Pieces of 2^{d}:

Existence and uniqueness for Barnes-Wall and Ypsilanti lattices.

(today is August 11, 2004 in Ann Arbor.)

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

Dedicated to Donald G. Higman.

Abstract

We give a new existence proof for the rank 2^{d} even lattices usually called the Barnes-Wall lattices, and establish new results on uniqueness, structure and transitivity of the automorphism group on certain kinds of sublattices. Our proofs are relatively free of calculations, matrix work and counting, due to the uniqueness viewpoint. We deduce the labeling of coordinates on which earlier constructions depend.

Extending these ideas, we construct in dimensions 2^{d}, for d >> 0, the Ypsilanti lattices, which are families of indecomposable even unimodular lattices which resemble the Barnes-Wall lattices. The number Υ(2^{d}) of isometry types here is large: log_{2}(Υ(2^{d})) has dominant term at least \frac{r}{4}d 2^{2d}, for any r ∈ [0, \frac{1}{2}). Our lattices may be the first explicitly given families whose sizes are asymptotically comparable to the Siegel mass formula estimate (log_{2}(mass(n)) has dominant term \frac{1}{4}log_{2}(n)n^{2}).

This work continues our general uniqueness program for lattices, begun in Pieces of Eight [20]. See also our new uniqueness proof for the E_{8}-lattice [15].

Contents

1 Notation and terminology 4 2 Introduction 6 3 Statement of Results 8 4 Background on Codes 10 5 Background on Lattices 11 6 Actions of 2-groups and endomorphisms on lattices 15

6.1 Commutator density, 3/4-generation and 2/4-generation … 19 7 Sultry twists and the NextBW procedure 21 8 The groups R_{2^{d}}, G_{2^{d}} and invariant lattices 25 9 Sultriness 29 10 Proof of uniqueness 29 11 Minimal vectors, the zoop2 property and Aut(BW_{2^{d}}) 31 12 Orbits on norm 4 frames in L_{E_{8}}. 38 13 Clean pictures, dirty pictures and transitivity 40 14 The Ypsilanti lattices 42

14.1 Michigan lattices and Washtenawizations . . . . . . . . . . . . 42 14.2 Overlattices of direct sums of 2-special lattices . . . . . . . . . 44 14.3 Avoidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 14.4 Down Washtenaw Avenue to Ypsilanti . . . . . . . . . . . . . 47 14.5 From dimensions 2^{d} to arbitrary dimensions . . . . . . . . . . 49 14.6 Number of Ypsilanti cousins compared with the mass formula 49

15 Appendices 51 15.1 A1. Group orders . . . . . . . . . . . . . . . . . . . . . . . . . 51 15.2 A2. Aut^{0}(2_{ε}^{1+2d}), Out^{0}(2_{ε}^{1+2d}) and BRW^{0}(2^{d}, ε). . . . . . . . . 51 15.3 A3. Indecomposable integral representations for a group of

order 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1 Notation and terminology

annihilator, self annihilating  Section 4
A_{ij} and other diagonal notation  5.2
admissible  14.10
ancestors and generations, ancestral  13.5, 13.7
BW_{2^{d}}, the Barnes-Wall lattice in dimension 2^{d}  3.4
lattice of BW-type  3.4
BRW^{0}(2^{d}, ±)  Bolt, Room and Wall group, 15.2
classification  10.2
coelementary abelian subgroup, p-coelementary abelian a subgroup B ≤ A so

A/B is p-elementary abelian

D, a lower dihedral group  7.2
defect of an involution  6.10
density, commutator density  6.17,6.16
D(L), discriminant group of a lattice L  D(L) = L^{∗}/L
determinant of a lattice, L  |D(L)|
d-invariant  12.1
duality level  6.7
double basis  5.3
DT, DTL  14.15
eigenlattice, total eigenlattice, T el  6.8
f, f_{i}, f_{12}; various fourvolutions  7.2
F_{i}  7.2
fourvolution  6.1
frame, plain frame PF, sultry frame SF  6.15, 8.7
G_{2^{d}}  8.2
Hamming codes  4.2,4.4
I(d, p, q)  11.12
labeling  11.16
lower  8.3
mass formula, mass(n)  14.30
L, an integral lattice of rank n  Section 5
L_{i}, L_{i}[k]  7.2

L^{∗}, the dual of the lattice L  Section 5
ε_{A}, sign changes, monomial group  5.5
M_{i}, M_{i}[k]  7.2
minimal vectors, MinV ec(L), μ(L)  7.7
nextbw  7.2
power set, even sets  Section 4
Q_{i}  7.2
r-modular  5.10
R  8.2, 7.2
R_{2^{d}}  8.2
R_{2^{i}}  7.2
Scalar(G, M), the scalar subgroup  6.14
SSD, semiselfdual, RSSD, relatively semiselfdual  5.11
sultry frame  8.7
sultry transformation, twist  6.3
sBW, ssBW  7.11
t_{i}, t_{ij}, t_{ij′}  7.2
upper  8.3
Condition X(2^{d})  3.3
𝔛  10.1
𝔜  14.11
Ypsilanti lattices, cousins, etc.  Section 14
zop2 , zoop2  11.4
ε_{A}, sign changes, monomial group  5.5
ψ_{i}  7.2

Ω, index set (often identified with an affine space 𝔽_{2}^{d})

Ω, universe, 1, the “all ones vector” (1, 1, …, 1) in 𝔽_{2}^{Ω} Section 4

Conventions. Our groups and most endomorphisms act on the right, often with exponential notation. Group theory notation is mostly consistent with [12, 23, 19]. The commutator of x and y means [x, y] = x^{−1}y^{−1}xy and the conjugate of of x by y means x^{y} := y^{−1}xy = x[x, y]. These notations extend to actions of a group on an additive group; see 6.16, ff.

Here are some fairly standard notations used for particular extensions of groups: p^{k} means an elementary abelian p-group; A.B means a group extension with normal subgroup A and quotient B; p^{a+b+...} means an iterated group extension, with factors p^{a}, p^{b}, … (listed in upward sense); A:B, A·B mean, respectively, a split extension, nonsplit extension.

2 Introduction

All lattices in this article are positive definite. A sublattice is simply an additive subgroup of a lattice (no requirement on the rank).

We prove existence and uniqueness of the Barnes-Wall lattices of rank 2^{d} by induction and establish properties of them and their automorphism groups, including some new ones. In particular, the uniqueness theorem seems to be new. With future classifications (and discoveries!) of lattices in mind, we promote systematic study of uniqueness for important lattices. In [20], we used scaled unimodular lattices and SSD involutions to give a new uniqueness proof of the Leech lattice and revise the basic theory of the Leech lattice, Conway groups and Mathieu groups. There is a new and elementary uniqueness proof for the E_{8} lattice in [15].

The Barnes-Wall lattices BW_{2^{d}} are even lattices in Euclidean space of dimension 2^{d}. They have minimum norm 2^{⌊ \frac{d}{2} ⌋} and remarkable automorphism groups [4] isomorphic to BRW^{0}(2^{d}, +) ≅ 2_{+}^{1+2d}Ω^{+}(2d, 2), d ≥ 4.

Various terms have been applied to these abstract groups and their analogues over finite fields in general. We think that BRW group for the groups which occur here would be most appropriate since Bolt, Room and Wall seem to have been the first to determine their structure [4]. Compare the later articles [7], [16], [13], [17]. See Appendix A2.

These lattices (and related ones) were defined in [2]. Independently, these lattices were rediscovered and their groups analyzed by Broué and Enguehard in [7]. This coincidence does not seem well recognized in the literature. We first noticed [7], then [2] only years later. The beautiful and definitive analysis of Broué and Enguehard [7] was the main inspiration for this article.

We shall abbreviate Barnes-Wall by BW.

For ranks 2^{d} ≤ 16, the BW lattices are well-known in several contexts. For d = 1, we have a square lattice, and, depending on scaling, BW_{2^{2}} is the D_{4} or F_{4} root lattice. We have BW_{2^{3}} ≅ L_{E_{8}}, though in [2], we find \sqrt{2}L_{E_{8}}. As sublattices of many of the Niemeier lattices, there are scaled copies of BW_{2^{3}} and BW_{2^{4}}. See also [20]. About the BRW groups, there are further details in Appendix A2.

We prove existence and uniqueness of the BW lattices of rank 2^{d} by induction and establish basic properties of them and their automorphism groups. We start not with a frame (a double orthogonal basis) but an orthogonal sum of two scaled BW lattices of rank 2^{d−1}, then show how, by choosing overlattices, to enlarge this to a BW lattice of rank 2^{d}. Analysis of choices and induction give suitable existence and uniqueness theorems, structure of the set of minimum norm vectors, properties of automorphism groups, transitivity on certain sublattices, etc. The uniqueness and transitivity theorems are new. (The referee called our attention to a characterization of Barnes-Wall lattices in [21].)

Our program emphasizes elementary algebra and involves very little of special calculations, matrix work and combinatorial arguments. We heavily exploit commutator density and equivalent properties, like 3/4-generation and 2/4-generation, which are quite useful for manipulating sublattices and lessening computations. As far as we know, these properties are new.

Reflections on the uniqueness theory led us naturally to the Ypsilanti lattices, a very large family of BW-like lattices. The Ypsilanti lattices are fairly explicit and represent a nontrivial share of all the even unimodular lattices of dimension 2^{d}. Their existence also clarifies the need for some hypothesis like (e) in 3.3, as we now explain.

Let n > 0 be an integer divisible by 8. If L is a rank n even, unimodular lattice, the theta function of L lies in a vector space of dimension roughly \frac{n}{24} (see [31], p. 88). For n = 8 or 16, the dimension is 1, so the condition constant term 1 determines the theta function. For n = 24, the two conditions constant term 1 and no roots determines the theta function. In these cases, one can use arithmetic information about norms to determine structure.

Now take n to be 2^{d} for d large and L a BW lattice. The condition minimum norm μ(L) = 2^{⌊ \frac{d}{2} ⌋} represents 2^{⌊ \frac{d}{2} ⌋} linear demands on the theta function. This number is much less than \frac{2^{d}}{24}. It is unclear how knowledege of some higher coefficients can be used effectively to determine structure. The family of Ypsilanti lattices shows that many isometry types in a given dimension have the same minimum norm. To characterize these, or ones like them, we probably need more than hypotheses about their theta functions. We guess that for the Ypsilanti lattices, given theta functions may be shared by large sets of isometry types, and similarly for automorphism groups.

The numerous Ypsilanti lattices have nontrivial normal 2-subgroups (products of extraspecial 2-groups) in their automorphism groups. Their enumeration should be compared with [1], which shows that the portion of even unimodular lattices in rank divisible by 8 which have nontrivial automorphism group goes to 0 as the rank goes to infinity. See also the Mathematical Review article for [1] by Scharlau.

We acknowledge helpful conversations with Alex Ryba, Leonard Scott, Jean-Pierre Serre and Kannan Soundarajan.

The author has been supported by NSA grant USDOD-MDA904-03-1-0098.

3 Statement of Results

First, we give some notation, then state the main results.

Definition 3.1. Given a lattice, L, define μ(L) := min{(x, x)|x ∈ L, x ≠ 0}.

Definition 3.2. Given a lattice L, we define the dual lattice to be L^{∗} := {x ∈ ℚ ⊗ L|(x, L) ≤ ℤ}. Given an integral lattice, L, we define the discriminant group of L to be D(L) := L^{∗}/L, a finite abelian group. A set of invariants of an integral lattice are the orders of the cyclic summands in a direct product decomposition of D(L). (This depends on choice of decomposition.)

Definition 3.3. Condition X(2^{d}): This is defined for integers d ≥ 2. Let s ∈ {0, 1} be the remainder of d + 1 modulo 2.

We say that the quadruple (L, L_{1}, L_{2}, t) is a an X-quadruple if it satisfies condition X(2^{d}) (or, more simply, condition X), listed below:

(a) L is a rank 2^{d} even integral lattice containing L_{1} ⊥ L_{2}, the orthogonal direct sum of sublattices L_{1} ≅ L_{2} of rank 2^{d−1};

(b) When d = 2, L ≅ L_{D_{4}} ≅ BW_{4} and L_{1} ≅ L_{2} ≅ L_{A_{1}^{2}}; when d ≥ 3, 2^{− \frac{s}{2}} L_{1} and 2^{− \frac{s}{2}} L_{2} are initial entries of quadruples which satisfy condition X(2^{d−1}).

(c) μ(L) = 2^{⌊ \frac{d}{2} ⌋}.

(d) D(L) ≅ 2^{2^{d−1}}, 1 as d is even, odd, respectively.

(e) There is an isometry t of order 2 on L which interchanges L_{1} and L_{2} and satisfies [L, t] ≤ L_{1} ⊥ L_{2}, i.e., acts trivially on L/[L_{1} ⊥ L_{2}].

Definition 3.4. Also, we say that the lattice L is a lattice of Barnes-Wall type or a Barnes-Wall type lattice if there exist sublattices L_{1}, L_{2} of L and an involution t ∈ Aut(L) so that (L, L_{1}, L_{2}, t) satisfies condition X(2^{d}).

Theorem 3.5. Let d ≥ 2. A Barnes-Wall type lattice of rank 2^{d} exists and is unique up to isometry.

Corollary 3.6. (i) For every integer d ≥ 2, there is an integral even lattice L, unique up to scaled isometry, such that (a) the rank is 2^{d};

(b) Aut(L) contains a group G_{2^{d}} ≅ 2_{+}^{1+2d}Ω^{+}(2d, 2);

(ii) For such a lattice, the group of isometries is isomorphic to W_{E_{8}} if d = 3 and is just G_{2^{d}} for d ≥ 4. Also, D(L) ≅ 1 or 2^{2^{d−1}}, as d is odd, even, respectively. Also, μ(L) = 2^{⌊ \frac{d}{2} ⌋}.

We mention that the much-studied lattice L_{E_{8}} is the case d = 3 of the above. The author has recently given an elementary uniqueness proof for L_{E_{8}}. See [15], where previous uniqueness proofs are discussed. Also, a uniqueness proof for BW_{4} was given in [20].

In addition we prove transitivity results for certain types of sublattices made of scaled Barnes-Wall lattices, including frames. See 12.4, 13.1, 13.3.

A final application of our theory is the construction of the Ypsilanti lattices or the Ypsilanti cousins, built in a similar style. (Their definition is a special case of 𝔜 14.11, which is in turn a natural extension of the notation 𝔛 3.3; the idea came during a pleasant moment in Ypsilanti, Michigan.)

Let j ≥ 1, d = 5 + 3j. The Ypsilanti lattices are indecomposable, even, unimodular in dimension 2^{d}, and BW-like in the sense of minimum norm. For large dimensions, they become quite numerous. The following easily stated results give a sample of what we proved.

Theorem 3.7. For c ∈ [0, \frac{1}{8}) and integer j > 0 so that \frac{1}{16}(2 − 2^{1−j} + 3 · 2^{−1−2j}) > c, there is a family Y psi(2^{d}, j) of rank 2^{d} indecomposable, even unimodular lattices, defined for all d >> 0, so that log_{2} of the number of isometry types in Y psi(2^{d}, j) has dominant term at least c d 2^{d} (in other language, at least (\frac{1}{8} + o(1))d 2^{d}.

(i) There is an integer m so that for d >> 0 and L ∈ Y psi(2^{d}, j), μ(L) = 2^{m}.

(ii) the minimal vectors of L span a proper sublattice of finite index in L; (iii) Aut(L) has a normal 2-subgroup U of order divisible by 2^{1+2d}

The quotient Aut(L)/U is generally small. The integer m in (iii) is roughly ⌊\frac{d−j}{2} ⌋. Like the BW lattices, the minimum norms go to infinity roughly like the square root of the dimension.

Corollary 3.8. Let b ∈ [0, \frac{1}{8}). The number Υ(n) of isometry types of even unimodular lattices of dimension n ∈ 8ℤ which contain a Ypsilanti lattice as an orthogonal direct summand satisfies: log_{2}(Υ(n)) is asymptotically at least b · log_{2}(mass(n)) for n >> 0, where mass(n) is the number provided by the Siegel mass formula.

4 Background on Codes

Definition 4.1. An (n − k) × n matrix of the form H = (A|I_{n−k}), where A is an (n − k) × k matrix, is a parity check matrix for the code C if C is defined as the set of row vectors x ∈ F^{n} which satisfy Hx^{tr} = 0 [26], p.2.

Definition 4.2. The Hamming code H_{r} is defined (up to coordinate permutations) by the parity check matrix H_{r} which is the r × (2^{r} − 1) matrix consisting of the 2^{r} − 1 nonzero column vectors of height r over 𝔽_{2}. The binary simplex code S_{r} is the annihilator of the Hamming code H_{r}.

Remark 4.3. The code H_{r} can be interpreted as the subsets of nonzero vectors in 𝔽_{2}^{r} which sum to zero. It has parameters [2^{r} − 1, 2^{r} − 1 − r, 3] [26], p. 23. The minimum weight elements of H_{r} are simply the nonzero elements of a 2-dimensional subspace. Therefore, a nonzero codeword A in the annihilator meets every such 3-set in 0 or 2 elements. Equivalently, the complement A′ of A meets every such 3-set in a 1-set or the whole 3-set. It is clear that A′ with the zero vector is a codimension 1 linear subspace of 𝔽_{2}^{r}, whence A is an affine codimension 1 subspace. It follows that every nonzero element of S_{r} has weight 2^{r−1}, so S_{r} has parameters [2^{r} − 1, r, 2^{r−1}]. Note that for r ≥ 2, H_{r} ≥ S_{r} and that S_{r} contains 1, the all-ones vector, an odd set. Also, H_{r} is spanned by the affine planes with 0 removed.

Definition 4.4. The extended Hamming code is obtained by appending an overall parity check, so has parameters [2^{r}, 2^{r}−r−1, 4]. For r ≥ 2, it contains the all-ones vector. It is denoted H_{r}^{e}. Its annihilator is the extended simplex code S_{r}^{e}, which has parameters [2^{r}, r + 1, 2^{r−1}]. We have for r ≥ 2, that H_{r}^{e} ≥ S_{r}^{e} contains 1^{2^{r}}. Also, H_{r} is spanned by the affine planes.

Proposition 4.5. If r ≥ 1 is an integer, the Hamming code and simplex code of length 2^{r} have automorphism group isomorphic to AGL(r, 2).

Proof. This is well known. Since these two codes are mutual annihilators, they have a common group. A recent proof was given in an appendix of [11]. □

Lemma 4.6. If S is a subset of 𝔽_{2}^{d} of cardinality 2^{r} > 1 so that for every affine hyperplane H of 𝔽_{2}^{d}, |H ∩ S| = 0, 2^{r} or 2^{r−1}, then S is an affine subspace.

Proof. This is a result of Rothschild and Van Lint, [30]; it is given in [26], Chapter 13, Section 4, Lemma 6, page 379. □

Remark 4.7. The Reed-Muller codes are present in our analysis (the codes C_{X} in 11.16) but play a small role.

Definition 4.8. A code 0 ≠ C ≤ F^{X} is decomposable if there is a nontrivial partition X = Y ∪ Z of the index set, so that C = C_{Y} ⊕ C_{Z} is a nontrivial direct sum, where C_{W} means the set of vectors in C with support contained in W ⊆ X. If a code C ≠ 0 is not decomposable, it is indecomposable.

Lemma 4.9. For all t ≥ 3, there is a length 2^{t} indecomposable doubly even self orthogonal binary code.

Proof. For t = 3, take the extended Hamming code. Suppose t ≥ 4 and set u = t − 3. Take a partition of an index set S of size 2^{t} into 2^{u} parts S_{i} of size 8, for i = 1, …, 2^{u}. Let H_{i} be an extended Hamming code on S_{i}. Take a vector v_{i} of weight 2 with support A_{i} in S_{i} and define v = ∑_{i} v_{i}. Form the code C spanned by v_{i} and the codimension 1 subspace of ∑_{i} H_{i} which annihilates v. Then wt(v) = 2.2^{u} ∈ 4ℤ, whence C is even. □

5 Background on Lattices

Lemma 5.1. Let L be a positive definite integral lattice. Then L has a unique orthogonal decomposition into indecomposable summands. More precisely, let X(L) be the set of nonzero vectors of L which are not expressible as the orthogonal sum of two nonzero vectors of L. Generate an equivalence relation on X(L) by relating two elements if their inner product is nonzero. An orthogonally indecomposable summand of L is the sublattice spanned by an equivalence class in X(L). In fact, an orthogonal direct summand is a sum of a subset of this set of sublattices.

Proof. (See [25] and [27], which credits [25].) Let X_{i}, i = 1, …, t be the equivalence classes in X = X(L) and let L_{i} be the sublattice spanned by X_{i}. Positive definiteness implies that L is the sum of the L_{i}. Also by taking inner products, we deduce L_{i} ∩ L_{j} = 0 for i ≠ j. So, we have an orthogonal direct sum.

Let M be an arbitrary orthogonal direct summand. Let N be the annihilator of M in L. We show for each i that L_{i} ≤ M or L_{i} ≤ N. For x ∈ X, write x = x_{M} + x_{N}, where x_{M} ∈ M, x_{N} ∈ N. Indecomposability implies that one of these components is 0.

Now, suppose that M ∩ L_{i} is nonempty. Then, there exists x ∈ X_{i} so that (x, M) ≠ 0. The last paragraph implies that x ∈ M. We then deduce X_{i} ⊂ M and L_{i} ≤ M. If M ∩ L_{i} = ∅, then L_{i} ⊂ N. □

Notation 5.2. Given a lattice L, the ambient vector space is ℚ ⊗ L, with natural extension of the symmetric bilinear form on L.

Take isometries ψ_{i} : ℚ ⊗ L → V_{i} of rational vector spaces. From these, we get isometries ψ_{ij} = ψ_{i}^{−1}ψ_{j} from V_{i} to V_{j}. Priming on an index means replacement of the corresponding map by its negative.

For a subset A ⊆ ℚ ⊗ L, define the following subsets of V := V_{1} ⊥ V_{2}: A_{ij} := {x^{ψ_{i}} + x^{ψ_{j}}|x ∈ A}, A_{ij′} := {x^{ψ_{i}} − x^{ψ_{j}}|x ∈ A}, A_{i′j} := {−x^{ψ_{i}} + x^{ψ_{j}}|x ∈ A}, A_{i′j′} := {−x^{ψ_{i}} − x^{ψ_{j}}|x ∈ A}.

Notation 5.3. Given a basis B of Euclidean space and binary code C ≤ 𝔽_{2}^{B}, we define L_{B,C} := \{∑_{b∈B} \frac{1}{2}a_{b}b|a_{b} ∈ ℤ, (a_{b} + 2ℤ)_{b∈B} ∈ C\}. (This lattice is sometimes integral.) Note that L_{B,C} = \{∑_{b∈B} \frac{1}{2}a_{b}b|a_{b} ∈ ℤ, ∑_{b∈B}(a_{b}+2ℤ)c_{b} = 0 + 2ℤ for all (c_{b})_{b∈B} ∈ C^{⊥}}.

Notation 5.4. Let α_{i}, i ∈ Ω = 𝔽_{2}^{3} be vectors in ℝ^{Ω} which satisfy (α_{i}, α_{j}) = 2δ_{ij}. Let H_{8}^{e} be the extended Hamming code 4.4.

Define the A_{1}^{8}-description of L_{E_{8}} or the 2-twisted version of L_{E_{8}} to be the ℤ-span of all α_{i} and all \frac{1}{2}α_{c}, for c ∈ H_{8}^{e}. In the Notation 5.3, this is L_{{α_{i}|i=1,...,8},H_{8}^{e}}.

Notation 5.5. Suppose that Ω is an index set and {v_{i}|i ∈ Ω} is a basis of a vector space. For a subset A of Ω, define v_{A} := ∑_{i∈A} v_{i}. The linear transformation ε_{A} sends v_{i} to −v_{i} if i ∈ A and to v_{i} if i ∉ A. The group of such maps is E_{P (Ω)}. If C is a subset of P (Ω), E_{C} denotes the set of maps ε_{A} for A ∈ C. This is a subgroup if C is a subspace of the vector space P (Ω).

Proposition 5.6. For an integer d ≥ 2, define m := ⌊\frac{d}{2}⌋. Let Ω be an index set, identified with 𝔽_{2}^{d}. Take a basis B := {v_{i}|i ∈ Ω} where (v_{i}, v_{j}) = 2^{m}δ_{ij} of ℝ^{Ω}. Form L_{B,H}^{e}, as in 5.3; it is integral for d ≥ 3. Then, if d ≥ 4, Aut(L_{B,H_{d}^{e}}) is in the monomial group on B and in fact Aut(L_{B,H_{d}^{e}}) = E_{Ω}F, where F is a natural AGL(d, 2) subgroup of the group of permutation matrices. If d = 3, Aut(L_{B,H_{d}^{e}}) ≅ W_{E_{8}}.

Proof. For d = 3, we have the lattice 5.4 and for d = 2, we have the F_{4} lattice, spanned by vectors of shape (±1, 0, 0, 0), (±\frac{1}{2}, ±\frac{1}{2}, ±\frac{1}{2}, ±\frac{1}{2}). These automorphism groups are well known to be W_{E_{8}} and W_{F_{4}}, respectively.

For any d ≥ 2, the set of minimal vectors of L := L_{B,H_{d}^{e}} is just ±v_{i}, i ∈ Ω and \frac{1}{2}v_{S}ε_{T}, for S ∈ H_{d}^{e} an affine plane and T a subset of S. These span L since affine planes span H_{d}^{e}. All these minimal vectors have norm 2^{m}, m ≥ 1.

We now assume d ≥ 4. The set of these which are in 2^{m−1}L^{∗} is exactly ±v_{i}, i ∈ Ω, for if A is an affine plane there exists an affine plane B so that A∩B is a 1-set (because d ≥ 4), whence (\frac{1}{2}v_{A}ε_{S}, \frac{1}{2}v_{B}ε_{T}) = ±2^{m−2}. It follows that Aut(L) is contained in the monomial group based on B. Clearly it contains E_{Ω}F, described above, and maps to the stabilizer of the code L/Q in \frac{1}{2}Q/Q ≅ 𝔽_{2}^{Ω}, where Q is the square lattice with basis B. Since Aut(H_{d}^{e}) is a natural AGL(d, 2) subgroup of the symmetric group (4.5), we are done. □

Definition 5.7. Let c = (c_{i}) ∈ 𝔽_{2}^{n}. The Euclidean lift of c is the vector in {0, 1}^{n} ⊂ ℤ^{n} which reduces modulo 2 to c. When p is an odd prime and c = (c_{i}) ∈ 𝔽_{p}^{n}, we have a similar definition of lift, using the subset \{−\frac{p−1}{2}, −\frac{p−3}{2} …, −1, 0, 1, … \frac{p−3}{2}, \frac{p−1}{2}\}^{n} ⊂ ℤ^{n}.

Lemma 5.8. Let L be a lattice with sublattice of finite index M which is a coelementary abelian p-group for some prime p. Let F := 𝔽_{p}. Suppose that C is an error correcting code in F^{n} with minimum weight w. Suppose that J is the lattice between M^{n} and L^{n} corresponding to C, i.e. spanned by all (c_{1}x, c_{2}x, …, c_{n}x) for x ∈ L and (c_{i}) is the Euclidean lift of a codeword in C.

(i) If (y_{1}, …, y_{n}) ∈ J \M^{n}, the weight of (y_{1}, …, y_{n})(mod M^{n}) is at least w.

(ii) Suppose that M is indecomposable and C is indecomposable. Then Aut(J) ∩ Aut(M ⊥ ⋯ ⊥ M) ∩ Aut(L ⊥ ⋯ ⊥ L) factorizes as the product of subgroups A_{1}A_{2}, where A_{1} is the subgroup which fixes each direct summand isometric to M and acts diagonally on (L/M)^{n}, and where A_{2} is the subgroup of the natural group of block permutation matrices of degree n which fixes the code C.

Proof. (i) We take a basis v(1), …, v(d) of C. For a codeword v = (v_{i}) and x ∈ L/M, let vx be the vector in (L/M)^{n} whose i^{th} component is the Euclidean lift of v_{i} times x + M. So, J/M^{n} = ∑_{i=1,...,d;x∈L} v(i)x + M^{n} = ⨁_{i=1,...,d} v(i)(L/M).

Suppose that y = (y_{1}, …, y_{n})(mod M^{n}) represents the minimal weight k in J/M^{n}. Write it as a linear combination y = ∑_{i=1}^{d} v(i)x(i), where x(1), …, x(d) is a sequence of elements of L/M and the product v(i)x(i) is as in the previous paragraph.

Take any linear functional f : L/K → F and extend it componentwise to g : (L/M)^{n} → F^{n}. Then g(J/M) = C. Given a nonzero x(i), take an f so that f(x(i)) = 1. Then g(y) ∈ C has nonzero coefficient 1 at v(i), whence 0 ≠ g(y) = (f(y_{1}), …, f(y_{n})) ∈ C has at least w nonzero entries, whence so does y = (y_{1}, …, y_{n}). Therefore, k ≥ w.

(ii) First, suppose that h ∈ Aut(J) ∩ Aut(M ⊥ ⋯ ⊥ M) ∩ Aut(L ⊥ ⋯ ⊥ L). We claim that h determines a unique element of the code group, up to scalars. For any v ∈ C, x ∈ L \ M, h takes vx to an element of J/M. This means that for linear functionals f, g as in (i) where f(x) = 1, we have that g(h(vx)) is a codeword. Since h takes vx to another element of similar form v′x′, it follows that there is a block permutation matrix b so that, for all codewords x, the action of hb stabilizes each direct summand isometric to M and takes vx to an element of the form vx′, for all x ∈ L/M. Since the code is indecomposable, we use the property that for any v, w ∈ C, if hb takes vx to vx′, then hb takes wx to wx′. In other words, the actions of hb on the summands of (L/M)^{n} are identified. □

Lemma 5.9. Suppose that J is a lattice and that SMV (J), the sublattice spanned by the minimal vectors, has finite index. Suppose that SMV (J) = J_{1} ⊥ ⋯ ⊥ J_{n}, where the J_{i} are indecomposable lattices.

Let K satisfy SMV (J) ≤ K ≤ J and K is homogeneous with respect to the rational vector spaces spanned by the summands, i.e., K = ∑_{i=1}^{n} K_{i} where K_{i} := K ∩ (ℚ ⊗ J_{i}).

Suppose that J/K corresponds to an indecomposable code 4.8. Then J is orthogonally indecomposable.

Proof. (See 5.1.) Let x be an indecomposable vector of J which is not in K and let S be the indecomposable summand of J which contains it. Let A be the support of x + K in J/K, i.e., those indices where x + K projects nontrivially to ℚ ⊗ K_{i}/K_{i}. For i ∈ A, there exists a minimal vector y ∈ K_{i} so that (x, y) ≠ 0. Therefore, J_{i} ≤ S, for all i ∈ A. The indecomposability assumption on the code implies that all J_{i}, i = 1, …, n are in S and since SMV (J) has finite index in J and S is a summand of J, S = J. □

Definition 5.10. Let r > 0 be an integer. An integral lattice L is r-modular if L ≅ \sqrt{r}L^{∗} Definition 5.11. The SSD concepts were established in [20]. Call a lattice M semiselfdual (SSD) if 2M^{∗} ≤ M ≤ M^{∗}. If the sublattice M of the integral lattice L is semiselfdual, we define the orthogonal transformation t_{M} on ℚ ⊗ L by −1 on M and 1 on M^{⊥}. Then t_{M} leaves L invariant and so gives an isometry of L of order 1 or 2; it has order 2 on L if M ≠ 0.

A more general notion is that of relatively SSD (RSSD): this is the condtion that the sublattice M of the integral lattice L satisfies the weaker condition 2L ≤ M + M^{⊥}. In this case, the orthogonal involution defined as above preserves L.

6 Actions of 2-groups and endomorphisms on lattices

We gather an assortment of results on this topic.

Definition 6.1. A fourvolution is a linear transformation whose square is −1. A fourvolution on a lattice is a lattice isometry whose square is −1. In case we have a lower group as in 8.3, we use the terms lower and upper fourvolution. We may call an element in a group a fourvolution with respect to a representation, and even with respect to more than one representation, for example by restriction of one action to a submodule.

Lemma 6.2. If f is a fourvolution of the lattice L, then the adjoint of 1 ± f is 1 ∓ f, (1 ± f)^{2} = ±2f, 1 ± f is an isometry scaled by \sqrt{2} and we have L ≥ L(1 + f) ≥ 2L and |L : L(1 + f)| = |L(1 + f) : 2L| = |L/2L|^{\frac{1}{2}} . In particular, rank(L) is even.

Proof. For x, y ∈ L, we have (x(1±f), y(1±f)) = (x, y)±(x, yf)±(xf, y)± (xf, yf) = 2(x, y) ± (x, yf) ± (xf^{2}, yf) = 2(x, y). For adjointness, just compute (x(1±f), y) = (x, y)±(xf, y) = (x, y)±(xf^{2}, yf) = (x, y)∓(x, yf). The other statements are easy to prove. □

Notation 6.3. Let L be a lattice and f a fourvolution in Aut(L). Define S[k] := S(1−f)^{k}, for S ⊆ ℚ⊗L and k ∈ ℤ. Note that this makes sense since the linear map 1−f is invertible, with inverse \frac{1}{2}(1+f). Call a transformation of the form 1 − f a sultry tranformation and call S[k] the k^{th} sultry (1 − f)- twist of S or the k^{th} sultry twist of S. (The terminology is explained in 9.1.)

We have S[0] = S. Note that for all k, (Sp)[k] = (S[k])p, where p is any polynomial expression in f. Also, S[k][ℓ] = S[k + ℓ].

Lemma 6.4. If S is an f-invariant lattice in ℚ ⊗ L, then for k ≤ ℓ, |S[k] : S[ℓ]| = 2^{\frac{1}{2} rank(S)(ℓ−k)}.

Proof. This follows since (1 − f)^{2} = −2f and because for all integers p, q and all integers r ≥ 0, S(1 − f)^{p}/S(1 − f)^{p+r} ≅ S(1 − f)^{q}/S(1 − f)^{q+r}. □

Lemma 6.5. Let S, T be subsets of ℂ ⊗ L. Then (i) (S[1], T) = −(S, T f[1]). Now assume that S and T are f-invariant. Then S = Sf = −S, T =

T f = −T and the following hold.

(ii) For all integers k, ℓ, we have (S[k], T [ℓ]) = 2(S[k − 1], T [ℓ − 1]) and (S[k], T [ℓ]) = 2(S[k − 2], T [ℓ]) = 2(S[k], T [ℓ − 2]).

(iii) S^{∗}[k] = S[k]^{∗}f^{−k}.

(iv) (S[k], T [ℓ]) = (S[k′], T [ℓ′]), for all integers k, k′, ℓ, ℓ′ such that k +ℓ = k′ + ℓ′; and

(v) Assume that the integer ℓ satisfies S^{∗} = S[ℓ]. Then S^{∗}[k] = S[k + ℓ].

Proof. (i) and (v) are clear.

(ii) follows since 1 − f is an isometry scaled by \sqrt{2}.

(iii) We have x ∈ S[k]^{∗} if and only if (x, S[k]) ∈ ℤ if and only if x(1+f)^{k} = (−1)^{k}xf^{k}(1 − f)^{k} ∈ S^{∗} if and only if x ∈ (−1)^{k}S^{∗}[−k]f^{−k} = S^{∗}[−k]f^{−k}.

(iv) is trivial for k = 0 and for k ≥ 1 it follows from (i) and easy induction. If k is negative, use (ii) and the case k ≥ 0.

Example 6.6. If L ≅ L_{D_{4}} then L[−1] ≅ L_{F_{4}}, where we take the latter to be the span of a standard version of the F_{4} root system: (±10^{3}), (±1^{2}0^{2}), (±\frac{1}{2}^{4}).

Definition 6.7. Let L be a lattice with fourvolution f. Suppose that there is an integer r such that L^{∗} = L[−r] (see 6.3). We call r the duality level of L. Such a modular lattice (see 5.10 and 6.4) is called an r-sultrified dual and is 2^{r}-modular (see 5.10).

Definition 6.8. Given a group E acting on a lattice L and character λ ∈ Hom(E, {±1}), define the eigenlattice L^{λ} to be {a ∈ L|ay = λ(y)a, for all y ∈ E}. Define the total eigenlattice to be T el(E, L) := ∑_{λ∈Hom(E,{±1})} L^{λ}. The notation extends naturally a set of automorphisms. When t has order 1 or 2, define L^{+}, L^{−} to be the lattice of fixed, negated points, respectively. To denote dependence on t, we write L(±, t) or L^{±}(t) for L^{±}.

Remark 6.9. In case E is 2-elementary abelian, L/T el(E, L) is finite, and is in fact a 2-group, but in general is not elementary abelian. For an example, let E be a fourgroup and L = ℤ[E], the regular representation. Then, L/T el(E, L) ≅ 2 × 2 × 4.

Lemma 6.10. Suppose that the involution t acts on the additive group A. Let A^{ε} := {a ∈ A|a^{t} = εa}. Suppose furthermore that the minimal number of generators of A as an abelian group is r < ∞. Define integers k, ℓ by 2^{k} := |A:A^{−} + A^{+}| and 2^{ℓ} := |A:B|, where B := {x ∈ A|x(1 − t) ∈ 2A}.

Then: (i) 2A ≤ A^{−} + A^{+} ≤ B, whence ℓ ≤ k;

(ii) ℓ ≤ r/2.

(iii) A^{−} ≥ A(1 − t) ≥ 2A^{−} and |A(1 − t)/2A^{−}| = 2^{k}, whence rank(A^{−}) ≥ k.

(iv) A^{+} ≥ A(1 + t) ≥ 2A^{+} and |A(1 + t)/2A^{+}| = 2^{k}, whence rank(A^{+}) ≥ k.

(v) If A is free abelian, A^{ε} is a direct summand of A and A^{+} + A^{−} = A^{+} ⊕ A^{−}.

(vi) If A is free abelian, k = ℓ (whence k ≤ r/2).

(vii) Suppose that multiplication by 2 is a monomorphism of A (e.g., A is free abelian). If k = 0 (i.e., if t is trivial on A/2A), A = A^{+} + A^{−}.

Proof. (i) The proof follows from the equation 2a = (a + a^{t}) + (a − a^{t}).

(ii) Let B := {x ∈ A|x(1 − t) ∈ 2A}. Then the map (1 − t) induces an injection of A/B ≅ 2^{ℓ} into B/2A, so in particular ℓ ≤ k. If x_{1}, x_{2}, …, x_{ℓ} ∈ A form a basis modulo B, then x_{1}, x_{1}^{t}, x_{2}, x_{2}^{t}, …, x_{ℓ}, x_{ℓ}^{t} are independent modulo 2A. Therefore, 2ℓ ≤ r.

(iii) For the first statement, notice that the kernel of the map ϕ : A → A^{−}/2A^{−}, x ↦ x(1 − t) is A^{+} ⊕ A^{−} and then use Im(ϕ) ≅ A/Ker(ϕ), which has rank k.

(iv) This follows from (iii) by replacing t with −t.

(v) Clearly, A/A^{ε} is torsionfree. The second statement follows from A^{+} ∩ A^{−} = 0.

(vi) This follows from the general classification of free abelian groups which are modules for cyclic groups of prime order, e.g., 15.11; (74.3) in [10]. (The result for a cyclic group of order 2 is easy to prove directly.) It states that such a module A has the form F_{1} ⊕ ⋯ ⊕ F_{p} ⊕ E_{1} ⊕ ⋯ ⊕ E_{q}, where each F_{i} is a copy of the regular representation ℤ⟨t⟩ and where each E_{j} is infinite cyclic. By reducing such a decomposition modulo 2, one deduces that k = ℓ.

(vii) This is easy to prove directly (of course it is a consequence of the nontrivial result mentioned in (vi)). Suppose that the involution t is trivial on A/2A. Then t = 1 + 2S for some S ∈ End(A). From 1 = t^{2} = 1 + 4(S + S^{2}), we deduce that S + S^{2} = 0. For a ∈ A, a = a(1 + S) − aS. One checks that aS ∈ A^{−} and a(1 + S) ∈ A^{+}. □

Definition 6.11. The defect of the involution t acting on the free abelian group A is the integer k = ℓ, as in 6.10. It is the number of nontrivial Jordan blocks for the action of t on A/2A.

Lemma 6.12. Let L be a unimodular lattice and t an involution acting on L. Then the eigenlattices L^{ε} := {x ∈ L|x^{t} = εx} satisfy D(L^{+}) ≅ D(L^{−}) ≅ 2^{k}, where k is the defect of t in the sense of Definition 6.11.

Proof. Since each L^{ε} is a direct summand of L, which is unimodular, the orthogonal projection takes L onto [L^{ε}]^{∗}. The kernel of the map from L to [L^{ε}]^{∗}/L^{ε} is L^{ε} + L^{−ε}, so from Lemma 6.10, we deduce that the image is elementary abelian, of order 2^{k}. □

Remark 6.13. The notion of RSSD involution is essentially the same as that of an involution on a lattice. Let L be a lattice. An involution t ∈ Aut(L) creates a pair of eigenlattices, L^{±}. Since L^{+} ⊥ L^{−} is 2-coelementary abelian in L and t acts trivially on \frac{1}{2}[L^{+} ⊥ L^{−}]/[L^{+} ⊥ L^{−}], t is a RsSSD involution which preserves L (see 6.10, 5.11 and [20]).

Definition 6.14. For a group G acting on the RG-module M, where R is a commutative ring, the scalar subgroup is

Scalar(G, M) := {g ∈ G|g acts on M as multiplication by an element of R^{×}}.

When M is a free abelian group, this is just the subgroup of group elements which act as ±1.

Definition 6.15. A frame or plain frame in a rank n lattice is a set of 2n vectors of common norm, two of which are linearly dependent or orthogonal.

Later in 8.7, we work with a special case of this.

6.1 Commutator density, 3/4-generation and 2/4-generation

The concepts 6.17, 6.20 and results in this section seem to be new. Commutator density is an unusual property which is very useful for controlling commutators of an extraspecial 2-group acting on a lattice.

Note that we will be mixing additive and multiplicative commutator notation.

Definition 6.16. We recall a few definitions involving groups and modules. Let Q be a (multiplicative) group and S a subset of Q. For s, t ∈ Q, as usual [s, t] = s^{−1}t^{−1}st. For a module M, [M, S] is the commutator submodule, meaning as usual the additive group spanned by all commutators [x, s] = x(s − 1), x ∈ M, s ∈ S. Higher commutators are interpreted by extending these definitions, for example [x, s, t] = x(s − 1)(t − 1), [s, t, x] = −[x, [s, t]] and [t, x, s] = −[x, t, s].

Definition 6.17. Let Q be a group and S a subset of Q.

S-CD: A module M for Q is called S-commutator dense if [M, Q] = [M, S]. (When S = {f}, a single element, every element of [M, Q] = M(f−1) is a commutator.)

S-kCD: As is common, for the natural number k, we use the notation [M, Q; k] for [M, Q, Q, …, Q] (k times). We say that M is degree k Scommutator dense or if [M, Q; k] = [M, S; k].

S-HCD: If M has such properties for all k ≥ 1, we say that M is S-higher commutator dense.

S-TCD: A module M is S-commutator dense on submodules if all its submodules are S-commutator dense. In this spirit, we define degree k CD and S-HCD on submodules.

When the set S is understood, we may drop S from the preceeding notations. Note that commutator density is inherited by quotient modules but may not be for submodules.

Lemma 6.18. Suppose that the group Q acts on the ℤQ-module L and that L is f-commutator dense for a fourvolution f ∈ Q such that [Q, f] is scalar on L. Then [L, Q; k] = [L, f; k], for all k ≥ 1, i.e., L is f-HCD. In fact, [L, Q; k] = [L, f; k] = 2^{\frac{k}{2}} L if k is even, and [L, Q; k] = [L, f; k] = 2^{\frac{k−1}{2}} [L, f] if k is odd.

Proof. We have [L, Q] = [L, f], whence [L, Q, f] = [L, f, f] = 2Lf = 2L. Also, [Q, f, L] ≤ [Scalar(Q, L), L] ≤ 2L. The Three Subgroups Lemma [12, 23] implies that [f, L, Q] ≤ 2L, or [L, Q, Q] ≤ 2L, which is [L, f, f]. The statements [L, Q; k] = [L, f; k], for all k ≥ 1 are proven by induction. □

Lemma 6.19. Suppose that the lattice L contains the orthogonal direct sum of sublattices L_{1} ⊥ L_{2}, that L_{1} and L_{2} have rank n = 2m ∈ 2ℤ and L/L_{1} ⊥ L_{2} is elementary abelian of order 2^{m}. Suppose that involutions t, u act on L so that L_{1} = L^{−}(t), L_{2} = L^{+}(t) (see 5.11) and u interchanges L_{1} and L_{2}. Then:

(i) u acts trivially on L/L_{1} ⊥ L_{2} if and only if det(L^{±}(u)) = det(L_{1}) = det(L_{2}).

(ii) If the conditions of (i) hold, then L is the sum of any three of the four sublattices L^{±}(t), L^{±}(u).

Proof. Clearly u acts on \frac{1}{2}L_{1} ⊥ \frac{1}{2}L_{2} and on \frac{1}{2}L_{1} ⊥ \frac{1}{2}L_{2}/L_{1} ⊥ L_{2}, it acts with n Jordan blocks of size 2. Also, det((L_{1} ⊥ L_{2})^{±}(u)) = det(L_{1})2^{n} and det(\frac{1}{2}(L_{1} ⊥ L_{2})^{±}(u)) = det(L_{1})2^{−n}.

(i) The equivalence of the two conditions follows from comparision of the determinants of the lattices L^{±}(u) ≥ (L_{1} ⊥ L_{2})^{±}(u). Let 2^{r} be the index |L^{±}(u):(L_{1} ⊥ L_{2})^{±}(u)|. Then det(L^{±}(u)) = 2^{n−2r}det(L_{1}). The second condition in (i) implies that n = 2r, whence L^{±}(u) covers L/[L_{1} ⊥ L_{2}]. Conversely, take x ∈ L. It is fixed by u modulo L_{1} ⊥ L_{2}, which is a free module. Therefore there is y ∈ L_{1} with x(u − 1) = y(u − 1). The coset x + [L_{1} ⊥ L_{2}] therefore contains the fixed point x − y.

(ii) The hypotheses imply that L_{1} + L_{2} + L^{+}(u) = L, and a similar statement applies to −u. Finally, we may interchange the roles of t and u to deduce the remaining statements. □

Definition 6.20. Let the dihedral group D of order 8 be generated by involutions t, u. An action of D on the abelian group L has the 3/4 generation property if the central involution of D acts as −1 on L and L is the sum of any three of L^{±}(t), L^{±}(u). An action has the 2/4-generation property if L is the sum of the fixed points of a pair of generating involutions.

Proposition 6.21. Suppose that the dihedral group D acts on the lattice L with the central involution acting as −1. For this action, equivalent are the properties of 3/4-generation, 2/4-generation and commutator density for a fourvolution in D.

Proof. Let f be an element of order 4 in D and t, u a pair of generating involutions. Set L_{1} := L^{+}(t), L_{2} := L^{−}(t). Note that rank(L) is even.

Assume the 3/4 generation property, and assume the notations of 6.20. Then L has even rank 2n and [L, D] ≤ (L_{1} + L_{2}) ∩ (L^{+}(u) ⊥ L^{−}(u)), which has index 2^{n} in L. Since (f − 1)^{2} = −2f, this intersection equals L(f − 1), whence density.

Assume density. Consider the action of D on \frac{1}{2}(L_{1} ⊥ L_{2})/L_{1} ⊥ L_{2}. The action of t is trivial and f acts as an involution with n Jordan blocks (as f^{2} = −1 = [t, u]).

We have L > L(f − 1) ≥ L(t − 1) + L(u − 1) + L(t + 1) + L(u + 1). Since ±t, ±u is a normal subset of generators of D, the right side is [L, Q] which by density equals L(f − 1). Note that L(t + 1) + L(t − 1) ≥ 2L and L(u − 1) + 2L = L(u + 1) + 2L. It follows that L(f − 1) = L(t − 1) + L(u − 1) + L(t + 1) ≤ L^{−}(t) + L^{−}(u) + L^{+}(u).

Similar arguments apply if we replace t, −t, u by any 3-subset of {t, −t, u, −u}. This completes the proof that density implies 3/4-generation.

Obviously, 2/4-generation implies 3/4-generation. Assume 3/4-generation and let t, u be any generating pair of involutions. Set M := L^{+}(t) + L^{+}(u), a sublattice of L. Since the central involution of D acts as −1, the summands meet trivially, whence M has rank 2n. Since L^{+}(t) is RSSD in L (see 5.11), it is RSSD in M, i.e. M is t-invariant. It follows that M contains L^{−}(u) = L^{+}(u^{t}), whence M = L by the 3/4-generation property. □

We can actually drop reference to the quadratic form in the previous result.

Proposition 6.22. Suppose that the dihedral group D acts on the free abelian group L with the central involution acting as −1. For this action, equivalent are the properties of 3/4-generation, 2/4-generation and commutator density for a fourvolution in D.

Proof. This follows from 6.21 once we define a D-invariant positive definite integer valued quadratic form. One uses the familiar trick of taking any integer valued positive definite quadratic form on L, then summing its transforms under D. □

7 Sultry twists and the NextBW procedure

We discuss some procedures for proving the main theorem. We continue to let BW abbreviate “Barnes-Wall”.

We first show how to start from a BW-type lattice of rank 2^{d−1} and create one of rank 2^{d}. Later, in 10.2 we show how a BW-type lattice of rank 2^{d} is uniquely determined by an ancestor of rank 2^{d−1}. Eventually, we use an induction argument which will show that a BW-type lattice is unique, so is the same (up to rescaling) as the lattices constructed in [2, 7].

An important technique here is to use the commutator density enjoyed by these lattices. The twisting by sultry transformations helps control the analysis.

Notation 7.1. Let M be a BW lattice of rank 2^{d−1} ≥ 3. Let Q be a lower group (see 15.2) in Aut(M), i.e. in some BRW^{0}(2^{d−1}, +) subgroup of Aut(M), which by induction is isomorphic to BRW^{0}(2^{d−1}, +) or d − 1 = 3 and Aut(M) ≅ W_{E_{8}}. Also, let f ∈ Q be a fourvolution, F := N_{Aut(M)}(Q) ≅ BRW^{0}(2^{d−1}, +); see 15.2. Now let r be duality level of M (see 6.7). Then r ∈ {0, 1} and r ≡ d(mod 2).

Definition 7.2. The Next BW Procedure. We use notations M, F, Q, f as in 7.1.

Form M_{1} ⊥ M_{2}, two orthogonal copies of M based on the isometries ψ_{i} : M → M_{i} and let V_{i} := ℚ ⊗ M_{i} be their ambient rational vector spaces. Set V := V_{1} ⊥ V_{2}. Also, we use ψ_{ij} := ψ_{i}^{−1}ψ_{j}, the natural isometry from M_{i} to M_{j}, extended to V_{i} → V_{j}. See Notation 5.2.

Define Q_{i}, F_{i} and f_{i} and the groups and element in End(V_{i}) corresponding to Q, F and f under ψ_{i}. Extend their actions to V in the natural way. Also, define the group Q_{12} as the natural diagonal subgroup of Q_{1}×Q_{2} and element f_{12} := f_{1}f_{2} ∈ Q_{12} acting on V (see 5.2).

For the SSD sublattices M_{i}[1 − r], M_{ij}[1 − r], M_{ij′}[−r], we denote the respective SSD involutions by t_{i}, t_{ij}, t_{ij′}. Observe that −1 = t_{1}t_{2} = t_{12}t_{12′}. For convenience and symmetry, we define t_{i′} := −t_{i}, t_{i′j} := t_{ij′}, t_{i′j′} := t_{ij}. Finally, we define D := ⟨t_{1}, t_{2}, t_{12}, t_{12′}⟩ ≅ Dih_{8} and R := ⟨Q_{12}, D⟩ ≅ 2_{+}^{1+2d}. So, R = Q_{12}D, a central product.

Define L_{d} := M_{1}[1−r]+M_{2}[1−r]+M_{12}[−r], and R-invariant lattice. We call L_{d} the type BW-successor to M. From 6.5 and M^{∗} = M[−r], we deduce that L_{d} is an integral lattice and since elements of the the above generating set have even norms, L_{d} is even.

Lemma 7.3. D(L_{d}) ≅ 1, 2^{2^{d−1}} as d is odd, even. Therefore, the duality level is the remainder of d + 1 modulo 2.

Proof. When d is even, L := L_{d} is the kernel of the epimorphism M_{1} ⊥ M_{2} → M/M[1], defined by (x^{ψ_{1}}, y^{ψ_{2}}) ↦ x + y + M[1]. Since M_{1} ⊥ M_{2} is unimodular, D(L) ≅ 2^{2^{d−1}}.

When d is odd, this is the same as the kernel of the epimorphism M_{1}^{∗} ⊥ M_{2}^{∗} → M^{∗}/M defined by (x^{ψ_{1}}, y^{ψ_{2}}) ↦ x + y + M. Since M_{1}^{∗} ⊥ M_{2}^{∗} has determinant 2^{−2^{d}}, L is unimodular.

The statement about duality level follows from 6.4. □

Lemma 7.4. We take L := L_{d} (in the notation 7.2). Here, r ∈ {0, 1}, d = rank(L) and r ≡ d (mod 2). Then:

(i) L is the sum of any three of the four lattices

M_{1}[1 − r], M_{2}[1 − r], M_{12}[−r], M_{12′}[−r].

(ii) L^{∗} is the sum of any three of the four lattices

M_{1}[1 − r], M_{2}[1 − r], M_{12}[−1], M_{12′}[−1].

Proof. This follows from 6.19. Here is a different proof. (i) Let i = 1 or 2. Define j by: {1, 2} = {i, j}.

Now, we observe that for any integer k,

(a) M_{i}[k] ≤ M_{j}[k] + M_{12}[k] ≤ M_{j}[k] + M_{12}[k − 1];

(b) M_{12′}[k] ≤ M_{12′}[k] + M_{12}[k] = M_{12}[k] + M_{i}[2 + k] ≤ M_{12}[k] + M_{i}[1 + k]. At once, (i) follows. For (ii), (a) and (b) prove equality of N = M_{1}[1−r] ⊥ M_{2}[1−r]+M_{12}[−1]

and N′ = M_{12}[−1] + M_{12′}[−1] + M_{i}[1 − r]. It is clear by taking dot products that N = N′ is in L^{∗}. If r = 0, L = N and if r = 1, N/L ≅ M_{12}[−1]/M_{12}[0] ≅ 2^{2^{d−1}}, whence N = L^{∗} (see 7.3). □

Corollary 7.5. In the notation of 6.17 and 7.2, the R-module L_{d} is commutator dense with respect to any fourvolution in R.

Proof. Since Q_{12} acts diagonally on V_{i}, we deduce [M_{i}[k], Q_{12}] = [M_{i}[k], f] = M_{i}[k + 1] for all k and i = 1, 2. Also, [M_{ij}[k], Q_{12}] = M_{ij}[k + 1]. Note also, that [M_{i}[k], t] = M_{ij}[k + 1] and [M_{ij}[k], t] = 0 or 2M_{ij}[k] = M_{ij}[k + 2] for t = t_{ij} or t_{ij′}. Similar statements hold for the M_{ij′}[k]. Since R is generated by Q_{12} and ⟨t_{1}, t_{ij}⟩, [L_{d}, Q_{12}] = [L_{d}, R].

We prove density first prove for a few special cases of f.

Take f = f_{1}f_{2}, which acts diagonally. Then, by induction, [M_{ij}[k], f_{1}f_{2}] = [M_{ij}[k], Q_{12}] = M_{ij}[k + 1] and [M_{i}[k], f] = M_{i}[k + 1] and similarly for M_{j}[k]. So we have density for this f.

Now, let f = t_{1}t_{12′}. Then for (x, y) ∈ V_{1} ⊥ V_{2}, (x, y)(1−f) = (x−y, y+x). Since (1 − f)^{2} = −2f, L_{d}(1 − f) contains 2L_{d} and the diagonal M_{ij}[1 − r], which generate [L_{d}, R] (see the first paragraph). So, we have density for this f.

Finally, let f be an arbitrary fourvolution in R. Then |L : L(f − 1)| has order |L : 2L|^{\frac{1}{2}} so L(f − 1) ≤ [L, Q] implies that L(f − 1) = [L, Q]. □

Corollary 7.6. For i = 1, 2 and for all integers j ≥ 0, M_{i}[1 − r] ∩ L[j] = M_{i}[1 − r + j].

Proof. Fix i. We choose f_{i} for the twisting since it preserves the M_{i}[k]. The equalities are valid for j even since L[2k] = 2^{k}L, for all k ≥ 0 and M_{i}[1−r] is a direct summand of L. Now, M_{i}[1−r]∩L[1] ≥ M_{i}[2−r]. Applying one more twist, which is a scaled isometry, we get M_{i}[2 − r] ∩ L[2] ≥ M_{i}[3 − r]. Since M_{i}[2−r]∩L[2] = M_{i}[2−r]∩2L = M_{i}[2−r]∩2M_{i}[1−r] = 2M_{i}[1−r] = M_{i}[3−r], whence all our containments are equalities. □

We give a fairly complete account of minimal vectors.

Lemma 7.7. (i) A minimal vector of L_{d} has norm 2^{⌊ \frac{d}{2} ⌋} and is in M_{1}[1 − r] or M_{2}[1 − r] or has the form x_{1} + x_{2}, where each x_{i} projects to a minimal vector of M_{i}[1−r], for i = 1, 2. Its norm is μ(M_{i}[1−r]) = 2^{1−r}μ(M) = 2^{⌊ \frac{d}{2} ⌋}.

(ii) The minimal vectors span L_{d}.

Proof. (i) Suppose that the minimal vector x is not in M_{1}[1−r] or M_{2}[1−r]. Write x = x_{1} + x_{2}, where x_{i} is the projection of x to V_{i}, i = 1, 2. Since x_{i} ∈ M_{i}[−r], x_{i} has norm at least \frac{1}{2}μ(M_{i}[1−r]), whence (x, x) ≥ μ(M_{i}[1−r]). It follows that these inequalities are equalities. The last statement follows from induction and 7.4.

Easily, (i) implies (ii) since L_{d} is the sum of three sublattices spanned by minimal vectors. □

Corollary 7.8. L_{d} is a lattice of BW-type.

Definition 7.9. A minimal vector in L_{d} has type 1, 2 or 3, respectively, as it is in M_{1}[1 − r], M_{2}[1 − r] or in neither. These three types partition MinV ec(L_{d}).

Lemma 7.10. For d ≥ 2, L_{d} is an indecomposable lattice.

Proof. As in the proof of 5.1, we see that the minimal vectors of L are partitioned into equivalence classes by membership in the L_{i}. However, it is clear from 7.7 and induction that there is just one equivalence class in the sense of 5.1. □

The following terminology will be useful. It applies to lattices used in 7.2 and later.

Definition 7.11. A lattice M is a scaled BW-lattice, abbreviated sBW lattice, if there is an integer s > 0 so that M ≅ \sqrt{s}BW_{2^{e}}, for some e > 0. A sublattice M of a BW-lattice L is called a suitably scaled Barnes-Wall sublattice (relative to L), abbreviated ssBW sublattice, if M is a sBW lattice and μ(M) = μ(L).

We use the notation BW_{2^{p,q}}, p ≤ q, for a scaled copy of BW_{2^{p}} whose isometry type is suitable as a sublattice of BW_{2^{q}}, i.e. a sBW lattice with minimum norm 2^{⌊ \frac{q}{2} ⌋}.

8 The groups R_{2^{d}}, G_{2^{d}} and invariant lattices

Notation 8.1. In this section, L = L_{d} and d ≥ 2 have the meaning of 7.1 and 7.2.

Definition 8.2. We define R := R_{2^{d}} := ⟨Q_{12}, t_{i}, t_{ij}⟩ ≅ 2_{+}^{1+2d}, where Q_{12} and the involutions are as in 7.2. We define G_{2^{d}} := N_{Aut(L)}(R_{2^{d}}).

Definition 8.3. Elements and subsets of G_{2^{d}} are called lower if in R and are otherwise called upper. In particular, a fourvolution 6.1 may be called upper or lower.

Theorem 8.4. For d ≥ 2, G_{2^{d}} ≅ BRW^{0}(2^{d}, +) ≅ 2_{+}^{1+2d}Ω^{+}(2d, 2).

Proof. The cases d ≤ 3 have been discussed earlier (and the case d = 4 was treated explicitly in [20]). We may assume that d ≥ 4. Since G_{2^{d}} is finite, containing R as a normal subgroup, G_{2^{d}} is contained in G̃ = BRW^{0}(2^{d}, +), the natural 2_{+}^{1+2d}Ω^{+}(2d, 2) subgroup of GL(2^{d}, ℂ) containing R; see Appendix A2.)

Let D be the dihedral group of order 8 described in 7.2. Then D ≤ G := Aut(L). Let t ∈ D be a noncentral involution. We claim that C_{G_{2^{d}}}(t)R/R corresponds to a maximal parabolic in G̃/R. For standard theory about parabolic subgroups, see [8].

Suppose t = t_{1} or t_{2}. By induction, Aut(M_{i}[1 − r]) contains a copy of G_{2^{d−1}} as N_{Aut(M_{i}[1−r])}(Q_{i}), and S := Stab_{G}(M_{1}[1 − r] ⊥ M_{2}[1 − r]) contains a group T of the form [2_{+}^{1+2(d−1)} × 2_{+}^{1+2(d−1)}].[Ω^{+}(2(d − 1), 2) × 2]. Also, since t_{ij} interchanges M_{1}[1 − r] and M_{2}[1 − r], it normalizes this group. Its image in G̃/R is a maximal parabolic, the stabilizer of a singular vector.

Suppose that t = t_{ij} or t_{ij′}. Then the above argument goes through with M_{i}[1−r], t_{i} replaced by M_{ij}[−r], t_{ij}, and gives a distinct subgroup of the form [2_{+}^{1+2(d−1)}×2_{+}^{1+2(d−1)}].[Ω^{+}(2(d−1), 2)×2] containing R. (Proof of distinctness: in both cases, the center of the respective stabilizer is {±1, ±t}.)

Therefore, G/R contains two different maximal parabolics of G̃/R, whence G = G̃, [8], so we are done. □

Remark 8.5. Note that the 8.4 uses only a basic result about orthogonal groups (maximality of certain stabilizers) but nothing very explicit about their interior structure, nor about particular elements. This is possible since we have a suitable uniqueness statement.

Lemma 8.6. The subgroup of Aut(L) which is trivial on L/L[1] is just R if d ≥ 3. If d = 2, it contains R with index 6 and modulo it, Aut(L) maps onto Sym_{3}. In the notation 7.2, L[1] = M_{1}[−r] + M_{2}[−r] + M_{12}[1 − r].

Proof. Let T be the subgroup trivial on L/L[1]. Note that [L, R] = L[1], 7.4. Therefore, T ≥ R.

Assuming T > R, we have a normal nontrivial 2-group T/R in G_{2^{d}}/R_{2^{d}}. If d = 2, the latter quotient is simple, the shape of G_{2^{d}} given in 8.4 shows that this is impossible. Assume that d = 2. Simplicity does not hold. The statement is an exercise (note that the isometry group is W_{F_{4}}, order 2^{7}3^{2}, which acts on L/L[1] as GL(2, 2)). □

Notation 8.7. For x ∈ MinV ec(L), let SF (x) := x^{R}. Call this the sultry frame containing x. (See 9.1). From 8.6 and the structure of R ≅ 2_{+}^{1+2d}, SF (x) is a double orthogonal basis, of cardinality 2^{d+1}.

Proposition 8.8. Let x, y ∈ MinV ec(L). Equivalent are (i) y ∈ SF (x); (ii) x − y ∈ L[1].

Proof. Trivially, (i) implies (ii). For (ii), we use a familiar argument. Let z ∈ SF (x). First we note that z ± y ∈ L[1] is 0 or has norm at least 2μ(L). Assuming y ≠ ±z, we have (z ± y, z ± y) = 2μ(L) ± 2(z, y) ≥ 2μ(L), whence (z, y) = 0. This is not the case for every z ∈ SF (x). □

Proposition 8.9. The number of minimal vectors is (2^{d}+2)(2^{d−1}+2) … (2^{2}+ 2)(2 + 2). The values for small d are:

d  |MinV ec(L)|  Prime Factorization
0  2  2
1  4  2^{2}
2  24  2^{3}3
3  240  2^{4}3.5
4  4320  2^{5}3^{3}5
5  146880  2^{6}3^{3}5.17
6  9694080  2^{7}3^{4}5.11.17
7  1260230400  2^{8}3^{4}5^{2}11.13.17
8  325139443200  2^{9}3^{5}5^{2}.11.13.17.43
9  167121673804800  2^{10}3^{5}5^{2}11.13.17.43.257
10  171466837323724800  2^{11}3^{8}5^{2}11.13.17.19.43.257
11  351507016513635840000  2^{12}3^{8}5^{4}11.13.17.19.41.43.257
12 1440475753672879672320000 2^{13}3^{9}5^{4}11.13.17.19.41.43.257.683

Proof. Use 7.7, 8.7, 8.8 and induction. □

Corollary 8.10. If x ∈ MinV ec(L), Stab_{G_{2^{d}}}(x + L[1])/R_{2^{d}} is a maximal

parabolic subgroup of G_{2^{d}}/R_{2^{d}} of the shape 2^{\binom{d}{2}}:GL(d, 2).

Proof. The pairs {±x} of minimal vectors in this coset is an orbit of R_{2^{d}} for which a point stabilizer E is elementary abelian of order 2^{1+d}; see 8.8. These pairs of vectors are exactly the minimal vectors of the total eigenlattice of E, so as a set are stable under N_{G_{2^{d}}}(E), which has the indicated properties. □

Corollary 8.11. When d is even, L ∩ 2L^{∗} = L[1], whence the lower group is normal in Aut(L) and G_{2^{d}} = Aut(L).

Proof. When d is even, the duality level is 1, whence L∩2L^{∗} = L∩2L[−1] = L ∩ L[1] = L[1] is invariant by the entire automorphism group. Now use 8.6. □

When d is even, this result essentially solves the problem of determining the automorphism group. The case d arbitrary is harder (it is finally proved in 11.9, which does not use 8.11).

Proposition 8.12. For an odd prime, p, L/pL is an absolutely irreducible module for G_{2^{d}}.

Proof. This is trivial, since R acts absolutely irreducibly. □

Lemma 8.13. For all k, L[k]/L[k +1] is an absolutely irreducible 𝔽_{2}-module for G_{2^{d}}.

Proof. This is easy to check for d ≤ 4, so we assume that d ≥ 5 and use induction.

We may assume that k = 0. Let D ≅ Dih_{8} be as in 7.2. For a noncentral involution t of D, we get by induction that C_{G_{2^{d}}}(t) acts irreducibly on each L^{±}(t)/[L^{±}(t), C_{R}(t)].

Since L is a sum of fixed point sublattices for the noncentral involutions of D 6.21, it follows that L/L[1] has two absolutely irreducible composition factors for C_{G_{2^{d}}}(t), each of dimension 2^{d−2}.

The group C_{G_{2^{d}}}(t), of shape [2_{+}^{1+2(d−1)} × 2_{+}^{1+2(d−1)}].Ω^{+}(2(d − 1), 2) (discussed in the proof of 8.4), acts on L/L[1] and has exactly one irreducible submodule, namely T el(L, t)/L[1], and two composition factors (this follows by induction on d, since on any eigenspace for t, we know the irreducible quotients for any lattice invariant under C_{G_{2^{d}}}(t)). These irreducible submodules distinct as t ranges over a set of generators for D (e.g. t_{1}, t_{12} for the group of 7.2).

It follows that L/L[1] is irreducible for the action of G_{2^{d}}. We now prove absolute irreducibility. If K is an extension field of 𝔽_{2} and K ⊗ L/L[1] decomposes, then its restriction to C_{G_{2^{d}}}(t) would have over 4 composition factors (since O_{2}(C_{G_{2^{d}}}(t)) acts nontrivially), which is impossible since, by induction, the composition factors for C_{G_{2^{d}}}(t) are absolutely irreducible of dimension 2^{d−2}. □

Proposition 8.14. Let M be a lattice in ℚ ⊗ L which is invariant under G_{2^{d}}. Then there is a rational number r so that rM = L or L[1].

Proof. We may assume that M ≤ L. By 8.12, we may assume that L/M is a power of 2. For some positive integer n, 2^{n}L ≤ M. Now use 8.13 and the fact that [L[k], R_{2^{d}}] = L[k + 1]. □

Later, in 11.9, we prove that G_{2^{d}} is all of Aut(L_{d}).

9 Sultriness

When f is a fourvolution on a lattice L, 1 − f (actually, any of ±1 ± f) is an endomorphism of L which is also an isometry scaled by \sqrt{2}. Next, we see that a sultry transformation is naturally interpreted as a scaled lift of a transvection, a point which suggested the term “sultry”.

Theorem 9.1. The function γ_{f} : Aut(L) → Aut(L), x ↦ (1 − f)^{−1}x(1 − f), normalizes R = R_{2^{d}} and G_{2^{d}}. Furthermore, γ_{f} is the identity on C_{R}(f) and if x ∈ R\C_{R}(f), γ_{f} takes x to fx ∈ R\C_{R}(f), hence normalizes and induces an outer automorphism on the dihedral group ⟨f, x⟩. Hence, on R/Z(R), γ_{f} acts as the transvection associated to the nonsingular point Z(R)f of R/Z(R).

Proof. Since \frac{1}{\sqrt{2}}(±1 ± f) is orthogonal, the image of γ := γ_{f} is a subgroup of the orthogonal group. Since any ±1 ± f carries each L[k] onto L[k + 1], the image of γ stabilizes L. We conclude that γ takes Aut(L) onto itself.

We calculate that x(1−f) = x−xf = x−f^{−1}x = x+fx = (−f +1)(fx), which proves the remaining statement. □

10 Proof of uniqueness

Notation 10.1. Given d ≥ 3 and L_{1}, L_{2}, we let 𝔛 := 𝔛(L_{1}, L_{2}) be the set of all X-quadruples of the form (L, L_{1}, L_{2}, t); see 3.3.

Theorem 10.2. We use the notation in 3.3, 7.1, 7.2, 8.2 and 10.1. Suppose that d ≥ 3 and (L_{1}, L_{2}) is an orthogonal pair of lattices, so that each L_{i} is BW-type of rank 2^{d−1}.

(i) 𝔛 is an orbit under the natural action of F_{1} × F_{2}, where F_{i} := Stab_{Aut(L_{i})}(L_{i}[1 − r]) (see 7.2; by induction, F_{i} ≅ G_{2^{d−1}}). Define Q_{i} := C_{F_{i}}(L_{i}/L_{i}[1]).

The elements of 𝔛 are in correspondence with each of the following sets.

(a) F_{1}/Q_{1};

(b) F_{2}/Q_{2};

(c) Pairs of involutions {s, −s} in the orthogonal group on V which interchange L_{1} and L_{2}.

(d) Dihedral groups of order 8 which are generated by the SSD involutions associated to L_{1}, L_{2} and involutions as in (c).

(ii) (a) The subgroup G_{L}^{0} of F_{1} × F_{2} which stabilizes L has structure Q_{1} × Q_{2} ≤ G_{L}^{0} and G_{L}^{0}/Q_{12} is the diagonal subgroup of F_{1}/Q_{1} × F_{2}/Q_{2} with respect to the isomorphism induced by s, an involution as in (i.c).

(b) The subgroup G_{L} of Aut(L_{1} ⊥ L_{2}) ≅ Aut(L_{i}) ≀ 2 which stabilizes L is G_{L}^{0}⟨s⟩. We have G_{L} ≅ [2_{+}^{1+2(d−1)} × 2_{+}^{1+2(d−1)}].[Ω^{+}(2(d − 1), 2) × 2].

(c) The subgroup of G_{L} which acts trivially on L/L[1] is R := ⟨Q_{12}, s, t_{i}⟩, where t_{i} is the SSD involution associated to L_{i}. The quotient G_{L}/R ≅ 2^{2d−2}:Ω^{+}(2d−2, 2) is a maximal parabolic subgroup of Out^{0}(2_{+}^{1+2d}) ≅ Ω^{+}(2d, 2). (See Appendix A.0).

The extension in (c) is split, despite G_{2^{e}} being nonsplit over R_{2^{e}} for e ≥ 4. See Appendix A2. Proof. (i) We prove the classification by induction. For d = 2, Aut(L_{D_{4}}) ≅ 2_{+}^{1+4}[Sym_{3} ≀ 2] and for d = 3, Aut(L_{E_{8}}) ≅ W_{E_{8}}. When d = 4, the main theorem follows from the arguments of [20].

For the rest of the proof, we assume that d ≥ 4. By induction, a lattice satisfying the X(2^{d−1}) condition is uniquely determined up to isometry. This applies to the lattices L_{1}, L_{2}.

Let L be any member of 𝔛 and set G := Aut(L). There is an X-quadruple (L, L_{1}, L_{2}, t). Then det(L) and |L:M_{1}[1 − r] ⊥ M_{2}[1 − r]| are determined.

Let p_{i} be the orthogonal projection of L to V_{i} := ℚ ⊗ L_{i}, for i = 1, 2. Then L^{p_{i}} is a lattice containing L_{i} with quotient isomorphic to 2^{2^{d−2}}. Since Q acts trivially on L/[L_{1} ⊥ L_{2}], Q_{i} acts trivially on L^{p_{i}}/L_{i}. Therefore, L^{p_{i}} is the −1 twist of L_{i} with respect to Q_{i}, i.e., L^{p_{i}} = L_{i}(1 − f_{i})^{−1}, for a suitable fourvolution f_{i} ∈ Q_{i}.

There is a dihedral subgroup D of R so that t ∈ D. If y ∈ D is an involution which does not commute with t, then y interchanges L_{1} and L_{2}. Also, if J^{±} are the eigenlattices for y, then L is the sum of any three of the four L_{1}, L_{2}, J^{+}, J^{−}, by 6.19.

It follows that L is determined by D in the sense that L = [L_{1} ⊥ L_{2}] + ([L_{1} ⊥ L_{2}]^{+}(y))(f − 1)^{−1}, where f = ty.

is the sum of the fixed point sublattices of the involutions of D (see 6.19, 7.4).

Now, to what extent does L_{1} ⊥ L_{2} determine D? The answer is: up to conjugacy in Aut(L_{1} ⊥ L_{2}) ≅ G_{2^{d−1}} ≀ 2 (note that d ≥ 2 here). Our group D is generated by the center of the natural index 2 subgroup of Aut(L_{1} ⊥ L_{2}) and a wreathing involution. In general, wreathing involutions in a wreath product of groups K ≀ 2 form an orbit under the action of either direct factor isomorphic to K in the base group of the wreath product. This proves correspondence with (c) and (d). The stabilizer subgroup is diagonal in the base group K × K, and either direct factor represents all cosets of the stabilizer (whence the equivalence with (a) and (b))

It follows that, up to isometry preserving L_{1} ⊥ L_{2}, D, hence L, is determined by the pair of indecomposable lattices L_{1} and L_{2}.

Proof of statements (ii) and (iii) are easy. The statement about parabolic subgroups is proven with a standard result from the theory of Chevalley groups, e.g. [8]. Independently of that theory, the maximality could be proved directly by showing that there is no system of imprimitivity on the set of isotropic points. This is an exercise with Witt’s theorem. □

11 Minimal vectors, the zoop2 property and Aut(BW_{2^{d}})

We continue to use the notations of 8.1 and 8.2.

Remark 11.1. For L = BW_{2^{d}} a Barnes-Wall lattice and k ∈ ℤ, we have MinV ec(L[k]) = MinV ec(L)[k] (see 6.5).

Theorem 11.2. We use notation of 7.2. The group G_{2^{d}} acts transitively on the set of minimal vectors.

Proof. We use notation of 7.9. It is clear that the minimal vectors of types 1 and 2 are in a single G_{2^{d}}-orbit, say O. Consider a minimal vector u + v of type 3. We assume that L corresponds to the involution s = t_{ij′} in the sense of 10.2 (i). Then v and u^{t_{i′j}} differ by an element of M_{2}[1−r], so by induction, these are in the same orbit under Q_{2}, equivalently under Q_{12}. Therefore, u and v are in the same R-orbit.

By induction, we have transitivity on the minimal vectors of type 3 by the group RF_{12}, where the second factor is the natural diagonal subgroup of F_{1} × F_{2}. Call O′ the orbit containing the type 3 minimal vectors.

Suppose that G_{2^{d}} is not transitive. First Contradiction. Then MinV ec(L) is the disjoint union of two orbits O and O′ and so G_{2^{d}} preserves the ℤ-span of O, which is just M_{1}[1 − r] ⊥ M_{2}[1 − r], an orthogonal sum of two orthogonally indecomposable lattices. Thus G′_{2^{d}} ≥ R leaves both summands invariant, which is impossible since R is irreducible on ℂ ⊗ L. Second Contradiction. The lower involutions form a conjugacy class in G_{2^{d}}, so there is g ∈ G_{2^{d}} which conjugates t_{1} to t_{12}. Then g takes the set of minimal vectors fixed by t_{1} (those of type 2) to those fixed by t_{12}, which are contained in those of type 3. Therefore O and O′ are not distinct orbits. Transitivity follows. □

Now we give a few results about stablilzers in G_{2^{d}}. These will be strengthened later.

Lemma 11.3. (i) If F is a sultry frame and x ∈ F, then {g ∈ R|x^{g} = ±x} = {g ∈ R|y^{g} = ±y for all y ∈ F } is a maximal elementary abelian subgroup of R. Call it R_{F}. The quotient R/R_{F} operates regularly on the eigenlattices.

(ii) Define C_{F} := C_{G_{2^{d}}}(F/{±1}) := {g ∈ G_{2^{d}}|y^{g} = ±y for all y ∈ F }.

This is elementary abelian and has shape 2^{1+d+\binom{d}{2}}.

(iii) Its normalizer N_{F} := N_{G_{2^{d}}}(C_{F}) = Stab_{G_{2^{d}}}(F) in G_{2^{d}} satisfies N_{F} /C_{F} ≅ AGL(d, 2). We have R_{F} ≤ C_{F}.

Proof. (i) Set P := {g ∈ R|x^{g} = ±x}, Q := {g ∈ R|y^{g} = ±y for all y ∈ F }. Observe that Q ≤ P and Q is elementary abelian. Transitivity of R on F and normality of P in R implies that P = Q has order 2^{d+1}.

(ii) This follows from (i) and the order of the unipotent radical for the stabilizer of a maximal totally isotropic subspace for Ω^{+}(2d, 2).

(iii) This follows from the actions of R on R_{F} together with the structure of the stabilizer of a maximal totally isotropic subspace for Ω^{+}(2d, 2). □

Definition 11.4. Suppose that F is a frame. A subset S ⊆ V has the zop2 property (with respect to F) if |(x, y)| is 0 or a power of 2, for all x ∈ F and y ∈ S. We say that S has the zoop2 property if is has the zop2 property and just one power of 2 occurs among the scalars |(x, y)|, x ∈ F, y ∈ S.

Lemma 11.5. For all integers p, q, any minimal vector of L[q] has the zoop2 property with respect to any sultry frame in L[p].

Proof. We may assume that p = 0 and q ∈ {0, 1}. The property is easy to check for d ≤ 3. We assume d ≥ 4. Say x ∈ MinV ec(L), y ∈ SF (x) and z ∈ MinV ec(L[q]) so that (x, z) ≠ 0 ≠ (y, z). Take a lower involution t so that t fixes x and y. Then x, y ∈ L^{+}(t), a ssBW lattice, and z projects to a minimal vector in L^{+}(t)[q], so we are done by induction. □

Definition 11.6. Given a sultry frame F of 2^{d+1} elements, there are 2^{d} subsets which form a basis. Suppose X is such a set. If v ∈ V, we write v = ∑_{x∈X} a_{x}x and define the support of v to be the set {x ∈ X|a_{x} ≠ 0}. This depends on the double basis F, not on the choice X ⊂ F.

Lemma 11.7. Suppose that d ≥ 2. Let x ∈ MinV ec(L) and A(x) := {y ∈ MinV ec(L)|(x, y) = \frac{1}{2}(x, x)\}. Then A(x) ∪ \{x\} spans a lattice isometric to the Hamming code lattice described in 5.6. In particular, there is a labeling of SF (x)/{±1} with 𝔽_{2}^{d} so that the elements of A(x) have support which is an affine 2-space.

Proof. Define J_{0} to be the square lattice spanned by SF (x) and J the lattice spanned by A(x) and SF (x). In a natural way, J/J_{0} corresponds to a nonzero code in \frac{1}{2}J_{0}/J_{0} ≅ 𝔽_{2}^{2^{d}}. Since μ(L) = (x, x), this code has minimum weight at least 4. For y ∈ A(x), supp(y) is a 4-set with respect to the double basis SF (x), 11.5. Therefore the minimum weight of C is 4.

Note that we have an action of AGL(d, 2) on \frac{1}{2}J_{0}/J_{0} by coordinate permutations. This follows from 11.3. This action is triply transitive. Since C has minimum weight 4, its weight 4 codewords forms a Steiner system with parameters (3, 4, 2^{d}) which is stable under this action of AGL(d, 2). Such a system is unique since in AGL(d, 2), the stabilizer of three points fixes a unique fourth point. Therefore, C is the code H_{d}^{e}, up to equivalence.

Finally, we must show that A(x)∪{x} spans J. If d = 2, L ≅ L_{D_{4}} and the result is easy to check directly. We assume d ≥ 3. Let y ∈ SF (x), y ≠ ±x. Since d ≥ 3, we may choose a lower involution t which fixes both x and y (in the notation of 11.3, t ∈ R_{F}). Let L^{+} be the sublattice of points of L fixed by t, a sBW lattice. Then, induction implies that the sublattice of L^{+} spanned by A(x) ∩ L^{+} contains y. We conclude that SF (x) ⊂ span(A(x) ∪ {x}), and we are done. □

Proposition 11.8. Suppose that d ≥ 4. For x ∈ MinV ec(L),

Stab_{Aut(L)}(x) ≤ Stab_{Aut(L)}(SF (x)).

Proof. Define A(x) := \{y ∈ MinV ec(L)|(x, y) = \frac{1}{2}(x, x)\}. By 11.7 lattice J := span(A(x)∪{x}) contains SF (x) and is a copy of the lattice in 5.6. Since d ≥ 4, given a weight 4 codeword, there exists another weight 4 codeword which meets it in a 1-set (this is not so for d = 3). Therefore, SF (x) = {z ∈ MinV ec(J)|(z, J) ≤ \frac{1}{2}(z, z)ℤ\}, we are done (see the proof of 5.6). It follows from 5.1 that Stab_{Aut(L)}(x) ≤ Stab_{Aut(L)}(J). □

Corollary 11.9. For d ≥ 4, Aut(L_{d}) = G_{2^{d}}.

Proof. This follows since G_{2^{d}} is transitive on minimal vectors and the stabilizer of some minimal vector in G is contained in G_{2^{d}}. □

Definition 11.10. Let x ∈ MinV ec(L) and SF (x) its sultry frame. Let q ∈ ℤ and k ∈ ℤ. Define A(L, x, q, k) := {z ∈ MinV ec(L[q])|(z, y) ∈ {0, ±2^{k}} for all y ∈ SF (x)}. This is is the level k layer in MinV ec(L[q]) with respect to x or SF (x).

Lemma 11.11. Suppose that the group G_{0} factorizes as G_{0} := GZ, where G, Z are subgroups so that [G, Z] = 1. Suppose that G_{0} acts on the set X and that G stabilizes and acts transitively on a set of Z-orbit representatives. Let S be the set of all G-invariant sets of orbit representatives. Then Z acts transitively on S.

Proof. A member of S is determined by any element of X which it contains. Therefore the members of S partition X.

Suppose that X_{1}, X_{2} ∈ S. Let A be a Z-orbit and let a_{i} be the unique element of A ∩ X_{i}, for i = 1, 2. Take z ∈ Z so that a_{1}^{z} = a_{2}. Then X_{2} and X_{1}^{z} are both G-invariant sets of orbit representatives and contain a_{2}, hence are equal. □

Note that the next result deals with minimal vectors in all sultry twists of L.

Notation 11.12. For integers d ≥ 2 and p, q ∈ ℤ, define I(d, p, q) to be the set of integers listed below. Here, r ∈ {0, 1} is the remainder of d modulo 2, m := ⌊\frac{d}{2}⌋ and s ∈ \{0, 1\} is the remainder of p − q modulo 2. We define

I(d, p, q) := ⌊\frac{p + 1}{2}⌋ + ⌊\frac{q + 1}{2}⌋ + \{−rs, 0, 1, …, m\}.

As usual a + {b, c, … }, means {a + b, a + c, … }. We call I(d, p, q) the interval of exponents for dot products of minimal vectors. (See 11.14 for an explanation of this term.)

Example 11.13. Some examples:

I(3, p, p) = \begin{cases}\{p, p + 1\} & p even; \\ \{p + 1, p + 2\} & p odd. \end{cases} ; I(3, p, p + 1) = \{p, p + 1, p + 2\};

I(4, p, p) = \begin{cases}\{p, p + 1, p + 2\} & p even; \\ \{p + 1, p + 2, p + 3\} & p odd. \end{cases} ; I(4, p, p+1) = \{p+1, p+2, p+3\}.

Lemma 11.14. The set of integers

{(x, y)|x ∈ MinV ec(L[p]), y ∈ MinV ec(L[q])}

is {0, ±2^{k}|k ∈ I(d, p, q)}; see 11.12.

Proof. Let m := ⌊\frac{d}{2}⌋ and r := d − 2m ∈ \{0, 1\}.

First we take p = q = 0. Then for x ∈ MinV ec(L), with respect to a basis Ω ⊆ SF (x) an element y ∈ MinV ec(L[q]) has the form y = 2^{−t} ∑_{u∈A} uε_{B}, where A ⊆ Ω is an affine subpace of Ω of dimension a ≤ d, and a satisfies m = m−2t+a, or a = 2t. Then (x, y) = 0 or ±2^{m−t}. Thus, the values of a, t, m−t which occur are {0, 2, … 2m}, {0, 1, …, m}, {0, 1, …, m}, respectively.

Next, if p = 0 and q = −1, a similar discussion applies, but here μ(L[−1]) = \frac{1}{2}μ(L), so we get the condition m − 1 = m − 2t + a, or a = 2t − 1, whence odd parity for a. Therefore, the values of a, t, m − t which occur are {1, 3, …, d + r − 1}, {1, …, d − m}, {2m − d, 2m − d + 1, …, m − 1}, respectively.

Suppose that p = −1 and q = 0. We get the condition m = m−1−2t+a, or a = 2t + 1 is odd. Therefore the values of a, t, m − t which occur are {1, 3, …, d + r − 1}, {0, 1, …, d − m − 1}, {2m − d + 1, 2m − d + 2, …, m}, respectively.

For the general case, just observe that I(d, p, q + 2) = 1 + I(d, p, q), I(d, p + 2, q) = 1 + I(d, p, q), and I(d, p + 1, q + 1) = 1 + I(d, p, q). □

Lemma 11.15. Let G = AGL(d, 2) act naturally on the permutation module A := 𝔽_{2}^{Ω}, where Ω := 𝔽_{2}^{d} with the natural G-action. Let B be the submodule generated by the affine subspaces of codimension 1. For d ≥ 3, H^{1}(G, B) = 0.

Proof. We have an exact sequence 0 → B → A → A/B → 0. From this, the long exact cohomology sequence gives the exact sequence H^{0}(G, A/B) → H^{1}(G, B) → H^{1}(G, A). The right term is, by the Eckmann-Shapiro lemma, isomorphic to H^{1}(G_{0}, 𝔽_{2}), where G_{0} ≅ GL(d, 2) is the stabilizer of 0 in G. This is isomorphic to Hom(G_{0}, 𝔽_{2}), which is trivial for d ≥ 3. The module A/B is indecomposable for AGL(d, 2), with a faithful module of dimension d as the socle and quotient the trivial 1-dimensional module. Since the fixed points are 0, H^{0}(G, A/B) = 0. Exactness implies that H^{1}(G, B) = 0. □

Theorem 11.16. Let d ≥ 2 and let F := SF (x), for a minimal vector x ∈ L[p] (see 8.7). Also, define H := Stab_{G_{2^{d}}}(F) (denoted N_{F} in 11.3 (ii) ).

(i) There exists a basis X contained in F and labeling of X by 𝔽_{2}^{d} so that with respect to X, H is the monomial group E_{C_{X}}:AGL(d, 2) (see 5.5), where C_{X} is the code generated by affine subspaces of codimension 2 in X and where AGL(d, 2) is the natural subgroup of permutation matrices. The code C_{X} has parameters [2^{d}, 1 + d + \binom{d}{2}, 2^{d−2}].

(ii) Any two labelings as in (i) are conjugate by the action of H.

(iii) For fixed q, k, the sets A(L, x, q, k) (see 11.10) are the elements of L[q] which are all linear combinations of X of the form 2^{−t} ∑_{x∈A} xε_{B}, where A is an affine subpace of X of dimension a, p + a − 2t = q and k = p + ⌊\frac{d}{2}⌋ − 2t = ⌊\frac{d}{2}⌋ + q − a and ε_{B} effects sign changes exactly at indices in B ⊆ A; here B is in the code C_{A}, which is spanned by all A ∩ S, where S is an affine subspace of codimension 2 in X.

(iv) For a fixed integer a, the sets A(L, x, q, k) are nonempty exactly for the indices k ∈ I(d, p, q) and they are the orbits of H on MinV ec(L[q]).

Proof. (i): We use notation of 7.2. We may and do assume that d ≥ 4. There is by induction a basis X_{1} of V_{1} contained in F and labeling of X_{1} by Ω_{1} := 𝔽_{2}^{d−1} so that we get an identification of the stabilizer of SF (x) ∩ V_{1} with E_{C_{Ω_{1}}}:AGL(d − 1, 2), in analogous notation.

The frame is a double basis for the total eigenspace of E_{1}, a maximal elementary abelian subgroup of a lower group R_{1} on M_{1}. Using our standard diagonal notation 5.2, 10.2, take involution s = t_{12′} in dihedral group D and the corresponding subgroup E_{12} of R_{12}. Then s interchanges M_{1} and M_{2}. Let t ∈ D be the SSD involution associated to M_{1}. Then E := ⟨E_{1}, t⟩ is a maximal elementary abelian group in R and its total eigenlattice has the frame F as a double basis. Identify Ω_{1} with a codimension 1 affine subspace of Ω := 𝔽_{2}^{d}. We define Ω_{2} to be the complement in Ω of Ω_{1}. Choose any vector v_{0} ∈ Ω_{2}. Let v_{1} ∈ X_{1} be a frame vector labeled by 0 and let v_{2} := v_{1}^{s} ∈ X_{2} := SF (x) ∩ V_{2}. Since the action of s is an isomorphism of the transitive C_{H}(s)-sets X_{1} and X_{2}, the labeling on X_{1} transfers uniquely to X_{2} and we translate this labeling to X_{2} via vector addition by v_{0} to make a labeling of X_{2} by Ω_{2}. The resulting labeling of X is uniquely determined (depending on v_{0}, s, X_{1}).

From 15.3, we see that in G_{2^{d}}, a frame stabilizer contains a subgroup J isomorphic to AGL(d, 2) in the normalizer of E which permutes a basis of the eigenlattice. Its intersection, K, with a natural G_{2^{d−1}} subgroup is an analogous AGL(d−1, 2) subgroup. Let Z be the group generated by {±1_{V} }.

There are just two J-invariant sets of Z-orbit representatives in F. When one of them is restricted to K, we get two orbits. If X_{1} is one of these, the other is X_{1}^{s} or −X_{1}^{s}. We replace s by −s if necessary to arrange for the other to be X_{1}^{s}. Then s ∈ J. The labeling on X_{1} now extends to all of X, which is an H-invariant set.

(ii): Let ℓ, ℓ′ be two labelings for which H is the indicated monomial group. We shall transform one to the other by action of H. Call the domain of a labeling to be the points of SF (x) which get a label.

The stabilizer H_{ℓ} in H of the labeling ℓ (equivalently, of its domain) is a complement to the normal subgroup of sign changes. Such a subgroup is isomorphic to AGL(d, 2). We first note that any two complements are conjugate. This follows from a cohomology argument, 11.15. From this, we may and do arrange for the two labelings to have the same domain, which we call D. Since H acts 3-transitively and leaves invariant a unique Steiner system with parameters [3, 4, 2^{d}], addition of labels of vectors is determined by H once an origin is chosen. Given an origin, a partial labeling of D by a basis of 𝔽_{2}^{d} determines the labeling. Any two such choices lie in one orbit under the action of H.

(iii) and (iv): It is clear from induction and the form of the types 1, 2 and 3 minimal vectors that a minimal vector has the zoop2 property 11.4 with respect to a given sultry frame. So, the nonempty sets A(L, x, q, k), for k ∈ I(d, p, q), partition MinV ec(L[q]). It remains to show that they are orbits for the frame stabilizer.

The action of AGL(d, 2) is transitive on affine subspaces of given dimension.

Write v = v_{1} + v_{2}, where v_{i} is the projection to V_{i}, i = 1, 2. Either v = v_{1}, v = v_{2} or v_{1} ≠ 0 ≠ v_{2} and there exist integers t_{i} and affine subspaces A_{i} of X_{i} and B_{i} ∈ C_{A_{i}} so that v_{i} = 2^{−t_{i}} ∑_{y∈A_{i}} yε_{B_{i}}. The zoop2 property implies that t_{1} = t_{2} and dim(A_{1}) = dim(A_{2}). Call these common values t, a, respectively. We assume that v_{1} ≠ 0 ≠ v_{2}.

If there exists an affine hyperplane X′ of X so that U := supp(v) ⊆ X′, we use induction since the v is a minimal vector in the sBW sublattice of rank d − 1 supported by U. Suppose that no such X′ exists. Then we are in the third case v_{1} ≠ 0 ≠ v_{2} and we use notation v_{1} ≠ 0 ≠ v_{2} as above. Let X′ be any affine hyperplane. We claim that |U ∩X′| = \frac{1}{2}|U|. Suppose otherwise. Then, replacing X′ by its complement, we may assume that |U ∩ X′| < \frac{1}{2}|U|. Then the sublattice S of L supported by X′ has a vector in S^{∗} of norm less than \frac{1}{2}μ(L), a contradiction. The claim follows. We get a final contradiction by using 4.6. □

Remark 11.17. The results 11.16 (iii), (iv), were proved in [7]; see Théorème I.5, Théorème II.2.

12 Orbits on norm 4 frames in L_{E_{8}}.

We give an application of our theory by giving a short proof that the Weyl group of E_{8} has just four orbits on plain frames 6.15 of norm 4 vectors in L_{E_{8}}, equivalently, of D_{1}^{8}-sublattices. This result can be deduced from a classification of ℤ_{4} codes [9].

Definition 12.1. Let L be any lattice. If M is a sublattice, 2L ≤ M ≤ L, the d-invariant of the frame F (relative to M) is the dimension of the span of F + M/M. Also, we say two plain frames E, F are congruent if and only if E + M = F + M.

The d-invariant of a plain frame F is the dimension of the subspace of L/2L spanned by F + 2L, i.e., the relative d-invariant for M = 2L.

Remark 12.2. Now suppose that L = BW_{2^{3}}. The d-invariant of a frame is a number between 1 and 4 since the image is not trivial and spans a totally singular subspace.

Remark 12.3. It is easy to see that the Weyl group of E_{8} is transtive on frames of roots. This follows from Witt’s theorem since the Weyl group induces the full orthogonal group on L_{E_{8}} modulo 2 and any frame of roots spans an index 16 sublattice with all even inner products, hence corresponds mod 2 to a totally isotropic subspaces with nonsingular vectors. The next result refers to action of the proper subgroup G_{2^{3}} on frames of roots and norm 4 vectors.

Proposition 12.4. (i) In the action of G_{2^{3}} on frames of norm 2 vectors, there are four orbits. They are distinguished by their d-invariants relative to the sultry twist L[1].

(ii) In the action of W_{E_{8}} on frames of norm 4 vectors, there are four orbits. They are distinguished by their d-invariants.

Proof. (i) It is easy to determine the orbits of G_{2^{3}} on frames of roots. They are represented by the following vectors with respect to x_{1}, …, x_{8}, a standard orthogonal basis of roots (see 5.4):

F_{1} : ±x_{1}, …, ±x_{8}.

F_{2} : ±x_{i}, i ∉ A; \frac{1}{2} ∑_{j∈A} ±x_{j}, where A is a 4-set of indices representing a Hamming codeword, and evenly many signs over A are minus.

F_{3} := ±x_{i}, i ∈ B; \frac{1}{2}(00aaaa00), \frac{1}{2}(0000pqrs), \frac{1}{2}(00tu00cc), where B is a 2-set of indices (which we take to be {1, 2}) and the indicated partition of the eight indices into 2-sets has the property that the union of any two of them is a Hamming codeword. Also, a, b, c, p, q, r, s, t, u ∈ {±1} and where p = −q, r = −s, t = −u.

F_{4} := ±x_{1} and ±\frac{1}{2}(01111000), ±\frac{1}{2}(0001, −1, 110), ±\frac{1}{2}(0, −1, 0, 0, 1, 0, 1, 1). The proof is an easy exercise with the action of the monomial group

H ≅ 2^{7}:AGL(3, 2), a subgroup of G_{2^{3}}, where the group of sign changes at evenly many indices is indicated by 2^{7}. Since G_{2^{3}} is transitive on roots, an orbit of such a frame has a member containing x_{1}. We now restrict ourselves to transformations by elements of H ≤ G_{2^{3}}. If the remaining members of the frame are the x_{i}, we are in case F_{1}. If not, one can arrange for the next member of the frame to be something of the form mentioned in case F_{2}, supported by a 4-set, A. If all remaining members of the frame are some ±x_{j} or supported by the same 4-set, we are in the orbit of F_{2}. If not, similar reasoning brings us to case F_{3} or F_{4}.

One must show that these frames represent different orbits, and that is accomplished by showing that their images in L/L[1] span subspaces of dimensions 1, 2, 3 and 4, respectively. (This is verified by Smith canonical forms, easy to do by hand or with a software package like Maple): in our notation, L[1] is the ℤ-span of the x_{i} ± x_{j} and \frac{1}{2}(x_{1} + ⋯ + x_{8}). These dimensions are the d-invariants of the original orbits.

(ii) Let O_{i}, for i = 1, …, r be the orbits. Since W_{E_{8}} induces the full orthogonal group on L/2L, any orbit has a representative contained in L[1] since L[1]/2L is a maximal totally singular subspace. Now consider the subgroup G_{2^{3}}, which is normalized by (the nonorthogonal transformation) 1 − f, where f is a fourvolution. The action of (1 − f) takes the set of 240 roots bijectively to the union of the nonempty sets O_{i} ∩ L[1], and this correspondence preserves orbits of G_{2^{3}}. We are done by (i). □

13 Clean pictures, dirty pictures and transitivity

We next prove transitivity results for certain kinds of sublattices. In particular, we can classify certain scaled embeddings of BW_{2^{k}} in BW_{2^{d}}, for certain k ≤ d. See 15.4 for the clean and dirty terminology.

Theorem 13.1. Let L = BW_{2^{d}}, for d ≥ 4. There is a G_{2^{d}}-invariant bijection between sublattices of L which are ssBW of rank 2^{d−1} and noncentral lower involutions, via the SSD correspondence.

Proof. Let M be such a sublattice and t = t_{M} the associated SSD involution. Since t normalizes R and has trace 0, it is dirty (see Appendix A2), whence there is an element g ∈ R so that [t, g] = −1. We may arrange for g to be an involution. Then g interchanges M and N := L ∩ M^{⊥}, whence N is a ssBW_{2^{d−1}}. By 6.19, the condition det(L^{±}(g)) = det(M) implies that L is part of an X-quadruple (L, L^{+}(g), L^{−}(g), t), whence the classification 10.2 implies that t is lower. □

Remark 13.2. There are cases of sublattices X of BW_{2^{d}} of rank 2^{d−1} which satisfy L/[X ⊥ X^{⊥}] elementary abelian, but X is not isometric to a scaled BW_{2^{d−1}}. For d = 3, one can take X to be the sublattice spanned by a root system of type A_{1}^{4} which is not contained in a D_{4} subsystem. Such a sublattice is SSD and corresponds to a SSD involution of trace 0 which is upper with respect to any conjugate of G_{2^{3}} which contains it. The noncentral involutions of R_{3} have trace 0 and fixed point sublattice isometric to L_{D_{4}}.

Theorem 13.3. Suppose that L = BW_{2^{d}} and that M, M′ are sublattices which are the fixed point lattices for clean isometries of order 2. If rank(M) = rank(M′), then there is an isometry g of L so that M′ = M^{g}.

Proof. Such sublattices correspond to SSD involutions with nonzero traces. Now use 15.8. □

The following is an application of 13.3.

Corollary 13.4. Suppose that d ≥ 5 is odd. Then in BW_{2^{d}} any two ssBW sublattices of rank 2^{d−2} are in the same orbit under G_{2^{d}}.

Proof. Such sublattices must be SSD. □

Definition 13.5. Let L = BW_{2^{d}}. A first generation sublattice of L is a sublattice L_{1} so that there exists a sublattice L_{2} and an involution t so that (L, L_{1}, L_{2}, t) ∈ 𝔛.

A chain of lattices L = L(0) ≥ L(1) ≥ ⋯ ≥ L(d) is a generational chain if there exists an elementary abelian group E ≤ R and a chain of subspaces E = E(d) > E(1) > ⋯ > E(0) = ⟨−1⟩ so that for each k, |E(k)| = 2^{k+1} and L(k) is the total eigenlattice of E(k), 6.8.

In each L(k), each orthogonally indecomposable summand is a ssBW sublattice, all of common rank 2^{d−k} if k ≤ d − 2, and L(d − 1) is a direct sum of isometric rank 1 lattices. Call L(k) a k^{th} generation sublattice and E(k) its defining lower group. A sublattice is ancestral if it is a k^{th} generation sublattice, for some k.

Theorem 13.6. Let d ≥ 4. If L = BW_{2^{d}} and Z is a k^{th}-generation sublattice, k ≤ d − 2, then the stabilizer of Z in Aut(L), is just N_{Aut(L)}(E), where E is its defining lower group, as in 13.5. It contains R_{2^{d}} and its image in G_{2^{d}}/R_{2^{d}} is a maximal parabolic which modulo the unipotent radical has shape GL(k, 2) × Ω^{+}(2(d − k), 2). The k^{th} generation sublattices are in G_{2^{d}}-equivariant bijection with the elementary abelian subgroups of R_{2^{d}} which contain Z(R_{2^{d}}).

Proof. The direct summands of Z realize all the linear characters of E which do not have −1 in their kernel. Thus, Z determines E. By definition of ancestral sublattices, E determines Z. □

Definition 13.7. A sublattice of L = BW_{2^{d}} is an k-generation ancestor lookalike if it is an orthogonal direct sum of 2^{k} copies of ssBW lattices, all of rank 2^{d−k}.

The transitivity situation for lookalikes is unclear. Here is a simple result.

Proposition 13.8. For L = BW_{2^{3}} , there is just one orbit of the automorphism group on third generation ancestral lookalike sublattices and there are four orbits for G_{2^{3}}. For BW_{2^{4}}, there are at least 4 orbits of the automorphism group on third generation ancestral lookalike sublattices.

Proof. For the case L = BW_{2^{3}} ≅ L_{E_{8}}, this was covered in 12.4.

Now take the case L = BW_{2^{4}}. Let F be such a frame. Then F + L[1] spans a totally singular subspace of L/L[1]. Since Aut(L) induces on L/L[1] its simple orthogonal group, we may assume that F lies in the ancestor sublattice L_{1} + L_{2} ≅ \sqrt{2}L_{E_{8}} ⊥ \sqrt{2}L_{E_{8}}.

Since norm 4 elements in L_{1}+L_{2} are indecomposable, we have F = F_{1}∪F_{2} where F_{i} := F ∩ L_{i}. By using the ideas in the proof of 12.4, we find that the dimension of the span of F_{2} + L_{1}[1] in L_{1}/L_{1}[1] can be 1, 2, 3 or 4. We conclude that the image of F in L/L[1] spans a space of dimension at most 8 and dimensions 1,2,3 and 4 actually do occur. This gives a lower bound of 4 on the number of orbits. □

14 The Ypsilanti lattices

We now set up a procedure for creating many isometry types of lattices in sufficiently large dimensions divisible by 8. Here is a rough idea. We take several isometric “good” lattices (indecomposable, high minimum norm, elementary abelian discriminant group) and study overlattices L of their orthogonal direct sum L_{1} ⊥ ⋯ ⊥ L_{s}. We consider conditions like X (3.3) but without (e). A suitable concept of avoidance allows us to build many lattices L with enough but not too many minimal vectors. We gain enough control over the automorphism groups to get a fairly high lower bound on the number of isometry types.

We start with a generalization of the maps f −1 where f is a fourvolution.

14.1 Michigan lattices and Washtenawizations

Definition 14.1. A 2-special endomorphism on a lattice L is an endomorphism p so that

(i) (xp, yp) = 2(x, y) for all x, y ∈ L;

(ii) Lp^{2} = 2L (thus, \frac{1}{2}p^{2} ∈ Aut(L));

(iii) there is an integer r so that L^{∗} = Lp^{−r} (r is called the duality level). If L has a 2-special endomorphism, call L a 2-special lattice. Call L

normalized if the duality level is 0 or 1.

Remark 14.2. A 2-special lattice is scale-isometric by a power of a 2-special endomorphism to a normalized lattice.

Notation 14.3. We adapt notations used earlier and set L[k] := Lp^{k}, for k ∈ ℤ. When C_{Aut(L)}(L[k]/L[k + 1]) is independent of k ∈ ℤ, we define Lower(L) := C_{Aut(L)}(L/L[1]) and Upper(L) := Stab_{Aut(L)}(L[−1])/Lower(L).

Notation 14.4. The sublattice of the lattice L spanned by the minimal vectors is denoted SMV (L). When L has a 2-special endomorphism, define SMV (L, L[1]) := SMV (L) + L[1]/L[1] and define mvd(L, L[1]) to be the dimension of SMV (L, L[1]). This number is called the mv-dimension and is positive if L ≠ 0. In case p or L[1] is understood, we write mvd(L) for mvd(L, L[1]) and note that this invariant could depend on choice of 2-special endomorphism.

Define the Washtenaw number or Washtenaw ratio of L ≠ 0 to be the ratio

W ashtenaw(L) := 2 mvdim(L)/rank(L) = mvdim(L)/dim(L/L[1]) ∈ (0, 1].

Definition 14.5. A Michigan lattice is a lattice M (i) with a 2-special endomorphism, p; (ii) SMV (M) has finite index in M; (iii) Aut(M) fixes each Mp^{k}, k ∈ ℤ; (iv) g ∈ Aut(M) is trivial on Mp^{k}/Mp^{k+1} if and only if g is trivial on

Mp^{ℓ}/Mp^{ℓ+1}, for all k, ℓ ∈ ℤ.

Note that a Michigan lattice L is indecomposable if SMV (L) is indecomposable.

Definition 14.6. We are given a normalized Michigan lattice M such that SMV (M) is indecomposable. Let t ≥ 3 be an integer.

Let M_{1}, …, M_{2^{t}} denote pairwise orthogonal copies of M, identified by isometries ψ_{i} : M → M_{i}, with 2-special endomorphism p_{i} corresponding to p by ψ_{i}. The direct sum has a 2-special endomorphism, q, which is the direct sum of the p_{i}.

A degree t Washtenawization of M is a lattice W contained in ℚ⊗(M_{1} ⊥ ⋯ ⊥ M_{2^{t}}) so that

(i) W contains (M_{1} ⊥ ⋯ ⊥ M_{2^{t}})[1 − r] and is a sublattice of (M_{1} ⊥ ⋯ ⊥ M_{2^{t}})[−r]; (r is the duality level of M) and the quotient M/(M_{1} ⊥ ⋯ ⊥ M_{2^{t}})[1 − r] is elementary abelian of dimension 2^{t−2} rank(M);

(ii) For all i, W ∩ (ℚ ⊗ M_{i}) = M_{i}[1 − r];

(iii) μ(W) = 2^{1−r}μ(M);

(iv) SMV (W) = ∑_{i=1}^{2^{t}} SMV (M_{i}) and W ashtenaw(W) = \frac{1}{2}W ashtenaw(M);

(v) Aut(W) has the form [∏_{i=1}^{2^{t}} Lower(M_{i})].[Upper(M) × Aut(C)], where C is an indecomposable (4.8) self orthogonal doubly even binary code of length 2^{t}; furthermore, Aut(M) embeds in Aut(W) by diagonal action.

A minimal Washtenawization is a degree 3 Washtenawization, using the extended Hamming code (which is essentially the only choice here). It is unique up to isometry.

Remark 14.7. By 5.1, Washtenawizations are indecomposable, since the code is indecomposable. In the notation of 14.6, the duality level of W is 1 − r and |Upper(W)| divides |Upper(M)|(2^{t}!). Also, Aut(W) permutes the set {M_{1}, …, M_{2^{t}}}.

Proposition 14.8. For all t ≥ 3, degree t Washtenawizations exist.

Proof. Let M be a normalized Michigan lattice. Take the lattice W between (M_{1} ⊥ ⋯ ⊥ M_{2^{t}})[1 − r] and (M_{1} ⊥ ⋯ ⊥ M_{2^{t}})[−r] which corresponds to some indecomposable doubly even self orthogonal code, C (for example, see 4.9). Since nonzero code words have weight at least 4, the minimal vectors of W lie in SMV ((M_{1} ⊥ ⋯ ⊥ M_{2^{t}})[1 − r]) (use 5.8).

Since q acts diagonally as p on (M_{1} ⊥ ⋯ ⊥ M_{2^{t}})[1 − r], the definition of W implies that the image of SMV ((M_{1} ⊥ ⋯ ⊥ M_{2^{t}})[1 − r]) in W/W p has dimension 2^{t−1} mvdim(M). This implies that W ashtenaw(W) = \frac{1}{2}W ashtenaw(M).

Since Aut(W) permutes the minimal vectors, it permutes the indecomposable direct summands of the lattice they generate, which are just the 2^{t} SMV (M_{i}), which in turn define the M_{i} as the summands of W (as abelian groups) which contain the SMV (M_{i}). It follows that Aut(W) is contained in a natural wreath product Aut(M_{i}) ≀ Sym_{2^{t}} which permutes {M_{1}, …, M_{2^{t}}}. Obviously, Aut(W) contains a group G_{0} of the form indicated in 14.6(v). Now, use 5.8(ii) and the fact that Aut(M) leaves each twist M[k] invariant. □

14.2 Overlattices of direct sums of 2-special lattices

Notation 14.9. Throughout this section, M is a normalized 2-special lattice (14.1) and M_{1}, M_{2} are pairwise orthogonal lattices isometric to M with duality level r ∈ {0, 1}. Let t be an isometry of order 2 which interchanges them.

Definition 14.10. The i^{th} admissible component group K_{i} is the full general linear group on M_{i}[−r]/M_{i}[1 − r] when the duality level of M is 0 and when the duality level of M is 1, it is the full orthogonal group on the nonsingular quadratic space M_{i}[−r]/M_{i}[1 − r], x + M[1 − r] ↦ 2^{r−1}(x, x)(mod 2).

Notation 14.11. Let d ≥ 5 be an integer and let M_{1}, M_{2} be isometric normalized 2-special lattices of ranks 2^{d−1} and duality level 1. Set V_{i} := ℚ ⊗ M_{i}. Let 𝔜 := 𝔜(M_{1}[1 − r], M_{2}[1 − r]) denote the set of even integral lattices M which contain M_{1}[1−r] ⊥ M_{2}[1−r] and satisfy M ∩V_{i} = M_{i}[1−r] for i = 1, 2 and whose projection to V_{i} is M_{i}[−r]. This is a set of rank 2^{d} unimodular lattices. (Note differences with 10.1, which results in unimodular lattices for ranks 2^{d}, d odd only. )

Remark 14.12. A member L of 𝔜 is determined by an isomorphism of vector spaces ζ : M_{1}[−r]/M_{1}[1 − r] → M_{2}[−r]/M_{2}[1 − r], namely L/(M_{1}[1 − r] + M_{2}[1−r]) is just the diagonal in the identification of the two M_{i}[−r]/M_{i}[1−r] based on ζ. We may write L/(M_{1}[1 − r] + M_{2}[1 − r]) = {(x + M_{1}[1 − r], (x + M_{1}[1 − r])^{ζ})|x ∈ M_{1}[1 − r]}.

Conversely, given a linear isomorphism ζ, we get an L ∈ 𝔜 by taking the diagonal as above provided (a) when d−1 is odd, no condition; (b) when d−1 is even, ζ is an isometry of nonsingular quadratic spaces M_{1}[−r]/M_{1}[1−r] → M_{2}[−r]/M_{2}[1 − r].

The reason for the isometry condition in (b) is that the nonsingular cosets (respectively, the singular cosets) of the two M_{i}[−r]/M_{i}[1 − r] must be matched to create a diagonal which gives an even lattice L. In (a), since the two M_{i}[−r] are even integral lattices, any matching by a linear isomorphism results in an element of 𝔜, whence no conditions are demanded. The requirement in (b) of taking M_{1}[−r]/M_{1}[1 − r] to M_{2}[−r]/M_{2}[1 − r] comes from the definition of 𝔜, 14.11.

Notation 14.13. We use the notations L ↦ ζ(L), ζ ↦ L(ζ) to express the bijection between 𝔜 and such isomorphisms.

Such ζ are in bijection with K_{1} and with K_{2} (see 14.10) by ζ ↦ ζ_{i} ∈ K_{i}, where the latter are defined by the formulas ζ : x + M_{1}[1 − r] ↦ (x^{t} + M_{2}[1 − r])^{ζ_{2}} = y^{t} + M_{2}, where y + M_{2} = (x + M_{1})^{ζ_{1}t}, where t is as in 14.9. Call ζ_{i} the K_{i}-component of ζ, or of L = L(ζ).

14.3 Avoidance

Definition 14.14. We say that two subspaces of a vector space avoid each other if their intersection is 0. If g : V → V′ is an invertible linear transformation, W ≤ V and W′ ≤ V′, we say that g is a (W, W′)-avoiding map if W^{g} ∩ W′ = 0. Let A(W_{1}, W_{2}) be the set of (W, W′)-avoiding maps in the set of linear isomorphisms from V onto V′.

We need some terminology for discussing asymptotic behavior.

Notation 14.15. Suppose that f(x) is a real-valued function on (0, ∞). The dominant term in f(x) (abbreviated DT (f(x)) is the expression of the form a_{0}log_{2}(x)^{a_{1}}2^{a_{2}x}x^{a_{3}} which is asymptotic to f(x) (the a_{i} are constants). We may indicate dependence on the variable x by DT_{x}. (This definition applies to a limited family of real-valued functions, but suffices for our purposes.)

Similarly, if f is as above, we define the dominant term of the logarithm (DT L or DT L_{x}) of 2^{f(x)} to be DT (f(x)). For example,

DT L(2^{(0.43)log_{2}(2x−3)2^{3x−4}+2^{2x}−log_{2}(x+1)^{5}x^{3}−log_{2}(x)^{7}(x^{2}+1)}) = \frac{0.43}{16} log_{2}(x)2^{3x−4}.

Proposition 14.16. Suppose that a ≤ b are positive integers. Suppose that V := 𝔽_{2}^{2b} has a maximal Witt index nonsingular quadratic form and that W_{1} and W_{2} are two a-dimensional totally singular subspaces. We set q := \frac{a}{b} and think of q as a constant and a as a function of b.

(i) Let H be the stabilizer in O(V) of W_{1}. Then,

DT L_{b}(|H|) = DT_{b}(\frac{1}{2}a(3a − 1) + 2(b − a)b) = b^{2}(2 − 2q + \frac{3}{2}q^{2}).

(ii) For an integer k, let A(W_{1}, W_{2}; k) be the set of avoiding maps as in 14.14 so that dim(W_{1}^{g} ∩ W_{2}^{⊥}) = k. Then A(W_{1}, W_{2}; k) is nonempty precisely for k = 0, 1, …, min{a, b − a} and for each such k, A(W_{1}, W_{2}; k) is a regular orbit for the action of H.

Proof. (i): We have DT L_{k}(|Ω^{+}(2k, 2)|) = 2k^{2} − k. We may assume W_{1} = W_{2}. Let H be the subgroup of the orthogonal group which fixes W_{1} globally. It follows from 15.1 that DT L(|H|) is the DTL of \frac{1}{2}a(3a−1)+2(b−a)b−(b−a).

(ii) : By Witt’s theorem, two nonavoiding maps g, g′ are in the same H-orbit if the dimensions of the images of W_{1} under g, g′ intersect W_{1}^{⊥} in spaces of the same dimensions. All H-orbits are regular. For a nonempty A(W_{1}, W_{2}; k), we have k ≤ dim(W_{1}) = a and since the image of an avoiding W_{1} in V/W_{2}^{⊥} has dimension at most a = dim(V/W_{2}^{⊥}), we have k + a = k + dim(W_{2}) ≤ b, the dimension of any maximal totally isotropic subspace. The value k = min{a, b − a} can be achieved. □

Corollary 14.17. We use the notations of 14.16 and assume that q ≤ \frac{1}{2}. Then

DT L_{b}(|A(W_{1}, W_{2})|) = υ(q)log_{2}(b)b^{2}, where υ(q) := (2 − 2q + \frac{3}{2}q^{2}).

Proof. Note that q ≤ \frac{1}{2} means a = min\{a, b − a\} in 14.16. □

14.4 Down Washtenaw Avenue to Ypsilanti

We next create large families of lattices in dimensions 2^{d} >> 0.

Definition 14.18. Let W be a normalized Michigan lattice which has duality level r = 1 and Washtenaw ratio q ≤ \frac{1}{2}.

Take orthogonal copies M_{1}, M_{2} of W and consider the set 𝔜 := 𝔜(M_{1}, M_{2}) as in 14.11. Consider the associated maps ζ(L), L ∈ 𝔜 (see 14.13) which are avoiding maps 14.14 for the subspaces SMV (M_{1}[−1], M_{1}), SMV (M_{2}[−1], M_{2}) of M_{1}[−1]/M_{1}, M_{2}[−1]/M_{2}, respectively. The corresponding lattices form a subset 𝔜_{av}(M_{1}, M_{2}) of 𝔜(M_{1}, M_{2}) in the notation of 14.11. Their ranks are 2 rank(W). They are called Ypsilanti lattices. Let IsomT ypes(M_{1}, M_{2}) be the set of isometry types of lattices in 𝔜_{av}(M_{1}, M_{2}).

When W is a Washtenawization of a BW lattice, the Ypsilanti lattices of rank 2^{d} = 2 rank(W) are called the Ypsilanti cousins of BW_{2^{d}}.

Lemma 14.19. We use the notations of 14.18.

(i) If N ∈ 𝔜_{av}(M_{1}, M_{2}), SMV (N) = SMV (M_{1}) ⊥ SMV (M_{2}).

(ii) N ∈ 𝔜_{av}(M_{1}, M_{2}) is indecomposable.

Proof. (i) Obviously, μ(N) ≥ μ(M_{i}), i = 1, 2. Consider a vector x = x_{1} + x_{2} ∈ N \ (M_{1} ⊥ M_{2}). Then the x_{i} have norms at least μ(M_{i}[−1]) = \frac{1}{2}μ(M_{i}). For (x, x) to equal μ(M_{i}), we need x_{i} to be a minimal vector of M_{i}[−1] for i = 1, 2. This is not the case since N was defined with an avoiding map.

(ii) Use 5.9. □

Lemma 14.20. Suppose that we are given q = 2^{−j} for some j > 0. For all k ≥ 5 + 3j, there exists a Michigan lattice W (k) so that rank(W (k)) = 2^{k}, W ashtenaw(W (k)) = q and the duality level of W (k) is 1. We may also arrange for μ(W (k)) = 2^{1−r+⌊ \frac{j}{2} ⌋+⌊ \frac{e}{2} ⌋}, where e = k − 3j if k is even and e = k − 3j − 1 if k is odd.

Proof. If we start with BW_{2^{e}} and perform the minimal Washtenawization procedure s times, we get a lattice W (e, s) of rank 2^{e+3s}. We may take a degree 4 Washtenawization to W (e, s) and get a lattice W′(e, s + 1) of rank 2^{e+3s+4}.

Each Washtenawization changes duality level. We define W (k) according to the following cases. When k is even, we require k − 3j ≥ 4, which means k is at least 8. When k is odd, we require k − 3j − 1 ≥ 4, which means that k is at least 9.

W (k) := \begin{cases}W (k − 3j, j) & k even; \\ W′(k − 3j − 1, j) & k odd. \end{cases} □

Definition 14.21. We call a sequence of lattices as in 14.20 the j-Washtenaw series, for the fixed ratio q = 2^{−j}. It starts at rank 2^{5+3j}. The isometry types of certain members of the series depend on choice of indecomposable doubly even code of length 16. Ypsilanti cousins associated to such series are called Ypsilanti j-cousins. The set of such isometry types is denoted Y psi(2^{d}, j).

Lemma 14.22. We use the notations of 14.18, 14.20 and let W (k) be the Washtenaw series.

(i) If N and N′ are two cousins of rank 2^{d}, Isom(N, N′) is contained in the group G_{0}(e, h) of orthogonal transformations which stabilize L_{1} ⊥ ⋯ ⊥ L_{2^{h}}, the indecomposable direct summands of SMV (N) = SMV (N′) (in fact, the L_{i} are the pairwise isometric scaled Barnes-Wall lattices, of rank 2^{e}, on which the Washtenawizations M_{1}, M_{2} were based; the notation means d = k + 1 = e + h, with h = 3j or 3j + 1).

(ii) DT L_{d}(|G_{0}(e, h)|) is bounded above by a constant times d^{2}.

Proof. (i) Given N, N′ ∈ 𝔜_{av}(M_{1}, M_{2}), an isometry of N to N′ takes SMV (N) to SMV (N′). Both of these equal SMV (M_{1} ⊥ M_{2}).

(ii) This follows from DT L_{f}(|Ω^{+}(2f, 2)|) = 2f^{2} and boundedness of h. □

Notation 14.23. When W (k) runs through the j-Washtenaw series 14.21, we let Υ(2^{d}, j) := |Y psi(2^{d}, j)|.

Lemma 14.24. Υ(2^{d}, j) ≥ |𝔜_{av}(M_{1}, M_{2})|/|G_{0}(e, h)|, whence DT L_{d}(Υ(2^{d}, j)) ≥ \frac{1}{16}υ(2^{−j})d 2^{2d}, as in 14.17.

Proof. In 14.17, take b = 2^{d−2}, because the admissible component group 14.10 is O^{+}(2^{d−1}, 2) since the duality level has been arranged to be 1. Then use 14.22(ii) and 14.23.□

Remark 14.25. For a fixed large value of d, we can make the families Y psi(2^{d}, j), q = 2^{−j}, for all 1 ≤ j ≤ ⌊\frac{d−5}{3}⌋. This would make roughly d/3 times as many as one of the Y psi(2^{d}, j), so would not increase the DTL.

We summarize our counting in dimensions 2^{d}.

Theorem 14.26. For any j > 0, the number of isometry types of Ypsilanti lattices in dimension 2^{d} has DTL at least \frac{1}{16}υ(2^{−j})d 2^{2d}. In particular, the number of indecomposable even unimodular lattices in dimensions 2^{d} has DTL at least c d 2^{2d}, for any c ∈ (0, \frac{1}{8}).

Remark 14.27. With a bit more work, we could define lattices like Ypsilanti cousins for d < 9, though we would not expect them to represent more than a fraction of mass(2^{d}) isometry types. In dimension 32, the mass formula gives value about 10^{7} and the number of isometry types (still not known) has been bounded below by about 10^{10} (see [24]).

14.5 From dimensions 2^{d} to arbitrary dimensions

Notation 14.28. For an integer n > 0 divisible by 8, let 2^{d} be the largest power of 2 less than or equal to n. Fix some q^{−j}, j > 0. Let Y psi(n, j) be the set of isometry types of even integral unimodular lattices which contain a Ypsilanti j-cousin of rank 2^{d} as an orthogonal direct summand. Clearly, Υ(n, j) := |Y psi(n, j)| ≥ Υ(2^{d}, j).

Corollary 14.29. We use the notation of 14.28. For any constant c ∈ [0, \frac{1}{32}), we take j > 0 so that q = 2^{−j} satisfies 2 − q + \frac{3}{2}q^{2} > 64c.

Then log_{2}(Υ(n, j)) ≥ c log_{2}(n) n^{2}.

Proof. Take the integer d which satisfies 2^{d} ≤ n < 2^{d+1}. Then d < log_{2}(n) ≤ d + 1 and 2^{d} > \frac{n}{2}. We have Υ(n, j)| ≥ Υ(2^{d}, j) and DT L(Υ(n, j)|) ≥ DT L(Υ(2^{d}, j)) > 4c d 2^{d} ≥ 4c(log_{2}(n)−1)(\frac{n}{2})^{2} whose DT is at least c log_{2}(n)n^{2}. □

14.6 Number of Ypsilanti cousins compared with the mass formula

Notation 14.30. We follow the notations of [31], pp. 54, 90, except we write mass(n) instead of “M_{n}”. Stirling’s formula (n! ∼ n^{n+ \frac{1}{2}} e^{−n}(2π)^{\frac{1}{2}} ) implies that DT L(n!) = log_{2}(n)n. Let B_{j} be the j^{th} Bernoulli number. Let n ∈ 8ℤ, k := \frac{n}{8}.

Proposition 14.31. DT L_{n}(mass(n)) = \frac{1}{4}log_{2}(n) n^{2}.

Proof. We have mass(n) = \frac{B_{2k}}{8k} ∏_{j=1}^{4k−1} \frac{B_{j}}{4j}. Because ζ(2j) ∈ (1, 2) for all j ≥ 1, the formula B_{j} = 2ζ(2j).(2j)!/(2π)^{2j} shows that DT L_{j}(B_{j}) = DT (log_{2}(2j)2j).

We have

log_{2}(mass(n)) = log_{2}(B_{2k})+ ∑_{j=1}^{4k−1} log_{2}(B_{j})−(8k+1)−log_{2}((4k−1)!)−log_{2}(k).

Since DT L_{k}(B_{2k}) = DT L_{k}((4k)!), DT L_{n}(mass(n)) = DT_{n}(∑_{j=1}^{4k−1} log_{2}(2j)2j). The latter summation can be thought of as Riemann sums, which can be estimated with integrals (think of ∫ 2x ln(2x) dx = ∫ 2x ln(x) dx+ln(2) ∫ 2x dx = x^{2} ln(x)− \frac{1}{2}x^{2} +ln(2)x^{2} +c, which has dominant term x^{2} ln(x)). We conclude that DT L_{n}(mass(n)) = DT_{n}(log_{2}(8k)(4k)^{2}) = \frac{1}{4}log_{2}(n)n^{2}. □

Proposition 14.32. For positive integers n, q, define A(n, q) := ∑_{i≥0}⌊\frac{n}{q^{i}(q−1)}⌋ and let P be the set of prime numbers at most n+1. Set f(n) := ∏_{q∈P} q^{A(n,q)}. Then a finite subgroup of GL(n, ℚ) has order dividing f(n).

Proof. This is a result of Minkowski [28]. See the discussions in exercises for Section 7 of [6]. □

Lemma 14.33. DT L_{n}(f(n)) = n log_{2}(n).

Proof. Well known? A proof may be deduced from [29], Th. 8.8(b), p. 369. □

Remark 14.34. The DTL of n log_{2}(n) is small compared to DT L(mass(n)). It follows that the DTL of the number of isometry types of rank n even unimodular lattices is the same as that of DT L(mass(n)).

We summarize:

Corollary 14.35. For any a ∈ (0, \frac{1}{8}), there is an integer j so that DT L_{n}(Υ(n, j)) ≥ a·DT L(mass(n)). Furthermore, when n is a power of 2, and b ∈ (0, \frac{1}{2}), there is an integer j so that DT L_{n}(Υ(n, j)) ≥ b · DT L(mass(n)).

Remark 14.36. We conclude with some numerical comparisions.

Asymptotics for Υ(2^{d}, j), Υ(n, j)| and mass(n).
j  q = 2^{−j}  υ(q) = constant coefficient of  Lower bound for
16 DT L(Υ(2^{d}, j))(see 14.24) DT L(Υ(2^{d}, j))/DT L(mass(2^{d}))
1 .5000000000  1.375000000  .3437500000
2 .2500000000  1.593750000  .3984375000
3 .1250000000  1.773437500  .4433593750
4 .06250000000  1.880859375  .4702148438
5 .03125000000  1.938964844  .4847412109
6 .01562500000  1.969116211  .4922790527
7 .007812500000  1.984466553  .4961166382
8 .003906250000  1.992210388  .4980525970
9 .001953125000  1.996099472  .4990248680
10 .0009765625000  1.998048306  .4995120764

15 Appendices

15.1 A1. Group orders

Proposition 15.1. (i) For q a power of 2, the order of Ω^{+}(2n, q) is q^{n(n−1)}(q^{n}− 1) ∏_{i=1}^{n−1}(q^{2i} − 1).

(ii) The stabilizer in Ω^{+}(2n, q) of an isotropic point has shape q^{2(n−1)}:[Ω^{+}(2(n− 1), q) × q − 1].

(iii) The stabilizer in Ω^{+}(2n, q) of a maximal totally singular subspace has shape q^{\binom{n}{2}}:GL(n, q), and this is a maximal subgroup.

(iii) The stabilizer in Ω^{+}(2n, q) of a totally singular subspace of dimension m < n has the form RL, where the unipotent radical has order 2^{\binom{m}{2}+2m(n−m)} and L ≅ Ω^{+}(2(n − m), 2) × GL(m, 2). These are maximal subgroups.

Proof. These are well-known properties of the orthogonal groups. Proofs may be obtained from [8, 14]. □

15.2 A2. Aut^{0}(2_{ε}^{1+2d}), Out^{0}(2_{ε}^{1+2d}) and BRW^{0}(2^{d}, ε).

Basic theory of extraspecial groups extended upwards by their outer automorphism group has been developed in several places. We shall use [16, 18, 13, 17, 23, 3, 4, 5].

Notation 15.2. Let R ≅ 2_{ε}^{1+2d} be an extraspecial group which is a subgroup of G := GL(2^{d}, 𝔽), for a field 𝔽 of characteristic 0. Let N := N_{G}(R) ≅ 𝔽^{×}.2^{2d}O^{ε}(2d, 2). The Bolt-Room-Wall group is a subgroup of this of the form 2_{ε}^{1+2d}.Ω^{ε}(2d, 2). If d ≥ 3 or d = 2, ε = −, N′ has this property. For the excluded parameters, we take a suitable subgroup of such a group for larger d. We denote this group by BRW^{0}(2^{d}, +) or D(d). It is uniquely determined up to conjugacy in G by its isomorphism type if d ≥ 3 or d = 2, ε = −. It is conjugate to a subgroup of GL(2^{d}, ℚ) if ε = +. Let R = R_{2^{d}} denote O_{2}(G_{2^{d}}). We call R_{2^{d}} the lower group of BRW^{0}(2^{d}, +) and call G_{2^{d}}/R_{2^{d}} the upper group of BRW^{0}(2^{d}, +).

For g ∈ N, define C_{R mod R′}(g) := {x ∈ R|[x, g] ∈ R′}, B(g) := Z(C_{R mod R′}(g)) and let A(g) be some subgroup of C_{R mod R′}(g) which contains R′ and complements B(g) modulo R′, i.e., C_{R mod R′}(g) = A(g)B(g) and A(g) ∩ B(g) = R′. Thus, A(g) is extraspecial or cyclic of order 2. Define c(d) := dim(C_{R/R′}(g)), a(g) := \frac{1}{2}|A(g)/R′|, b(g) := \frac{1}{2}|B(g)/R′|. Then c(d) = 2a(d) + 2b(d).

Corollary 15.3. Let L be any ℤ-lattice invariant under H := BRW^{0}(2^{d}, +). Then H contains a subgroup K ≅ AGL(d, 2) and L has a linearly independent set of vectors {x_{i}|i ∈ Ω} so that there exists an identification of Ω with 𝔽_{2}^{d} which makes the ℤ-span of {x_{i}|i ∈ Ω} a permutation module for AGL(d, 2) on Ω.

Proof. In H, let E, F be maximal elementary abelian subgroups such that R = EF and let K be their common normalizer. It satisfies K/R ≅ GL(d, 2).

Now, let z generate Z(R) and let E_{1} complement ⟨z⟩ in E and F_{1} complement ⟨z⟩ in F. The action of K on the hyperplanes of E which complement Z(R) satisfies N_{K}(E_{1})F = K, N_{F} (E_{1}) = Z(R). Now consider the action of N_{K}(E_{1}) on the hyperplanes of F which complement Z(R). We have that K_{1} := N_{K}(E_{1})∩N_{K}(F_{1}) covers N_{K}(E_{1})/E. Therefore, K_{1}/Z(R) ≅ GL(d, 2). Let K_{0} be the subgroup of index 2 which acts trivially on the fixed points on L of E_{1}, a rank 1 lattice. So, K_{0} ≅ GL(d, 2). Let x be a basis element of this fixed point lattice. Then the semidirect product F_{1}:K_{0} is isomorphic to AGL(d, 2) and {x^{g}|g ∈ F_{1}} is a permutation basis of its ℤ-span (and the x^{g} are eigenvectors for all the characters of E which are nontrivial on Z(R)). □

Definition 15.4. We use the notation of 15.2. An element x ∈ N is dirty if there exists g so that [x, g] = xz, where z is an element of order 2 in the center. If g can be chosen to be of order 2, call x really dirty or extra dirty. If x is not dirty, call x clean.

Lemma 15.5. Let 𝔽_{2}^{2d} be equipped with a nondegenerate quadratic form with maximal Witt index. The set of maximal totally singular subspaces has two orbits under Ω^{+}(2d, 2) and these are interchanged by the elements of O^{+}(2d, 2) outside Ω^{+}(2d, 2).

Proof. This is surely well known. For a proof, see [14]. □

Theorem 15.6. We use the notation of 15.2, 15.4. Let g ∈ N. Then T r(g) = 0 if and only if g is dirty. Assume now that g is clean and has finite order. Then T r(g) = ±2^{a(g)+b(g)}η, where η is a root of unity. If g ∈ BRW (d, +), we may take η = 1. Furthermore, every coset of R in BRW (d, ε) contains a clean element and if g is clean, the set of clean elements in Rg is just g^{R} ∪ −g^{R}.

Proof. [18]. □

Lemma 15.7. Suppose that t, u are involutions in Ω^{+}(2d, 2), for d ≥ 2. Suppose that their commutators on the natural module W := 𝔽_{2}^{2d} are totally singular subspaces of the same dimension, e. Suppose that e < d or that e = d and that [W, t] and [W, u] are in the same orbit under Ω^{+}(2d, 2). Then t and u are conjugate. If e = d, t and u are conjugate if and only if [W, t] and [W, u] are in the same orbit under Ω^{+}(2d, 2), and there are two such orbits.

Proof. Well know. Use induction on d. □

Corollary 15.8. Suppose that t, u are clean involutions in H with T r(t) = T r(u) ≠ 0. Then t and u are conjugate in G_{2^{d}} unless possibly d is even and T r(t) = T r(u) = ±2^{⌊ \frac{d}{2} ⌋}. If the latter holds, t, u lie in one of two classes of involutions.

Proof. We may assume that t, u are noncentral. These involutions are not lower and have the same dimension of fixed points on R/R′ ≅ 𝔽_{2}^{2d}. Let T, U ≤ R be their respective centralizers in R. Since both t, u are clean, [R, t] and [R, u] are elementary abelian subgroups of T, U, respectively. From 15.7, we deduce that Rt and Ru are conjugate in G_{2^{d}} unless possibly T r(t) = T r(u) = ±2^{⌊ \frac{d}{2} ⌋}.

We may assume that Rt = Ru. Now use 15.6 to deduce that t is Rconjugate to u or −u. The trace condition implies that t is conjugate to u.

The exceptional case T r(t) = T r(u) = ±2^{⌊ \frac{d}{2} ⌋} leads to two classes; see 15.7. □

Corollary 15.9. Suppose that t, u ∈ H are dirty involutions. Then t and u are conjugate if and only if Rt and Ru contain clean elements of the same trace.

Proof. Necessity is clear, by 15.6. Suppose that Rt and Ru contain clean elements of the same trace. We may assume that Rt = Ru. Let c ∈ Rt be a clean involution. We may write t = abc, u = a′b′c, for a, a′ ∈ A(c), b, b′ ∈ B(c). Since t, u are dirty, a, a′ are noncentral involutions in A(c).

Note also that A(c) > Z(R) since Rc contains dirty elements.

We argue that the action of RC_{G}(c) on C_{R}(c) has t, u in one orbit. Clearly, we may assume that a = a′ since C_{R}(c) ∩ N_{G}(A(c)) induces Aut(A(c)) on A(c). We may assume that both b, b′ are nonidentity due to the action of O_{2}(C_{G}(c)) (think of the stabilizer of a nonmaximal nonzero singular subspace in the orthogonal group). Finally, note that the action of RC_{G}(⟨c, A(c)⟩) on B(c) is that of the subgroup of Aut(B(c)) which stabilizes Z(R) (isomorphic to AGL(B(c)/Z(R))). Conjugation by an element of R which takes b to b′ also takes t to u. □

Remark 15.10. The extension 1 → R_{2^{d}} → G_{2^{d}} → Ω^{+}(2d, 2) → 1 is nonsplit for d ≥ 4. This was proved first in [4], then later in [7] and in [16] (for both kinds of extraspecial groups, though with an error for d = 3; see [13] for a correction). The article [16] gives a sufficient condition for a subextension 1 → R_{2^{d}} → H → H/R_{2^{d}} → 1 to be split, and there are interesting applications, e.g. to the centralizer of a 2-central involution in the Monster. A general discussion of exceptional cohomology in simple group theory is in [17].

15.3 A3. Indecomposable integral representations for a group of order 2

Proposition 15.11. Let G be a cyclic group of order 2 and M a finitely generated ℤ-free G-module. Then M is a direct sum of modules isomorphic to ℤ[G], the group algebra; the ℤ-rank 1 trivial module; the ℤ-rank 1 nontrivial G-module.

Proof. [10], Section 74. The case where G has order any prime number is treated. □

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