; if L is integral, L^{∗} need not be; we have (det L^{∗})^{−1} = det L. If L is integral, L^{∗}/L decomposes into a direct sum of cyclic groups whose orders are the invariants. A root is a lattice vector of squared length 2. The radical mod p of L is the sublattice {x ∈ L | (x, L) ≤ pℤ} = pL^{∗} ∩ L. □

(2.2) Definition. We use the abbreviations EL, EUL for an even lattice, respectively, even unimodular lattice. An integral lattice L is called a scaled unimodular lattice (SUL) iff there is a unimodular lattice U and a scalar s so that L ≅ sU. In other words, there is an isomorphism of abelian groups θ : L ≅ U so that for all x, y ∈ L, (x, y)_{L} = s^{2}(x^{θ}, y^{θ})_{U}. We call s the scale or scale factor of L; clearly, s is the square root of an integer, and we take it to be positive. □

(2.3) Lemma. (i) If L is integral and unimodular and M is a direct summand (as abelian groups), the natural map L → Hom(M, ℤ) is onto; the nontrivial invariants of M and M^{⊥} ∩ L are the same.

(ii) If M is a sublattice of the integral lattice L, and M is SUL with scale factor s, then {x ∈ L | (x, M) ≤ s^{2}ℤ} = [M^{⊥} ∩ L] ⊥ M.

(2.4) Definition. Call an integral lattice L semiselfdual (SSD) iff 2L^{∗} ≤ L; in this case, 2L^{∗} = L ∩ 2L^{∗} is the radical mod 2 (2.1). □

The following idea seems to be new.

(2.5) Proposition. (Semiselfdual involutions.) If L is an integral lattice and M a semiselfdual lattice (2.4) contained in L, then the linear map on the ambient rational vector space which is −1 on M and 1 on M^{⊥} is an automorphism of the lattice L.

Proof. Let t be the linear map on V := ℚL. Let V^{±} denote the eigenspaces for ±1. Then t acts on the abelian group V/M ≅ V^{+} ⊕ (V^{−}/M) by +1 on the first summand and −1 on the second. The second summand contains the subgroup M^{∗}/M, left invariant by t, and on it t acts as +1 since M is SSD. Since L lies between M and V^{+} ⊕ M^{∗}, L is t-invariant since t acts as +1 on V^{+} ⊕ (M^{∗}/M). □

(2.6) Lemma. If the lattice L is even and integral, has no roots and M ≅ \sqrt{2}L_{E_{8}} is a sublattice, it is a direct summand of L, as an abelian group. Its image in L/2L is a totally singular subspace, of dimension 8.

Proof. Since det(M) = 2^{8}, the only torsion in L/M occurs for the prime 2. If M is not a summand, there is an element u ∈ L so that u ∈ L \ M and 2u ∈ M. Then we may assume that 2u is a shortest element in the coset 2u + 2M, whence (2u, 2u) = 4 or 8 (A.2). Since L is an even unimodular lattice, we get (u, u) = 2, a contradiction since L has no roots. The last statement follows easily. □

(2.7) Lemma. Suppose that the lattice L is generated by the subset S. Then L is integral if (S, S) ⊆ ℤ and it is even if it is integral and (x, x) ∈ 2ℤ for all x ∈ S.

(2.8) Remark. For basic coding concepts, see [20], [14]. Binary codewords may be interpreted as subsets of the alphabet, the index set for coordinates. The only special codes we refer to will be the (extended) Hamming code (A.3), with parameters (length, dimension, minimum weight) [8,4,4], and the binary Golay code, with parameters [24,12,8]. Existence and uniqueness proofs for the Hamming code are elementary exercises, but not so for the Golay code. We will study a Golay code, any binary code with parameters [24,12,8], and deduce its existence and uniqueness in (4.2). □

(2.9) Proposition. (Tower of Scaled E_{8}-lattices.) Let L ≅ L_{E_{8}} and M ≤ L, M ≅ \sqrt{2}L_{E_{8}}. Let G := Aut(L) ≅ W_{E_{8}} and let P := Stab_{G}(M).

(i) P ≅ 2_{+}^{1+6}GL(4, 2).

(ii) O_{2}(P)/⟨−1⟩ operates regularly on 𝒦, the set of sublattices K of L such that K ≅ \sqrt{2}L_{E_{8}} and K/2L complements M/2L in L/2L. If K ∈ 𝒦, Stab_{G}(M) ∩ Stab_{G}(K) = Stab_{P} (K) ≅ 2^{.}GL(4, 2) (nonsplit).

Proof. (i) and (ii) follow from basic properties of O^{+}(8, 2); see (A.3) and Appendix B; also (3.1). □

3. The automorphism group of the Leech lattice.

Thoughout this article, we let Λ be a Leech lattice, that is, an EUL lattice in dimension 24 with no vectors of squared length 2. Such lattices are easily seen to exist. The standard description involves the Golay code (A.4), but this can be avoided; we give a version in (3.2), (3.3.i); this is not new: cf. [19][25][26]. Assuming only existence of Λ, we prove its uniqueness (3.7) and get significant information about its automorphism group.

(3.1) Lemma. Let s be the positive square root of a positive integer. (i) Given a lattice L ≅ sL_{E_{8}}, there are 135 sublattices M of L such that M ≅ \sqrt{2}sL_{E_{8}}; if M is such, M contains 2L and the invariant factors of M are ((2s^{2})^{8}). The set of such M forms an orbit under Aut(L). For the natural nonsingular bilinear form on L/2L (which takes the pair (x+2L, y +2L) to s^{−2}(x, y)+2ℤ), M/2L may be interpreted as a maximal totally singular subspace.

(ii) Given a lattice L ≅ sL_{E_{8}}, there are 135 overlattices M of L such that M ≅ \frac{s}{\sqrt{2}}L_{E_{8}}; the set of such forms an orbit under Aut(L); if M is such and s^{2} ∈ 2ℤ, the invariant factors of M are ((\frac{1}{2}s^{2})^{8}).

Proof. Write f for the given bilinear form: f(x, y) = (x, y). Without loss, we may assume s = 1.

(i) If M is between L and 2L and corresponds to a maximal totally singular subspace, we deduce det(M) = 2^{8} from |L : M| = 16. Since (x, x) ∈ 4ℤ, for all x ∈ M, we get (M, M) ≤ 2ℤ and so M is an even unimodular integral lattice for the bilinear form \frac{1}{2}f. From (A.9), we get M ≅ L_{E_{8}}.

Let M be any such a sublattice of L. Then M^{∗}/M ≅ 2^{8} and L/M is a subspace. Then (M^{∗}, 2f) has determinant 1 and so in the nonsingular space M^{∗}/M with quadratic form x + M ↦ f(x, x) (mod 2), L maps to a totally singular subspace. Since det(L) = 1 and det(M^{∗}) = 2^{−8}, M^{∗}/L ≅ 2^{4}. This forces L to contain 2M^{∗} = M and L/2M^{∗} to be a maximal totally singular subspace of M^{∗}/2M^{∗}. Since the action of Aut(L) on L_{E_{8}}/2L_{E_{8}} is that of O^{+}(8, 2), we have transitivity.

(ii) Take dual lattices and apply (i). □

We next give results, (3.3.ii) and (3.7), which characterize Leech lattices by certain internal data. Note that we ignore 1s when referring to the invariants of a lattice (2.1), e.g., T below has rank 16 and invariants (2^{8}) (the full set of 16 invariants is (2^{8}1^{8}) but we drop the 1s (2.1)).

(3.2) Definition. Consider the 6-tuple (M_{1}, M_{2}, M_{3}, θ, N_{1}, ζ), which satisfies the following conditions:

The M_{i} ≅ \sqrt{2}L_{E_{8}}, for i = 1, 2, 3, are pairwise orthogonal sublattices in ℚ^{24}; for each i, we let p_{i} be the orthogonal projection to ℚM_{i}; W_{i} := Aut(M_{i}).

θ : M_{1} → M_{2} is an isometry (this is independent of the identifications fixed in the previous line);

M_{12} := M_{12,θ} := {(x, x^{θ}) | x ∈ M_{1}} ≅ 2L_{E_{8}};

N_{1} is a sublattice of M_{1}; N_{1} ≅ 2L_{E_{8}};
N_{2} := N_{2,θ} := N_{1}^{θ}; N_{12} := N_{12,θ} := \{(x, x^{θ}) | x ∈ N_{1}\} ≅ 2\sqrt{2}L_{E_{8}};

T := M_{1} + M_{2} + \frac{1}{2}N_{12}; its invariants are (2^{8}) and the radical mod 2 (2.1) of T is U := N_{1} +N_{2} +M_{12}; U/2T ≅ 2^{8} and U ≅ \sqrt{2}T (to see this, just replace M_{i}, N_{i} by N_{i}, 2M_{i}, respectively, in the definition of T and (3.1.ii)); we remark that T^{∗} = \frac{1}{2}U ≅ \frac{1}{\sqrt{2}}T);

ζ : T^{∗}/T → M_{3}^{∗}/M_{3} is an isometry with respect to the nonsingular quadratic forms x + T ↦ (x, x) + 2ℤ and x + M_{3} ↦ (x, x) + 2ℤ;

L is the sublattice between T ⊥ M_{3} and T^{∗} ⊥ M_{3}^{∗} = \frac{1}{2}U ⊥ \frac{1}{2}M_{3} which is diagonal with respect to the isometry ζ (note: for all i, L^{p_{i}} = M_{i}^{∗} ≅ \frac{1}{\sqrt{2}}L_{E_{8}});

Call such an 6-tuple a Leech 6-tuple. An ordered triple of pairwise orthogonal rank 8 lattices M_{i} ≅ \sqrt{2}L_{E_{8}}, for i = 1, 2, 3, in ℚ^{24} is called a Leech trio. We extend above maps on lattices to the ambient rational vector spaces. □

(3.3) Theorem. (i) L is a Leech lattice.

(ii) Any Leech lattice containing a Leech lattice containing a Leech trio comes from a Leech 6-tuple as above.

(iii) The set of Leech 6-tuples which extends a given Leech trio forms a single orbit under the natural action of Aut(M_{1}) × Aut(M_{2}) × Aut(M_{3}). A stablizer has the shape [2 × 2^{1+6}]GL(4, 2); the first factor is ⟨−1⟩ and the second factor is a diagonal embedding of the stabilizer of a maximal totally singular subspace for the action of Aut(M_{i}) on M_{i}/2M_{i}. Its projection to Aut(M_{i}) is 2_{+}^{1+6}GL(4, 2), for i = 1, 2, and its projection to Aut(M_{3}) is 2^{.}GL(4, 2).

Proof. We begin by observing that T is the sublattice T := M_{1} + M_{2} + \frac{1}{2}N_{12} of \frac{1}{2}N_{1} ⊥ \frac{1}{2}N_{2} ≅ L_{E_{8}} ⊥ L_{E_{8}}. Since M_{i} ≅ \sqrt{2}L_{E_{8}} and T/M_{1} ⊥ M_{2} is diagonal in M_{1}^{∗}/M_{1} ⊕ M_{2}^{∗}/M_{2}, it is clear that the minimum squared length in T is 4, so T has no roots.

(i) Use (2.1) (applied to T ⊥ M_{3} ≤ L ≤ T^{∗} ⊥ M_{3}^{∗}) to get that L is unimodular. Since T^{∗} ≅ \frac{1}{\sqrt{2}}T, every vector has squared length an integer, even though T^{∗} is not an integral lattice. Similarly, M_{3}^{∗} ≅ \frac{1}{\sqrt{2}}L_{E_{8}}, whence every vector has squared length an integer. We must show that L has no roots.

Suppose that r ∈ L is a root. Let r_{i} := r^{p_{i}}, for i = 1, 2, 3. Then n_{i} := (r_{i}, r_{i}) ∈ ℤ and n_{1} + n_{2} + n_{3} = 2. For all i, n_{i} < 2 (else r = r_{i} ∈ M_{i}^{∗} ∩ L = M_{i}, which has no roots). So, n_{i} ∈ {0, 1}, for all i.

Suppose that n_{3} = 1 and {n_{1}, n_{2}} = {0, 1}. The latter forces T to have a root since r_{1}, r_{2} ∈ L^{p_{1}+p_{2}} = T^{∗} ≅ \frac{1}{\sqrt{2}}T a contradiction to the first paragraph.

Suppose that n_{3} = 0. Then r ∈ M_{3}^{⊥} ∩ L = T, again a contradiction to the first paragraph.

(ii) Let L be a Leech lattice. We are given a trio and identifications M_{1} ≅ M_{2} ≅ M_{3}. Define J := M_{3}^{⊥} ∩ L; then det(J) = det(M_{3}) = 2^{8} (2.1). For i = 1, 2, M_{i} ≤ J ∩ ℚM_{i} ≤ M_{i}^{∗}. Since L has no roots and M_{i} ≅ \sqrt{2}L_{E_{8}}, we have M_{i} = J ∩ ℚM_{i} = L ∩ ℚM_{i} (2.6). Since the invariants for K := M_{1} ⊥ M_{2} are (2^{16}), J lies between K and \frac{1}{2}K, and it follows that J has invariants (2^{8}) and J/K corresponds to a 4-dimensional subspace of K^{∗}/K ≅ 2^{8} which is diagonally embedded with respect to the decomposition K^{∗}/K ≅ M_{1}^{∗}/M_{1} × M_{2}^{∗}/M_{2}. We claim that J^{p_{i}} ≅ L_{E_{8}} for i = 1, 2, which is the same as saying (3.1) that for the natural nonsingular quadratic form on M_{i}^{∗}/M_{i}, J^{p_{i}} corresponds to a maximal totally isotropic subspace. If not, there is x ∈ J which maps to a nonsingular vector of M_{i}^{∗}/M_{i}; write x_{j} := x^{p_{j}} for j = 1, 2. Then (x_{i}, x_{i}) is an odd integer, which means that both x_{1} and x_{2} correspond to nonsingular vectors since (x, x) ∈ 2ℤ. Since both cosets x_{j} + M_{j} contain vectors of squared length 1 (A.2), we deduce that J contains a root, a contradiction. So, J^{p_{j}} ≅ L_{E_{8}}. Define N_{j} := 2J^{p_{j}} ≤ M_{j}, for j = 1, 2. It follows that J = M_{1} + M_{2} + D, where 2D is a diagonal subgroup of N_{1} ⊥ N_{2} defined by an isometry, θ, which carries M_{1} to M_{2} and N_{1} to N_{2} (use (3.1.ii)). We have D ≅ \sqrt{2}L_{E_{8}}.

Define N := J ⊥ M_{3}; det(N) = 2^{16}. Since M_{3} is a direct summand of L (2.6) and its invariants are (2^{8}), the same is true for M_{3}^{⊥} ∩ L. Since J ≤ M_{3}^{⊥} ∩ L, J = M_{3}^{⊥} ∩ L.

Since the invariants of M_{1} ⊥ M_{2} ⊥ M_{3} are (2^{24}) and those of J are (2^{8}), L/N corresponds to a subgroup of N^{∗}/N ≅ 2^{16} of dimension 8 which is diagonally embedded with respect to the decomposition N^{∗}/N ≅ J^{∗}/J × M_{3}^{∗}/M_{3}. In fact, with respect to the natural nonsingular bilinear form on N^{∗}/N, defined by x + N ↦ (x, x) + 2ℤ, L/N is diagonally embedded with respect to an isometry ζ : J^{∗}/J → M_{3}^{∗}/M_{3} since L is even.

(iii) We consider the proof of (ii). Fix identifications M_{1} ≅ M_{2} ≅ M_{3} and corresponding copies of W_{i} := Aut(M_{i}) ≅ W_{E_{8}}. The choices for θ form an orbit under the natural action of W_{1} or of W_{2}; its stabilizer is a diagonal subgroup W_{12} of W_{1} × W_{2}.

The subgroup S_{12} of W_{12} stabilizing the sublattice T is isomorphic to a subgroup of W_{E_{8}} of shape 2^{1+6}GL(4, 2), (A.3); S_{12} acts on T^{∗}/T as GL(4, 2) (since [M_{i}, O_{2}(S_{12})] = N_{i} and [N_{i}, O_{2}(S_{12})] = 2M_{i}, for i = 1, 2; see (3.2)). The set of isometries T^{∗}/T → M_{3}^{∗}/M_{3} forms a single orbit under the natural action of W_{3}. It follows that the isometry of (ii) has stabilizer S contained in S_{12} × W_{3} and is isomorphic to [2 × 2_{+}^{1+6}]GL(4, 2); the projection of S to W_{1} × W_{2} is S_{12} and the projection to W_{3} is isomorphic to 2.GL(4, 2) since . □

(3.4) Corollary. If L is a Leech lattice and M is a sublattice isometric to \sqrt{2}L_{E_{8}}, then isomorphism type of M^{⊥} is determined.

Proof. Since L is unimodular and M is a direct summand, det(M^{⊥}) = det(M) = 2^{8}. We finish with (3.3.ii) if we find a sublattice of M^{⊥} isometric to \sqrt{2}L_{E_{8}} ⊥ \sqrt{2}L_{E_{8}}. This follows from (A.6,7,8). □

(3.5) Proposition. Choose a quadratic form so that ℚ^{24} contains a Leech lattice, day Λ. Then ℚ^{24} = ℚΛ and O(ℚ^{24}) acts transitively on the following sets:

(i) pairs (L, M), where L is a Leech lattice in ℚΛ and M is a sublattice of L isometric to \sqrt{2}L_{E_{8}};

(ii) triples (L, M_{1}, M_{2}), where L is a Leech lattice in ℚΛ and M_{1}, M_{2} is a pair of orthogonal sublattices, each isometric to \sqrt{2}L_{E_{8}};

(iii) quadruples (L, M_{1}, M_{2}, M_{3}), where L is a Leech lattice in ℚΛ and (M_{1}, M_{2}, M_{3}) is a Leech trio in L.

Proof. (iii) Use (3.3.iii).

(i) A proof may be obtained from the ideas in the proofs of (3.3.ii,iii) and (3.4).

(ii) It suffices, by (iii), to prove that this ordered pair is part of a Leech trio. Define Q := M_{1} ⊥ M_{2}, M_{3} := Q^{⊥} ∩L. Since the invariants of Q are (2^{16}), the invariants of both M_{3} and R := M_{3}^{⊥} ∩L are (2^{a}), for some a ≤ 16. Since rank(M_{3}) = 8, a ≤ 8. We have |R : Q| = 2^{8−a/2} (2.1). If a < 8, then for some k ∈ {1, 2}, R_{k} := R ∩ ℚM_{k} > M_{k}. Since L is an even lattice, R_{k}/M_{k} must be a totally singular subspace with respect to the natural nonsingular bilinear form on M_{k}^{∗}/M_{k}; by (A.2.i) (applied to M_{k}^{∗} ≅ \frac{1}{\sqrt{2}}L_{E_{8}}), R_{k} \ M_{k}, hence L, contains a root, a contradiction. So, a = 8, whence M_{3} = 2M_{3}^{∗} is doubly even and so M_{3} ≅ \sqrt{2}L_{E_{8}} (A.9). □

(3.6) Corollary. Aut(Λ) acts transitively on the set of sublattices isometric to \sqrt{2}L_{E_{8}}. We also have a conjugacy class of involutions in Aut(Λ), the SSD involutions (2.5) associated to these sublattices.

(3.7) Corollary. Any two Leech lattices are isometric.

Proof. Any Leech lattice has a Leech trio (A.7). Now use (3.5). □

At this point, we know little about Aut(L) beyond some transitivity properties. We need to study the sublattices which occur as M^{⊥}, for a sublattice M ≅ \sqrt{2}L_{E_{8}}.

(3.8) Theorem. (i) In a Leech lattice, Λ, let M be a sublattice isometric to \sqrt{2}L_{E_{8}} and set T := M^{⊥}. Then Aut(T) is an extension 2^{1+8}Ω^{+}(8, 2).

(ii) The noncentral involutions of Aut(T) which lie in O_{2}(Aut(T)) form a single Aut(T) conjugacy class. For such an involution, the sum of the fixed point sublattice and the negated sublattice is isometric to \sqrt{2}L_{E_{8}} ⊥ \sqrt{2}L_{E_{8}}.

(iii) There is a bijection between the involutions of (ii) and ordered pairs of orthogonal sublattices of T, each isometric to \sqrt{2}L_{E_{8}}. If M_{1} ≅ M_{2} ≅ \sqrt{2}L_{E_{8}} are orthogonal sublattices of T, then Stab_{Aut(T)}(M_{1}) ∩ Stab_{Aut(T)}(M_{2}) ≅ 2 × 2^{1+6}GL(4, 2) and the image of this group in Aut(M_{i}), for i = 1, 2, is of the form 2^{1+6}GL(4, 2).

(iv) The unordered pairs of sublattices as in (ii) is in bijection with the set of maximal totally singular subspaces of T/2T (all of which have dimension 12 and contain the 8-dimensional radical).

(v) T ∩ 2Λ = 2M_{1} + 2M_{2} + N_{12} = 2T and T/T ∩ 2Λ ≅ 2^{16}; also T + M + 2Λ = T + 2Λ.

(vi) The actions of Stab(M) on T^{∗}/T and M^{∗}/M = \frac{1}{2}M/M ≅ M/2M are equivalent and may be identified with the action of Ω^{+}(8, 2) on the space 𝔽_{2}^{8} stabilizing a nondegenerate quadratic form. (We remark that these actions are equivalent to the irreducible action on U/U ∩ 2Λ ≅ U + 2Λ/2Λ but not to T/U, where U is the radical modulo 2 of T. )

Proof. First, we show that B := {a ∈ Aut(T) | [T, a] ≤ U} ≅ 2_{+}^{1+8}.

Take the sublattice Q := M_{1} ⊥ M_{2} of T, with M_{j} ≅ \sqrt{2}L_{E_{8}}, for j = 1, 2, as in (3.2). Note that U = N_{1} + N_{2} + M_{12}. Let A be the stabilizer in Aut(T) of Q. Note that A lies in a wreath product [Aut(M_{1}) × Aut(M_{2})]2 and that A has the form [2_{+}^{1+6} × 2_{+}^{1+6}].[GL(4, 2) × 2]; see (A.3) and consider the subgroup of the above wreath product which stabilizes N_{1} and N_{2}. In more detail, let R_{i} ≅ 2_{+}^{1+6}, for i = 1, 2 be the normal subgroup of Stab_{Aut(M_{i})}(N_{i}) as in (A.3); then B ≤ A has the form R_{12} ∘ E, a central product, where R_{12} is diagonally embedded in R_{1} × R_{2}, E is dihedral of order 8, E ∩ O_{2}(A) ≅ 2 × 2 and E contains an involution which interchanges R_{1} and R_{2} under conjugation. The statement about B follows.

Since B acts absolutely irreducibly on T, C(B) consists of scalar matrices and so the quotient Aut(T)/BC(B) embeds in Out(B) ≅ O^{+}(8, 2) (C.3). Our subgroup A contains B and maps onto a parabolic subgroup P of Out(B)′ of the form 2^{6}: GL(4, 2).

We may do the above for any sublattice of T which is isometric to M_{1} ⊥ M_{2}. One such sublattice is \frac{1}{2}N_{12;θ} ⊥ \frac{1}{2}N_{12;−θ}, and we thereby get a subgroup isomorphic to P and distinct from P. Since P is a maximal parabolic subgroup of Out(B)′ = Ω^{+}(8, 2) [4], these two subgroups generate Out(B)′. It follows that Aut(T)/B ≅ O^{+}(8, 2) or Ω^{+}(8, 2).

There are several ways to see that Aut(T)/B ≅ O̸^{+}(8, 2): (a) a subgroup H of GL(ℂT) which contains B as a normal subgroup and with quotient O^{+}(8, 2) has the property that certain elements of H \ H′ have traces of the form ε2^{c/2}, where ε is a root of unity and c is an odd integer (C.4); since the representation of Aut(T) on T is rational, this does not happen; (b) study the centralizer in A of B_{1}, a 2_{+}^{1+6} subgroup of B; C(B_{1}) lies in GL(2, ℚ) and contains a copy of C_{B}(B_{1}) ≅ Dih_{8}, which is a maximal finite 2-subgroup in GL(2, ℚ); (c) in case Aut(T)/B were O^{+}(8, 2), we would get a contradiction when we examine the structure of the frame group; see (A.5).

We now have (i). For (ii), it is clear since Aut(T) has index 2 in a holomorph of an extraspecial group (Appendix C), that the noncentral involutions in O_{2}(Aut(T)) form a single conjugacy class. Since two such involutions are the central involutions of R_{1} and R_{2}, the connection with the stated sublattices follows. For (iii), note that such a decompostion (with ordered summands) leads to a SSD involution t (−1 on M_{1}, 1 on M_{2}) which preserves T. It remains to show that t satisfies [T, t] ≤ U. It is clear from the proof of (3.3.ii) that there are appropriate sublattices N_{k} ≤ M_{k} and exactly two isometries ±θ : M_{1} → M_{2} so that T = M_{1} + M_{2} + \frac{1}{2}N_{12} as in (3.2). Then we get [T, t] = N_{1} ≤ U = N_{1} + N_{2} + M_{12}, as required.

For (iv), note that Aut(T) induces Ω^{+}(8, 2) on T/U ≅ 2^{8} and that M_{1} ⊥ M_{2}/U represents a maximal totally singular subspace.

The first part of (v) follows since T is a direct summand of Λ. From (2.3), we get that T + M ≥ 2Λ and T + M/2Λ has codimension 8 in Λ/2Λ. Since the image of T in Λ/2Λ has dimension 16, T + 2Λ = T + M + 2Λ.

For the first part of (vi), just note that Λ projects onto each of M^{∗}/M and T^{∗}/T. Since T^{∗} = \frac{1}{2}U and U ∩ 2Λ = T ∩ Λ = 2T, U/2T ≅ T^{∗}/T as modules for Stab(M); these modules are self-dual. Clearly, Λ/2Λ has a composition series with factors U/2T ≅ U +2Λ/2Λ, T/U (T +2Λ/2Λ is the annihilator of U + 2Λ/2Λ) and finally Λ/T (isomorphic to Λ/2Λ modulo the annihilator of U + 2Λ/2Λ). The first and third are dual, hence isomorphic. The middle factor turns out to be not isomorphic to these, but the proof is perhaps not easy (this fact is not a necessary part of our theory; anyway, here is a nonelementary proof: we take an element x of order 3 in Stab(M) for which 1 occurs as an eigenvalue in the first and third factor with multiplicity 6; so, on Λ, the multiplicity of 1 is at least 12; since we know the classes of elements of order 3 in Aut(Λ) [14], [1]), we deduce that x has 1 with multiplicity exactly 12 and so on the middle composition factor, x does not have 1 as an eigenvalue). □

(3.9) Corollary. Notation as in (3.8.i). The group Stab_{Aut(Λ)}(M) = Stab_{Aut(Λ)}(M^{⊥}) is of the form 2_{+}^{1+8}.W′_{E_{8}} (though we do not need it, we mention that this is a nonsplit extension (C.3)) and it induces W′_{E_{8}} on M and Aut(T) ≅ 2_{+}^{1+8}Ω^{+}(8, 2) on M^{⊥}.

Proof. Because of the decompositon (3.2), it is clear that Stab(M) = Stab(M^{⊥}) induces exactly W′_{E_{8}} on M (3.8). The normal extraspecial group is generated by its involutions. By (3.8.ii) and how the involutions may be interpreted as SSD maps that act trivially on M, the normal subgroup of Stab(M) of shape 2_{+}^{1+8} acts trivially on M □

(3.10) Corollary. If (M_{1}, M_{2}, M_{3}) is a Leech trio in Λ, its stabilizer in Aut(Λ) is of the form 2^{3+12}GL(4, 2), of order 2^{21}3^{2}5.7. The stabilizer of the unordered trio has the form 2^{3+12}[GL(4, 2) × Sym_{3}], a group of order 2^{22}3^{3}5.7. This subgroup determines the Leech trio by the three nontrivial linear characters of the normal eights group which occur in ℚ^{24} = ℚΛ (each with multiplicity 8).

Proof. Let B and A be these respective subgroups. We have A/B ≅ Sym_{3} (3.5.iii) and C_{B}(M_{3}) ≅ 2 × 2^{1+6} which acts on M_{1}, M_{2} with respective kernels the two direct factor of order 2; see the proofs of (3.5.iii) and (3.8). The result follows since A embeds in the subgroup H ≅ [2_{+}^{1+6}GL(4, 2)]≀Sym_{3} of Aut(M_{1} ⊥ M_{2} ⊥ M_{3}), where the wreathing is done with the natural degree 3 action; in other words, B is forced to have shape 2^{3+12} and be the unique normal subgroup of A of index 2^{6} in O_{2}(H). The last statement is trivial. □

(3.11) Definition. Write Λ_{n} := {x ∈ Λ | (x, x) = 2n}, the set of lattice vectors of type n. □

(3.12) Lemma. (i) Aut(Λ) acts transitively on the set of pairs (x, M), where M is a sublattice isometric to \sqrt{2}L_{E_{8}} and x ∈ M has type 4;

(ii) Stab((x, M)) ≅ [2^{1+8+6}]Alt_{7}(≤ 2^{1+1+8+6}Ω^{+}(6, 2)) and its order is 2^{18}3^{2}5.7; (iii) Ω^{+}(6, 2) ≅ Alt_{8}.

Proof. By (3.6), it suffices for (i) to prove transitivity of Stab(M) on M ∩ Λ_{4}. Define an equivalence relation on M ∩Λ_{4} by congruence modulo 2M. Each class consists of 16 vectors, two of which are orthogonal or opposite (A.2). Now, Stab(M) induces W′_{E_{8}} on M (3.9), which is transitive on the set of equivalence classes since they correspond to singular points in M/2M. Let K be a class and a, b, c, d linearly independent vectors in K. Then \frac{1}{2}(a − b) and \frac{1}{2}(c − d) are roots and the product of the corresponding reflections interchanges a and b. Transitivity follows since all “even” transformations on M come from the action of Stab(M) (3.9). Note that the group Stab(M) has a permutation representation of degree 8 on K/{±1} and that the simple group Ω^{+}(6, 2) has order 8!/2. □

(3.13) Proposition. Let T_{n} := T ∩ Λ_{n}. Then

(i) Aut(T) has one orbit on T_{2} (length 240 = 2^{4}3.5, stabilizer of shape 2.2_{+}^{1+6}2^{6}Ω^{+}(6, 2)) and it has two orbits on T_{4}; one orbit in U, the radical modulo 2 (length 240 = 2^{4}3.5, stabilizer of shape 2.2_{+}^{1+6}2^{6}Ω^{+}(6, 2)), and the second orbit outside U (length 1920 = 2^{7}3.5, stablizer of shape Sp(6, 2)).

(ii) Aut(T) has one orbit on T_{3} (length 61440 = 2^{12}3.5).

Proof. (i) Since Aut(T) induces the full orthogonal group on T/U (3.8.vi), any singular coset of U lies in a natural sublattice of the form M_{1} ⊥ M_{2}, with M_{i} ≅ \sqrt{2}L_{E_{8}}. Since U ∩ M_{i} ≅ 2L_{E_{8}}, (3.5.iii), (3.10) and (A.3.ii) imply the statement about elements of type 2 and show that the elements x of type 4 in M_{i} \ N_{i} form one Aut(T)-orbit. Transitivity on elements in U ∩ T_{4} follows since U ≅ \sqrt{2}T. We are done if we show that every element of type 4 is in the Aut(T)-orbit of an element of M_{i} \ N_{i}. Clearly every element of type 4 in M_{1} ⊥ M_{2} has the form x = x_{1} + x_{2}, where (a) one of the summands is 0 (whence x has the desired form); or (b) each x_{i} has type 2. Assume the latter and, by way of contradiction assume that (*) x is not in the orbit of an element from (a). Then by (3.8.ii), the orbit of x under O_{2}(Aut(T)) has length 2^{9} and lies in a single coset of U. Now, the singular vector x + U of T/U lies in 35 sublattices of the form M′_{1} ⊥ M′_{2} as in (3.8.ii). For each associated SSD involution, t (2.5), we have an expression x = y_{1} + y_{2} such that y_{1}^{t} = y_{1} and y_{2}^{t} = −y_{2}; because of (*), each y_{i} has type 2. We therefore find that t fixes x modulo 2Λ. This set of 70 involutions generates a subgroup R of O_{2}(Aut(T)) of order at least 2^{7}. We have a map from R to the frame containing x, defined by r ↦ x^{r}. Since the map is injective, we have a contradiction to (*) since |R| ≥ 2^{7} and a frame has cardinality 48.

(ii) An element of T_{3} maps to a nonsingular vector in T/U, where Aut(T) acts as Ω^{+}(8, 2), so is transitive on such vectors. Therefore, it suffices to show that the only element of O_{2}(Aut(T)) ≅ 2_{+}^{1+8} which stabilizes {x, −x}, for x ∈ T_{3}, is {±1}. But this is clear from (3.8.ii) where it is shown that the noncentral involutions of O_{2}(H) have, for their action on T, fixed point sublattices isometric to \sqrt{2}L_{E_{8}}; these contain no vectors of odd type. □

(3.14) Lemma. If x ∈ Λ_{2} ∪ Λ_{3}, A(x) := {y ∈ Λ |(x, y) ∈ 2ℤ}, the “annihilator mod 2”, contains a sublattice isometric to \sqrt{2}L_{E_{8}}.

Proof. Let (M_{1}, M_{2}, M_{3}) be a Leech triple and let x = x_{1} +x_{2} +x_{3}, where x_{i} is the projection of x to the rational span of M_{i}. Then each x_{i} ∈ M_{i}^{∗} = \frac{1}{2}M_{i}, whence n_{i} := (x_{i}, x_{i}) is a nonnegative integer. If some n_{i} is 0, we are done, so assume all are positive.

Since Λ is even, we may reindex to assume n_{3} > 0 is even. Let y := x_{1} + x_{2}. Each x_{i} is nonzero. If x has type 2, (y, y) = 2 and y ∈ T^{∗} ≅ \frac{1}{\sqrt{2}}T; we quote (3.13.i) to transform x by Stab(M_{3}) to an element where some n_{i} is 0. If x has type 3, y has type 1 or 2 and a similar use of transitivity (3.13) works (there are cases: y ∈ T implies x_{3} ∈ M_{3} and this is impossible since minimum squared lengths in T and M_{3} are 4; if y ∉ T, then x_{3} has type 1 or 2 and y has type 2 or 1). □

(3.15) Lemma. (i) Suppose that x ∈ Λ_{n} and M ≅ \sqrt{2}L_{E_{8}} satisfies (M, x) ≤ 2ℤ (such M exist if n ≤ 3, by (3.14)). Then x ∈ M ⊥ (M^{⊥} ∩ Λ), so we write x = u + v, u ∈ M, v ∈ M^{⊥}.

(ii) If n = 2, u = 0 or v = 0. In case u = 0, there is a sublattice M_{1} ⊥ M_{2} of M^{⊥} as in (3.2) containing x.

(iii) If n = 3, x = v ∈ M^{⊥}.

Proof. For (i), use (2.3.ii). The remaining statements follow from (3.13). □

(3.16) Theorem. Aut(Λ) is transitive on Λ_{2}.

Proof. From (3.13.ii), a type 2 vector lies in a \sqrt{2}L_{E_{8}} sublattice, say M, of Λ. From (3.9), we know that Stab(M) induces W′_{E_{8}} on M, so all vectors of type 2 in M lie in a single orbit under Stab(M). □

(3.17) Theorem. Aut(Λ) is transitive on (i) pairs (M, x), where M ≅ L_{E_{8}} is a sublattice and x ∈ Λ_{3} ∩ M^{⊥};

(ii) Λ_{3};

(iii) quadruples (M_{1}, M_{2}, M_{3}, x), where (M_{1}, M_{2}, M_{3}) is a Leech trio and x ∈ Λ_{3} ∩ M_{3}^{⊥}. The stabilizer of pair as in (i) has the form 2.Sp(6, 2) , order 2^{10}3^{4}5.7, and the stabilizer of a

quadruple as in (iii) has the form 2.2^{3+3}.GL(3, 2), order 2^{10}3.7.

Proof. (i) follows from (3.13). Note that (3.15) tells us that any type 3 vector is part of a pair as in (i), so (ii) follows from (i). For (iii), use Witt’s theorem to see that H, the stabilizer in Aut(T) of a maximal totally singular subspace in T/U, is transitive on nonsingular vectors. As in the proof of (3.13), we know that if x ∈ T has type 3, O_{2}(Aut(T)) is transitive on the type 3 vectors in x + U, so we get the stabilizer for (i). The stabilizer for (iii) has index 135 in the stabilizer for (i), so by surveying the maximal parabolics for Sp(6, 2), we get the indicated subgroup. □

(3.18) Lemma. Let x ∈ Λ_{4}. Then the set of sublattices M ≅ \sqrt{2}L_{E_{8}} which contain x form an orbit of length 253 under Stab_{Aut(Λ)}(x).

Proof. (A.4), (A.7), (3.12.i). □

(3.19) Theorem. Aut(Λ) is transitive on frames (A.4) and on Λ_{n}, for n = 0, 2, 3, 4; |Aut(Λ)| = 2^{22}3^{9}5^{4}7^{2}11.13.23.

Proof. By (3.16) and (3.17), it suffices to prove transitivity on Λ_{4}. Let G = Aut(Λ). Any type 4 element lies in a frame and any frame gives rise to a Golay code (A.4) and so we may take our element x of type 4 and embed it in a pair (M, x) as in (A.7)(3.12.i). Transitivities on Λ_{4} and frames follow (3.12). The number of such pairs containing x is 253 (3.18). Then, we have |G : G_{(M,x)}| = |G : G_{x}||G_{x} : G_{(M,x)}|. The first factor on the right side is u_{4} = 2^{4}3^{7}5^{3}7.13 (A.1.ii) and the second is 253 (A.6). We conclude that |G : G_{(M,x)}| = 2^{4}3^{7}5^{3}7.11.13.23, whence (3.12.ii) |G| = 2^{22}3^{9}5^{4}7^{2}11.13.23. □

(3.20) Theorem. Let Λ be any Leech lattice. Then, for any frame Σ, there is an ordered basis Σ^{+} ⊂ Σ and a code so that Λ is as described in (A.4). The stabilizer Σ in Aut(Λ) has the form D : P (A.4.1), (A.5).

(3.21) Remark. The preceeding statement is not trivial. Moving from a frame plus Golay code to a containing Leech lattice involves work over ℤ/4ℤ, where the sign problems are harder to deal with than over ℤ/2ℤ, e.g. [14], Appendix 9A. □

(3.22) Remark. (The TU-tower.) We call attention to the chain ⋯ < 2T < U < T < \frac{1}{2}U < ⋯ whose every member is a scaled copy of T by a power of \sqrt{2} (3.2). This is an analogue of (2.9). □

We get another nice uniqueness result, which makes the Barnes-Wall lattice a member of the Broué-Enguehard series; see (A.10) and (A.11).

(3.23) Theorem. There is a unique even integer lattice which has rank 16, theta series which begins like 1 + 4320q^{2} + … and has no vectors of type 2 in its radical modulo 2.

Proof. Let T be such a lattice and U := {x ∈ T | (x, T) ≤ 2ℤ}, the radical modulo 2. As usual, for an even integral lattice, L, we write L_{n} for {x ∈ L | (x, x) = 2n}. Our assumption may be expressed: T_{1} = ∅ = U_{2} and rank(T) = 16.

The main thing we have to establish is that the relation “congruence modulo U” is an equivalence relation on T_{2} whose classes consist of “frames”, that is, 16-sets of type 2 vectors, two of which are proportional or orthogonal. So, let x, y be nonproportional members of T_{2} such that x + U = y + U. Then, x + y = u ∈ U and so (x, x + y) ∈ 2ℤ, whence (x, y) ∈ 2ℤ. We want (x, y) = 0. If nonzero, we may replace y by −y if necessary to arrange (x, y) < 0. Then (x, y) ≤ −2 and 0 < (x + y, x + y) ≤ 4 + 4 − 4 = 4, whence x + y ∈ U_{2} = ∅, a contradiction. So, each class has at most 16 vectors.

Since 4320/16 = 135 > 2^{7}, it follows that the finite abelian 2-group T/U is elementary abelian and that the nonsingular quadratic form on it inherited from T has maximal Witt index (the other possibility, a quadratic form of nonmaximal Witt index, would have exactly 119 singular points). It follows that the 4320 elements of T_{2} are distributed among these 135 singular cosets of U in T, whence each class has exactly 16 members.

In case \frac{1}{2}U/T has maximal Witt index (for its form taking values in \frac{1}{2}ℤ/ℤ) , we follow an idea from Section 3. Let Q ≅ \sqrt{2}L_{E_{8}}. There is an isometry of \frac{1}{2}Q/Q and \frac{1}{2}U/T, so can take Λ to be the preimage in \frac{1}{2}Q ⊥ \frac{1}{2}U of the diagonal subgroup of \frac{1}{2}Q/Q × \frac{1}{2}U/T. Then, Λ is an even unimodular lattice with no roots (since the minimum norm in T is 4), so L is isometric to the standard Leech lattice (3.7). From (3.4), we deduce that our T is isometric to the lattice T of (3.2), and we are done.

The isometry type of \frac{1}{2}U/T has not been established. However, we did prove that T/U has maximal Witt index, so if we use \frac{1}{\sqrt{2}}T, \frac{1}{\sqrt{2}}U for \frac{1}{2}U, T, respectively, in the argument of the last paragraph, we get a rootless even unimodular lattice since the minimum norm in T is 4 and we deduce from (3.4) that the isometry type of \frac{1}{\sqrt{2}}U is determined. Uniqueness of T then follows from (3.22). □

4. The Mathieu group, with the Golay code (finally!).

(4.1) Definition. A Golay code, 𝒢, is a binary code of length 24, dimension 12 and minimum weight at least 8. A group M_{24} is the automorphism group of a binary Golay code. We use the notations of (A.4), (A.5). □

(4.2) Theorem. (Uniqueness of the binary Golay code.) There is a unique (up to equivalence, i.e., coordinate permutations) binary code of length 24, dimension 12 and minimum weight at least 8 (hence equal to 8).

Proof. Existence of such codes comes from (A.4.3) and existence of a Leech lattice. To prove uniqueness, we suppose that 𝒞 is any Golay code. We define a lattice, L with 𝒞 following the recipe in (A.4). Let Ω := {1, …, 24} be an index set and {x_{i} | i ∈ Ω} be a basis of Euclidean space ℝ^{Ω} such that (x_{i}, x_{j}) = 2δ_{i,j}. For A ⊆ Ω, define x_{A} := ∑_{i∈A} x_{i}, ν_{i} := −x_{i} + \frac{1}{4}x_{Ω}. Let L be the span of all 2x_{i}, ±x_{i} ± x_{j}, ν_{i} and all \frac{1}{2}x_{A}, A ∈ 𝒞. Then L is a EUL without vectors of type 2, hence is isometric to Λ (3.7). By transitivity of Aut(Λ) on frames (3.19), we may assume that {±2x_{i} | i ∈ Ω} corresponds by our isometry to the standard frame (A.4), whence 𝒞 ≅ 𝒢. □

(4.3) Proposition. Aut(𝒢) = P and |P | = 2^{10}3^{3}5.7.11.23 (see (A.5) for the definition of P; the big Mathieu group M_{24} is defined to be Aut(𝒢), so we get |M_{24}| = 2^{10}3^{3}5.7.11.23).

Proof. (A.5), (3.19). □

(4.4) Proposition. P acts transitively (i) on the set of octads; (ii) and on Ω.

Proof. (i) If 𝒪 and 𝒪′ are octads, then M(𝒪) (A.6) contains some x ∈ Σ, the standard frame (A.4), and M(𝒪′) contains some x′ ∈ Σ. Now use (3.12.i) and note that an element of Aut(Λ) which carries the first pair to the second stabilizes Σ.

(ii) Given i, j ∈ Ω, expand each to an octad to create pairs (M(𝒪), 2α_{i}) and (M(𝒪′), 2α_{j}). Now use (3.12.i) and (3.19). □

(4.5) Proposition. The stabilizer in P of an octad is isomorphic to AGL(4, 2), the affine general linear group, which is a semidirect product of the general linear group GL(4, 2) by the group of translations, isomorphic to 2^{4}.

Proof. Let H be the stabilizer in P of the octad 𝒪; form the sublattice M(𝒪) (A.6) and use (3.9) and (A.5) to conclude that the stabilizer of (M(𝒪), F), where F is a frame in the sense of (A.2.i), has the form DH, order 2^{16}|GL(4, 2)|, whence |D| = 2^{12} implies that |H| = 2^{4}|GL(4, 2)| = |AGL(4, 2)|. It is clear from (3.3.iii) that H has a normal 2-subgroup R of order 16 whose quotient is Ω^{+}(6, 2). We now display R ≅ 2^{4}.

The involutions of R can be associated to semiselfdual sublattices (2.4) as follows: the space W of Golay sets disjoint from 𝒪 consist of 30 octads, ∅ and 𝒪 + Ω (A.7). Given a codimension 1 subspace, W_{0} of W which contains ⟨Ω⟩, we define an involution t ∈ P as follows. Let 𝒪_{i}, i = 1, 2, 3 be octads in W_{0} which are linearly independent modulo 𝒪 + Ω. Define an associated T I (triple intersection) as a set of the form 𝒪′_{1}∩𝒪′_{2}∩𝒪′_{3}, where 𝒪′_{i} represents 𝒪_{i} or its complement 𝒪_{i}+[𝒪+Ω] in 𝒪+Ω; a TI is a 2-set. Similarly, call a set a DI, a double intersection, if it has the form 𝒪′_{1} ∩𝒪′_{2}, notation as above; a DI is a 4-set. The intersection of any two octads disjoint from 𝒪 is a 0-, 4- or 8-set. Define M := M(𝒪_{1}, 𝒪_{2}, 𝒪_{3}) as the span of all α_{i} − α_{j}, where {i, j} is an associated T I and all \frac{1}{2}[α_{i} − α_{j} + α_{k} − α_{ℓ}], where \{i, j\} and \{k, ℓ\} are TIs whose union forms a DI. (It is easy to check that the definition of M depends just on W_{0}.) For such an M, we have a SSD involution, t = t_{M} ∈ P. Its effect on the α_{i} is to interchange two whose indices form a TI. If W_{0}, W_{1} and W_{2} are three such subspaces of W with associated involutions t_{0}, t_{1}, t_{2}, then t_{0}t_{1}t_{2} = 1 if W_{0} ∩ W_{1} = W_{1} ∩ W_{2} = W_{2} ∩ W_{0}.

It is clear that we have 15 involutions which have cycle shape 1^{8}2^{8} and which, with the identity, form an elementary abelian group, R, which acts regularly on 𝒪 + Ω. At once, H splits over R. Let K be a complement. Since the action of K on T is faithful (because the action of Stab(M) on M^{⊥} ∩ Λ = Λ ∩ [ℚΛ, R] has kernel {±1} (3.8)), we are done. □

(4.6) Remark. (A trio of sporadic isomorphisms.) We note that the sporadic isomorphisms Ω^{+}(6, 2) ≅ GL(4, 2) ≅ Alt_{8} follow from our analysis (C.3), (A.3), (3.12), (4.5); this says more than GL(4, 2) ≅ Alt_{8}, which is observed in the traditional course of studying M_{24} and the octad stabilizer [8],[14]. □

(4.7) Remark. It follows from (3.13) that if x = 2α_{i} ∈ Λ_{4} is in the standard frame, there are two orbits of Stab(x) on the set of M ≅ \sqrt{2}L_{E_{8}} in x^{⊥} ∩ Λ. One is the set of all M(𝒪) (A.4)(A.6), where 𝒪 is an octad avoiding i, and a sublattice from the other orbit contains no vectors from the standard frame and is a sublattice of Λ ∩ ∑_{i∉𝒪′} ℚα_{i}, where 𝒪′ is an octad containing i. □

Next, we give a uniqueness argument for the ternary Golay code. We use existence of a ternary Golay code to prove uniqueness. If 𝒯𝒢 is such a code, namely, a [12, 6, 6] code over 𝔽_{3}, then we use a lattice M which is the orthogonal direct sum of 12 lattices isometric to L_{A_{2}}. Using rational linear combinations of elements of the M_{i} related to the elements of 𝒯𝒢, we get a rootless even unimodular rank 24 lattice, Λ, which by our characterization (3.7), must be unique up to isomorphism. We give our version here, which seems easier to use that the ones in the literature, e.g. [8].

(4.8) Ternary Construction of the Leech lattice. One may imitate the analogous construction of the E_{8}-lattice, described in [14], (8.22) . For i = 1, …, 12, take a base α_{i}, β_{i} of the A_{2} root system in M_{i} ≅ L_{A_{2}}, define ν_{i} := \frac{1}{3}[α_{i} − β_{i}]. Then M_{i}^{∗} = M_{i} + ℤν_{i}. Define M_{0} as the span of all 3M_{i}^{∗} and all ∑ c_{i}α_{i}, where c_{i} ∈ ℤ and ∑_{i} c_{i} ∈ 3ℤ. For c = (c_{i}) ∈ 𝔽_{3}^{6} and x one of the symbols α, β, ν, define x_{S} := ∑_{i∈S} c_{i}x_{i}, where we think of 𝔽_{3} as the subset {−1, 0, 1} of ℤ. Finally, define γ_{i} := \frac{1}{3}[4α_{i} + ∑_{j≠i} α_{j}]. We define the lattice Λ_{0} := M_{0} + ∑_{c∈𝒯𝒢} ℤν_{c}; then L_{0}/M_{0} ≅ 𝒯𝒢. We take Λ := L_{0} + +ℤγ_{i} (the definition of Λ is independent of i since γ_{i} − γ_{j} = α_{i} − α_{j} ∈ M_{0}); see (2.7). Since det(M_{i}) = 3 and det(M) = 3^{12}, we get det(M_{0}) = 3^{14}, det(L_{0}) = 3^{2} and det(Λ) = 1 by repeated use of the formula det(K) = det(J)|J : K|^{2}, for lattices K ≤ J (2.1). The rootless property of Λ is easy to verify. □

From this particular way of constructing Λ, we can see an important property of the Sylow 3-group of Aut(Λ), which has order 3^{9}.

(4.9) Lemma. A Sylow 3-group contains exactly one elementary abelian subgroup of order 3^{6}. Such a subgroup is therefore weakly closed in a Sylow 3-subgroup.

Proof. This follows from the fact that an element of order 3 in Aut(𝒯𝒢) ≅ 2 · M_{12} acts with Jordan canonical form 2J_{3} or J_{1}J_{2}J_{3}. Proof of this fact is an exercise using Section 7 of [14], (7.37). □

This Lemma can be proved with much less work than a general analysis of 𝒯𝒢 and Aut(𝒯𝒢). Since we deduce that |Aut(𝒯𝒢)|_{3} = 3^{3} from our knowledge of |Aut(Λ)| , it would suffice to display a group of automorphisms of order 3^{3} and just observe the Jordan canonical forms of its elements; this may be done by selective use of Section 7 of [14].

(4.10) Theorem. There is a unique ternary code with parameters of the form [12, 6, 6].

Proof. Any such code may be used with a lattice M := M_{1} ⊥ ⋯ ⊥ M_{12}, where M_{i} ≅ L_{A_{2}}, to construct a Leech lattice, Λ, as above. Though the code is unknown, we may still construct a group of automorphisms of Λ which is isomorphic to 𝒯𝒢 and preserves M, namely the maps ε_{s} := ∏_{i} ε_{i}^{s_{i}}, where s = (s_{i}) ∈ 𝒯𝒢 is a codeword and ε_{i} is the identity on M_{j} if j ≠ i and

ε_{i} : \{ \begin{matrix}α_{i} \\ β_{i} \end{matrix} ↦ \{ \begin{matrix}β_{i} \\ −α_{i} − β_{i} \end{matrix} , a rotation by 2π/3 on M_{i}. Note that α_{i}^{ε_{i}} = α_{i} − 3ν_{i}, β_{i}^{ε_{i}} = β_{i} − 3β_{i} − 3ν_{i}

and ν_{i}^{ε_{i}} = ν_{i} − β_{i}, whence ε_{i} satisfies |M_{i} : M_{i}(ε_{i} − 1)^{k}| = 3^{k}. Proof that the ε_{c} preserve Λ is routine (one must use the property that all inner products in 𝒯𝒢 are zero mod 3).

This group of shape 3^{6} is isomorphic to 𝒯𝒢 by the inverse of c ↦ ε_{c} and may be used to recover the lattices M_{0} ∩ M_{i} (in the notation of (4.8)) as as the sublattices affording 12 distinct rational characters of 3^{6}; one gets M_{i} from M_{0} ∩ M_{i} as M_{i} = 3[M_{0} ∩ M_{i}]^{∗}.

The weak closure property of such a group in a Sylow 3-subgroup of Aut(Λ) (4.9) then implies that any two such lattices M are in a single orbit under the action of Aut(Λ). From this, it follows that the associated code in M^{∗}/M is unique up to equivalence. □

5. Other consequences for the Leech lattice and its automorphism group.

We make no attempt to systematically derive the standard results about Mathieu groups and Conway groups using SSD theory, but merely give a sample to illustrate use of our theory. The next result analyzes an entry from the list of triangle stabilizers in Aut(Λ) [5] [14].

(5.1) Proposition. G is transitive on triangles of type 222. Let a, b, c form a triangle of type 222, that is, a triple of vectors of type 2 which sum to 0. Let H := Stab_{G}(a, b, c). Then, H contains 2-central involutions with centralizers of shapes 2_{−}^{1+6}Ω^{−}(6, 2).

Proof. Note that the image mod 2Λ of any triangle with edges of even type lies in a maximal totally singular subspace. By (3.16), we may assume that a ∈ M_{3}, where (M_{1}, M_{2}, M_{3}) is a Leech triple. If b ∈ M_{1} ⊥ M_{2} ⊥ M_{3}, we are done since (b, b) = 4, (b, a) ≠ 0 and M_{i} ≅ \sqrt{2}L_{E_{8}} imply that b ∈ M_{3}. It is therefore enough, by (3.6) to show that any triangle of type 222 lies in a sublattice isometric to \sqrt{2}L_{E_{8}} ⊥ \sqrt{2}L_{E_{8}} ⊥ \sqrt{2}L_{E_{8}} since Stab(M_{3}) induces W′_{E_{8}} on M_{3}.

For an index k, define T = T_{k} := Λ ∩ M_{k}^{⊥} and let U = U_{k}/2T be the radical mod 2 of T (2.3). For an index k and x ∈ Λ, define A_{k}(x) := {y ∈ T_{k} | (x, y) ∈ 2ℤ}, a sublattice of T_{k} of index 1 or 2.

If there is an index i ≠ 3 so that (M_{i}, b) = 0, we are also done because the images of a, b in T_{i}/2T_{i} lie in a common maximal totally singular subspace, so by (3.8.ii) and the first paragraph, we are done. So, we may assume that there is no such i.

We let b_{i} be the projection of b to ℚM_{i}; each (b_{i}, b_{i}) is a positive integer. Since 4 = (b, b) = (b_{1}, b_{1}) + (b_{2}, b_{2}) + (b_{3}, b_{3}), we conclude that the three summands on the right side are (1, 1, 2), in some order.

If (b_{3}, b_{3}) = 2, b′ := b_{1} + b_{2} ∈ T_{3}^{∗} \ T^{3}. By transitivity (3.13.i), (3.8.ii) and the singularity of b + T_{3} ∈ T_{3}^{∗}/T_{3} , we may assume that b′ ∈ \frac{1}{2}[N_{1} + N_{2}], whence b′ lies in one of the \frac{1}{2}N_{j}. If i is the remaining index, we have {a, b} ∈ M_{i}^{⊥} ∩ Λ = T_{i}, and we are done by another use of (3.8) and the first paragraph of this proof.

Finally, we assume that (b_{3}, b_{3}) = 1; let j be the index so that (b_{j}, b_{j}) = 2 and set b′ := b − b_{j} ∈ T_{j}^{∗} \ T_{j}; b′ + T_{j} is a singular vector in T_{j}^{∗}/T_{j}. Then the image in T_{j}^{∗}/T_{j} lies in a totally singular subspace, say R/T_{j}, where R = R′ ⊥ R′′ ≅ L_{E_{8}} ⊥ L_{E_{8}}. By transitivity (3.13.i), we may assume that a lies in one summand, say R′. Since (b′, b′) = 2, b′ lies in one summand, and since (a, b′) = (a, b) = −2 ≠ 0, this summand must be R′. Therefore, \{a, b\}^{⊥} ≥ R′′ ∩ Λ ≅ \sqrt{2}L_{E_{8}}. Now, Stab(R′′ ∩ Λ) will transform the set {a, b} into a sublattice of (R′′ ∩ Λ)^{⊥} which is isometric to \sqrt{2}L_{E_{8}} ⊥ \sqrt{2}L_{E_{8}}, and finally we are done.

The structure of the centralizer of an involution follows from making the choice of triangle of type 222 in the sublattice M of (3.9). □

(5.2) Remark. The proof of (5.1) can be adapted to show that the group Co_{2} has an involution with centralizer of the form 2_{+}^{1+8}Sp(6, 2) and by looking in \frac{1}{2}N_{12} (2.3), we can find another with centralizer of the form 2^{1+6}2^{4}.GL(4, 2). The stabilizer H of a triangle of type 222 should be isomorphic to P SU(6, 2), but no proof seems to exist in the literature; indeed, in [5], the table of stabilizers had a question mark at 222 (removed in later versions). Centralizer of involution characterizations of P SU(6, 2) in the literature seem to require more than the centralizer shape in (5.1). One can identify H by verifying that it has the 3-transposition property [10][11]. □

(5.3) Remark. There are four classes of involutions in Aut(Λ), [1][14]. All may be interpreted as SSD involutions. We know already (3.6) about involutions associated to M ≅ \sqrt{2}L_{E_{8}}; their negatives are also SSD, associated to M^{⊥} ∩Λ. The involutions of trace 0 are associated to the sublattice of Λ consisting of vectors supported at a dodecad; it is isometric to the halfspin lattice for D_{12} [14] and has invariants (2^{12}). Finally, the involution −1 is the SSD involution associated to Λ itself. □

(5.4) Remark. Something like SSD theory should work for elements of order greater than 2. We would expect a theory of existence and uniqueness of other codes used in other descriptions of the Leech lattice [14], e.g the ternary Golay code which is associated to the subgroup 3^{12}2 · M_{12}. □

(5.5) Proposition. A Leech trio stabilizer is a 2-local in Aut(Λ) of shape 2^{1+2+12}[Sym_{3} × GL(4, 2)] (3.10) and it acts absolutely irreducibly on Λ/pΛ, for all odd primes p.

Proof. If M is any sublattice isomorphic to \sqrt{2}L_{E_{8}}, O_{2}(Stab_{Aut(Λ)}(M)) acts on Λ/pΛ with an irreducible direct summand of dimension 16. Also, O_{2}(Stab_{Aut(Λ)}(M)) stabilizes any Leech trio containing M (the other two sublattices of a trio are just the fixed points of x and xz, where z is the SSD involution associated to M and x is a noncentral involution in O_{2}(Stab_{Aut(Λ)}(M)) (3.8.ii). If we let M range over members of a Leech trio, it follows that the trio stabilizer acts irreducibly on Λ/pΛ. □

(5.6) Corollary. Λ/pΛ is an absolutely irreducible G-module, for all prime numbers p.

Proof. If p is odd, we use (5.5). Suppose that p = 2. Let Q/2Λ be a proper submodule of Λ/pΛ, of dimension d > 0. By transitivity of G on vectors of type 3 (3.17) and d < 24, we have Q ∩ Λ_{3} = ∅ (A.2.ii). Then, Q/2Λ is totally singular, whence d ≤ 12. But then 2^{d} < min{u_{2}/2, u_{4}/24}, a contradiction to transitivity on Λ_{2} and Λ_{4}. So, we have irreducibility. Suppose that we do not have absolute irreducibility. Then we have a nontrivial centralizer algebra and an integer e > 1 so that when Λ/2Λ is extended to a splitting field, every irreducible for Aut(Λ) occurs with multiplicity divisible by e. If we take an element x of order 23 in Aut(Λ) and note that x^{23} −1 has a single nontrivial irreducible factor, we see that e > 1 is impossible, a contradiction. □

(5.7) Remark. We have absolutely irreducible action modulo all primes for W_{E_{8}} on L_{E_{8}} (easy to prove) and also that of the sporadic simple group F_{3} on a 248 dimensional lattice. These three examples seem to be all the known ones. □

Appendices. Background

Appendix A. Elementary lattice theory

(A.1) Proposition. Theta functions: for an even integral lattice L, the theta function is ∑_{k≥0} u_{k}q^{k}, where u_{k} is the number of lattice vectors of squared length 2k.

(i) The theta series for L_{E_{8}} begins 1 + 240q + 2160q^{2} + . . ..

(ii) The theta series for a Leech lattice, Λ, begins 1+196560q^{2}+16773120q^{3}+398034000q^{4}+......

Proof. [22], for instance; or [8], p.135. □

(A.2) Proposition. (The E_{8} lattice and Leech lattice mod 2). Let L be L_{E_{8}} or a Leech lattice Λ and let L_{n} := {x ∈ L | (x, x) = 2n}.

(i) For L = L_{E_{8}}, a coset of 2L meets L_{n} for exactly one value of n ∈ {0, 1, 2} and such a nonempty intersection has the form {±x}, except for n = 2 for which it is a “frame”, a set of 16 vectors, two of which are equal, opposite or orthogonal.

(ii) For L = Λ, a coset of 2L meets L_{n} for exactly one value of n ∈ {0, 2, 3, 4} and such a nonempty intersection has the form {±x}, except for n = 4 for which it is a “frame”, a set of 48 vectors, two of which are equal, opposite or orthogonal.

Proof. (i) is trivial. For (ii), which is almost as trivial, see [5] or [14].□

(A.3) Lemma. (i) In W_{E_{8}}, the stabilizer P of a maximal totally singular subspace in L_{E_{8}} mod 2 has the form 2_{+}^{1+6}GL(4, 2) ≅ 2_{+}^{1+6}Ω^{+}(6, 2); P splits over O_{2}(P)/Z(P) but not over O_{2}(P), i.e., P contains a perfect group of the form 2.GL(4, 2) but does not contain GL(4, 2). Also, the nontrivial cosets of this subspace each contain 16 roots.

(ii) W_{E_{8}} acts transitively on

(a) pairs (M, x) where M ≅ \sqrt{2}L_{E_{8}} is a sublattice of L = L_{E_{8}} and x ∈ L, (x, x) = 2;

(b) pairs (M, x) where M ≅ \sqrt{2}L_{E_{8}} is a sublattice of L = L_{E_{8}} and x ∈ L \ M, (x, x) = 4;

(c) pairs (M, x) where M ≅ \sqrt{2}L_{E_{8}} is a sublattice of L = L_{E_{8}} and x ∈ M, (x, x) = 4.

Proof. (i) Since all such stabilizers are conjugate, it suffices to examine a convenient one. The chief factors of such a group P are clear; the only issue is the structure of the maximal normal 2subgroup. Use the following description of L_{E_{8}}. Let v_{1}, …, v_{8} be a basis of 8-dimensional Euclidean space such that (v_{i}, v_{j}) = 2δ_{i,j} and let ℋ be a dimension 4, length 8, minimum weight 4 extended binary Hamming code, e.g. the span in 𝔽_{2}^{8} of (11110000), (11001100), (10101010), (11111111). Its group is a subgroup S ≅ AGL(3, 2) of the full group Sym_{8} of coordinate permutations. We may and do identify the index set with the normal eights group E := O_{2}(S) of S; the codewords of weight 4 may be identified with affine subspaces of dimension 3 in E. For A ⊆ E, define v_{A} := ∑_{i∈A} v_{i}. Define the lattice L to be the ℤ-span of all ±v_{i} ± v_{j}, \frac{1}{2}v_{A} for A ∈ ℋ and all −v_{i} + \frac{1}{4}v_{E}. This lattice is even and unimodular so is isometric to L_{E_{8}}. Denote by M the sublattice spanned by all ±v_{i} ± v_{j} and \frac{1}{2}v_{E}. Both L and M admit the group S by coordinate permutations and admit the group of sign changes consisting of maps ε_{A}, A ⊆ E, which are defined by v_{i} ↦ \{ \begin{matrix}−v_{i} & i ∈ A; \\ v_{i} & i ∉ A. \end{matrix} The group ⟨E, F ⟩ is extraspecial (C.1) and acts trivially on both L/M and M/2L. For a discussion of a related family of lattices, see [2],(A.11). For the statement about splittings in P, we observe that P has a subgroup of the form 2.GL(4, 2) which occurs as the stabilizer of a maximal totally singular subspace complementary to the one stablized by P.

To show that any subgroup 2.GL(4, 2) of P is nonsplit, we note that, if split, the ambient 8 dimensional complex representation would have irreducible constituents of degrees 1 and 7 [1]; on any invariant lattice taken modulo 2 therefore we would have the trivial module, which is not the case (we have two 4-dimensional irreducible constituents).

(ii) Let L_{n} := {x ∈ L | (x, x) = 2n}. By Witt’s Theorem and the fact that W_{E_{8}} induces the full orthogonal group on L/2L, there is a single orbit of W_{E_{8}} on singular vectors outside the subspace M/2L of L/2L and a single orbit on nonsingular vectors outside this subspace. These nonsingular vectors are the images in L/2L of L_{1} and a nonsingular vector corresponds to a pair {x, −x} in L_{1}. Since −1 ∈ P, the stabilizer in W_{E_{8}} of M, we clearly have one orbit, whence (a).

(b) Now, let X := M ∩ L_{2}. A singular vector outside M/2L corresponds via L → L/2L to a subset R of X of 16 vectors, two of which are equal, negatives or orthogonal. It suffices to show that Stab_{P} (R) is transitive on R. Since P induces GL(4, 2) on L/M, it suffices to take R to contain an “odd” vector, say −v_{i} + \frac{1}{4}v_{E}, and show R is contained in a single K-orbit. Since membership in R is determined by congruence modulo 2L, every vector in R is odd. Finally, it suffices to show that K has a single orbit on Y, the set of all odd vectors of squared length 4. It is easy to see that Y consists of [−v_{i} + \frac{1}{4}v_{E}]^{ε_{A}}, for all indices i ∈ E and all even subsets A ⊆ E. Since S is transitive on E and K contains D, the group of all ε_{A}, transitivity is clear.

(c) We can view the action of P as that of the stabilizer in Aut(M) of 2L, so the preceeding argument applies here since M ≅ \sqrt{2}L_{E_{8}} and 2L/2M is a maximal totally isotropic in the sense of the associated nonsingular quadratic form (3.1) on M/2M. We deduce (c) from (b). □

(A.4) (Frames, codes and a description of the Leech lattice.)

(A.4.1) (Frame and Code concepts. ) Given a Leech lattice, Λ, a frame is a set of 48 vectors in Λ of squared length 8, two of which are opposite or orthogonal; a frame is an equivalence class of sets of vectors of type 4 in which x and y are equivalence iff x − y ∈ 2Λ. An oriented frame or a frame basis is a subset of a frame which is a basis of ℚΛ.

(A.4.2) (Standard frame and standard description of a Leech lattice) We now assume the existence of a Golay code, 𝒢, and its use in describing a Leech lattice, Λ, in the standard way with a basis consisting of an orthogonal set of roots {α_{i} | i ∈ Ω}. The standard frame is Σ := {±2α_{i} | i ∈ Ω} and the standard oriented frame is Σ^{+} := {2α_{i} | i ∈ Ω}. We define Λ := span_{ℤ}{2α_{i}, ±α_{i} ± α_{j}, \frac{1}{2}α_{S}, ν_{i} | i, j ∈ Ω, S ∈ 𝒢\}, where, for S ⊆ Ω, α_{S} := ∑_{i∈S} α_{i} and ν_{i} := \frac{1}{4}α_{Ω} − α_{i}\} (this set of generators is unnecessarily large, but shows symmetry). The frame group is the stabilizer of a given frame in Aut(Λ). Clearly, in the standard frame group Stab(Σ), there is a natural subgroup of the form D : P, where D acts diagonally with respect to the frame and where P is the group of permuation matrices identified with the automorphism group of the code 𝒢. The orthogonal transformations which stabilize each 1-space spanned be elements of Σ have the form ε_{A}, A ∈ Ω,

which are defined by α_{i} ↦ \{ \begin{matrix}−α_{i} & i ∈ A; \\ α_{i} & i ∉ A. \end{matrix}

(A.4.3) (Deduction of a code from a Leech lattice.) Conversely, given a Leech lattice, Λ, and a frame, F (which exists, by (A.1.ii)), we find that a code occurs naturally. We define Λ(4) := span\{\frac{1}{2}(x − y) | x, y ∈ F \}, Λ(2) := \{x ∈ L | 2x ∈ Λ(4)\}. Then, by using the 24 coordinate spaces \frac{1}{2}ℤx mod ℤF, x ∈ F, Λ(2)/Λ(4) gives a binary code 𝒞 and, since Λ is a Leech lattice, it is straightforward to see that the code is doubly even of dimension 12 (whence the universe set is in 𝒞, making it closed under complementation), the code has minimum weight at least eight and that |Λ : Λ(2)| = 2.

We now prove that the minimum weight is eight. Let S ∈ 𝒞 have minumum weight, say w ≥ 8. Since 𝒞 is closed under complementation, we may assume that w ≤ 12. We suppose that that w ≥ 9 and obtain a contradiction. Consider the map ψ : 𝒞 → P (S), the power set, defined by A ↦ A ∩ S. Since Λ is integral, all intersections of pairs of sets in 𝒞 are even, so dim(Im(ψ)) ≤ w − 1 and dim(Ker(ψ)) ≥ 13 − w. Assuming that dim(Ker(ψ)) ≥ 2, we have A ∈ Ker(ψ), A ≠ 0, Ω + S. Since |A| ≥ 9, 18 ≤ |A + S| < 24, whence 0 < |A + S + Ω| ≤ 6, a contradiction. We have dim(Ker(ψ)) = 1 and w = 12 and so every set in 𝒞 is 0, Ω or a 12-set. Since Im(ψ) is the codimension 1 space of even sets in P (S), there is A ∈ 𝒞 so that A ∩ S is a 2-set. Since A is a 12-set, A + S is a 20-set, our final contradiction.

(A.4.4) (The Steiner system, octads and dodecads.) S(5, 8, 24) denotes a Steiner system with parameters (5, 8, 24), that is, a family of 8-sets in a fixed 24-set such that any 5-set is contained in a unique member of this family. Sets of weight 8 in 𝒞 are called octads (4.4). The octads in a Golay code as above form such a Steiner system. Sets of weight 12 in 𝒞 are called dodecads; their stabilizers are, by definition, the group M_{12}. □

(A.5) Proposition. Stab_{Aut(Λ)}(Σ) = D: P and D := {ε_{A} | A ∈ 𝒢} ≅ 2^{12} (see (A.4.2)).

Proof. Since the code 𝒢 is its annihilator in the power set P (Ω) (with addition the symmetric difference and bilinear form (A, B) ↦ |A ∩ B| mod 2), application of ε_{A} to lattice elements of the form \frac{1}{2}α_{S}, S ∈ 𝒢, shows that if it stabilizes Λ, then A ∈ 𝒢. Trivially, ε_{A} ∈ Aut(Λ) if A ∈ 𝒢.

Now, let g ∈ Stab(Σ); then g is a product dp, where d is diagonal and p is a permutation matrix. Applying g to a vector of the form \frac{1}{2}α_{S}, S ∈ 𝒢 gives a vector of the form \frac{1}{2} ∑_{i∈T} ±α_{i} in Λ, for some T ⊆ Ω. Since its inner product with every \frac{1}{2}α_{S}, S ∈ 𝒢 is an integer, we get |T ∩ S| even, for every S ∈ 𝒢, which implies that T ∈ 𝒢. This means that p is in the group of the code, whence both p and d stabilize Σ. □

(A.6) Notation. M(𝒪) is the set of lattice vectors supported at the octad 𝒪 (A.4). From (A.4.3), it is clear that Λ ∩ M(𝒪) is just the 112 ±α_{i} ± α_{j} and all 128 \frac{1}{2}α_{𝒪} ε_{A}, for all A ∈ 𝒢. □

(A.7) Lemma. For x ∈ Σ (A.4.2), the set of M ≅ \sqrt{2}L_{E_{8}} containing x is just the set of lattices of the form M(𝒪) where 𝒪 is an octad containing the index i, where x = ±2α_{i}. (There are 253=11.23 such.)

Proof. It is clear from (3.6) that the stabilizer of x is transitive on the set of such M which contain it. Since Stab(x) ≤ Stab(Σ), which acts monomially with respect to the double basis Σ, it follows that every such M has the form M(𝒪). (The count 253 = \binom{23}{4}/\binom{7}{4} follows from (A.4.4)). □

(A.8) Proposition. The Golay sets disjoint from an octad consist of 30 octads, the empty set and the octad complement. Any octad is part of a trio (a partition of Ω into three octads.

Proof. Let 𝒪 be an octad. Study the map 𝒢 → P (𝒪), as in (A.4). □

(A.9) Theorem. (Classifications of EULs.) An EUL has rank divisible by 8. In rank 8, there is, up to isometry, just one EUL, L_{E_{8}}[28]. In rank 16, there are two, the halfspin lattice of rank 16 and the direct sum of two copies of L_{E_{8}}[28]. In rank 24, there are 24, and they are distinguished by the systems formed by their sets of roots; the empty root system corresponds to the Leech lattice [21]. See also [27], [22], [18].

(A.10) [2] (The lattices of Broué-Enguehard.) Given an integer n ≥ 3, there is a lattice L_{n} of rank 2^{n} with the following properties:

(i) det(L_{n}) = \{ \begin{matrix}1 & n odd; \\ 2^{8} & n even; \end{matrix}

(ii) Aut(L_{3}) ≅ W_{E_{8}} ≅ 2^{·}O^{+}(8, 2); Aut(L_{n}) ≅ 2_{+}^{1+2n}Ω^{+}(2n, 2) if n ≥ 4;

(iii) the minimum squared length is 2^{[n/2]}. (iv) Aut(L_{n}) is transitive on minimal vectors.

(A.11) Remark. (i) These lattices are beautifully described by an error correcting code of length 2^{n} based on the action of GL(n, 2) on 𝔽_{2}^{n} and the minimal vectors are listed. In case n = 3, the expected group of automorphisms 2_{+}^{1+6}Ω^{+}(6, 2) is proper in the full group of automorphisms, the Weyl group for E_{8}.

(ii) The Barnes-Wall lattice [3] is a rank 16 lattice with determinant 256 and minimum squared length 4; the theta series begins 1+4320q^{4} +61440q^{6} +522720q^{8} +· · ·. In [8], p.131, its occurrence as our lattice T (3.2) and certain other properties are asserted but no reference or proof is given. This lattice turns out to be a member of the Broué-Enguehard series (3.23); though [2] is mentioned in [8], this series is not discussed in detail and its relationship to the Barnes-Wall lattice seemed to be unexplained. See (3.23). □

Appendix B. Orthogonal groups in characteristic 2.

For basic theory, see [9], [1]. We summarize what we need.

(B.1) The groups. We are in even dimension 2n ≥ 2 over the field F, which is perfect of characteristic 2 (this includes finite fields). There are two equivalence classes of quadratic forms, according to the Witt index, the dimension of a maximal totally singular subspace; the possibilities for the Witt index are n (plus type) and n − 1 (minus type) and the isomorphism types of the isometry groups are denoted O^{ε}(2n, F), ε = ±. This group is not simple since it has a normal subgroup Ω^{ε}(2n, F) of index 2, the kernel of the Dickson invariant [9], which is a simple group except in the case (n, ε) = (2, +).

(B.2) Permutation representations. By Witt’s theorem, which says that in V, a finite dimensional vector space with a nonsingular quadratic form, an isometry between subspaces extends to an isometry on V, the groups O^{ε}(2n, F) are transitive on the totally singular subspaces of a given dimension d ≤ n; when restricted to Ω^{ε}(2n, F) we still have transitivity, except for d = n, where we have two orbits. For one of these subspaces, the stabilizer in Ω^{ε}(2n, F) or O^{ε}(2n, F) is a parabolic subgroup of the form F^{\binom{n}{2}}: GL(n, F). The stabilizer in Ω^{ε}(2n, F) of a singular, resp. nonsingular vector, has the shape F^{2n−2}Ω^{ε}(2n − 2, F), Ω^{ε}(2n − 1, F) ≅ Sp(2n − 2, F).

(B.3) Weyl group of E_{8}. The Weyl group W := W_{E_{8}} satisfies Z(W) = {±1}, W/Z(W) ≅ O^{+}(8, 2) and |W | = 2^{14}3^{5}5^{2}7. □

Appendix C. Extraspecial p-Groups

(C.1) Definition. [12][17] Given a prime number p, an extraspecial p-group is a finite pgroup P such that Z(P) = P′ has order p. It follows that Φ(P) = Z(P) and that P/Z(P) is a vector space of dimension 2n over 𝔽_{p}, for some integer n ≥ 1 and that the map P/Z(P)×P/Z(P) → Z(P) based on commutation may be interpreted as a nonsingular alternating bilinear form. When p = 2, the squaring map induces a map P/Z(P) → Z(P) which may be interpreted as a nonsingular quadratic form. □

(C.2) Theorem. Given P as in (C.1), the irreducible representations consist of p^{2n} linear characters and an algebraically conjugate family of p − 1 irreducibles of dimension p^{n}; the latter are faithful.

(C.3) Definition. A holomorph of P, as in (C.1), is a group G so that P ◁ G and the natural map G → Aut(P) has kernel Z(P) and image C_{Aut(P)}(Z(P)); if the image is a proper subgroup, the G is a partial holomorph. A holomorph is standard if it exists as a subgroup of GL(2^{m}, ℂ); otherwise it is a twisted holomorph. For p = 2 extensions are generally nonsplit [15][16] but are split for p odd. In any holomorph, G, the conjugacy classes within P consist of the p elements of Z(P), plus one or two further ones, distinguished by their orders, p and p^{2} (either one or both orders may occur). □

(C.4) Theorem. Let G be a standard holomorph and χ the character of G in the p^{n} dimensional irreducible representation. For g ∈ G, define P_{g} by P_{g}/Z(P) := C_{P/Z(P)}(g) and d_{g} := dim(P_{g}/Z(P)). We say that g acts cleanly on P iff P_{g} = C_{P} (g) (the other possibility, left unnamed, is |P_{g} : C_{P} (g)| = p). Then χ(g) = εp^{d_{g}/2} for some root of unity ε, if g acts cleanly; and χ(g) = 0 otherwise.

Proof. [13]. □

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