SYMMETRIC GROUPS AND LATTICES

ANNE-MARIE BERGÉ AND JACQUES MARTINET

Abstract. This paper deals with various problems in lattice theory involving local extrema. In particular, we construct infinite series of highly symmetric spherical 3-designs which include some of the examples constructed in [9] in dimensions 5 and 7. We also construct new types of dual-extreme lattices. Résumé. Quelques applications de l’algorithme de Voronoï équivariant. Nous considérons dans cet article divers problèmes de la théorie des réseaux liés à des questions d’extrema locaux. En particulier, nous construisons des séries infinies de 3-designs sphériques qui englobent certains de ceux construits dans [9] en dimensions 5 et 7. Nous construisons également de nouveaux types de réseaux dual-extrêmes.

1. Introduction.

Let E be an n-dimensional Euclidean space equipped with scalar product x · y, and let L be the set of lattices (discrete subgroups of rank n) in E. For a lattice Λ ∈ L, we denote by min Λ its minimal norm min Λ = min_{x∈Λ∖{0}} x · x, and by det Λ the determinant of the Gram matrix (e_{i} · e_{j}) of any ℤ-basis (e_{1}, e_{2}, …, e_{n}) of Λ. The density of the sphere packing associated with Λ is measured by the Hermite invariant of Λ

γ(Λ) = \frac{min Λ}{det Λ^{1/n}} ;

another invariant attached to this packing is the sphere of Λ, i.e. the set

S(Λ) = {x ∈ Λ | x · x = min Λ}

of minimal vectors of Λ, and its kissing number 2s = |S(Λ)|.

A lattice Λ is called extreme if it achieves a local maximum of γ over L. In 1907, Voronoi showed that a lattice is extreme if and only if it is both perfect and eutactic. Geometrical definitions of these notions are as follows; we denote by End^{s}(E) the space of symmetric endomorphisms of E and for any non-zero x ∈ E, p_{x} ∈ End^{s}(E) is the orthogonal projection onto the line ℝx.

Euclidean lattices, groups, duality .

2000 Mathematics Subject Classification: 11H55, 11H56 .

Laboratoire A2X, UMR 5465 associée au C.N.R.S. & Université Bordeaux 1

A lattice Λ is called perfect if the p_{x}, x ∈ S(Λ) span End^{s}(E) . This notion belongs to linear algebra, as well as the following enlarged notion of eutaxy.

A lattice Λ is weakly eutactic if there exists real numbers ρ_{x}, x ∈ S(Λ) such that Id_{E} = ∑_{x∈S(Λ)} ρ_{x} p_{x}. We remark that this condition implies that the sphere S(Λ) spans the space E, and also that any system of eutaxy coefficients gives way, by averaging, to another such system invariant under the automorphism group Aut(Λ) of Λ. Now the lattice Λ is eutactic if it has a system of strictly positive eutaxy coefficients; note that this classical notion of eutaxy involves convexity, and coincides when s = n with the weaker one, characterized by the pairwise orthogonality of the n lines ℝ x, x ∈ S(Λ).

While revisiting Voronoi’s theory in the setting of spherical designs, Venkov raised a more restrictive notion of eutaxy: the lattice Λ is called strongly eutactic if we can choose a system of eutaxy coefficients that are all equal. Strong eutaxy amounts to the property of S(Λ) to be a spherical 2-design (or equivalently 3-design, since S(Λ) is symmetric); see [10], Prop. 6.2. Empirical observations in low dimension allow us to suspect that strong eutaxy is rare: for example in five dimensions, there are 127 weakly eutactic lattices (the complete enumeration was done by Batut in 2000, [1]), and only eight among them are strongly eutactic. However, no systematic enumeration is so far done beyond dimension 5; in [9], various examples of strongly eutactic lattices are given in dimensions 6 and 7, most of them related to root lattices or cubic lattices.

In Section 4, we construct, from the point of view of group representations, two infinite sequences of strongly eutactic lattices which include some of the “exotic” lattices in the lists above, and provide new ones. In particular, we obtain in all odd dimension n ≥ 5 two integral lattices B_{n} and C_{n} with minimal norms n − 2 and n respectively.

Another aim of this paper deals with duality, i.e. involves together with the lattice Λ ∈ L its dual Λ^{∗} = {x ∈ E | ∀ y ∈ Λ, x · y ∈ ℤ} . In 1989, we introduced the dual-Hermite invariant

γ′(Λ) = \sqrt{min Λ min Λ^{∗}}

and characterized dual-extremality, i.e. extremality with respect to γ′, by the following notions of perfection and eutaxy: Λ is dual-perfect if the p_{x}, x ∈ S(Λ) together with the p_{y}, y ∈ S(Λ^{∗}) span End^{s}(E) ; Λ is dual-eutactic if there exists a relation ∑_{x∈S(Λ)} ρ_{x} p_{x} = ∑_{y∈S(Λ^{∗})} ρ′_{y} p_{y} with strictly positive coefficients ρ_{x}, ρ′_{y} (see [7], Ch. 2 and 3, § 8).

Note that these conditions imply for the kissing numbers of Λ and Λ^{∗} the inequality

s + s^{∗} ≥ \frac{n(n + 1)}{2} + 1 .  (1)

It is clear that a pair of dual lattices both eutactic and one of them perfect is dual-extreme. At the time [8] was written, all known dualextreme lattices, to within the single exception of an extreme 7-dimen-sional lattice whose dual is not eutactic, were obtained using this trick. Hence they were rational, and moreover they satisfied the strongest

inequality s + s^{∗} ≥ \frac{n(n + 1)}{2} + n + 1 for all n ≥ 2. (Actually, these properties hold in dimensions 2, 3, 4.) Examples of dual-extreme pairs of dual lattices, none of them perfect, not both eutactic and with s +

s^{∗} = \frac{n(n + 1)}{2} +n were found for even n ≥ 8 in [8], then for n = 5 in [3]. The example of [3] is moreover the first example of an irrational dualextreme lattice (its field of definition is quadratic); a 7-dimensional example, defined over a cubic field, can be obtained in the same way.

We prove in Section 2 that for each even n ≥ 8, the densest section of the Coxeter lattice 𝔸_{n+1}^{(n+2)/2} is a dual-extreme, n-dimensional lattice which realizes the lower bound of (1); then, (1) is optimal for infinitely many dimensions.

Recall that a lattice Λ is symplectic if there is an isometry σ of E such that σ(Λ) = Λ^{∗} and σ^{2} = − Id. Note that such an isoduality gives E a complex structure; thus n must be even. Symplectic lattices are important in connection with complex and algebraic geometry, since they correspond to principally polarized Abelian varieties (see [6]). There is also a “Voronoi theory” for them: one can characterize symplecticextreme lattices (which achieve local maxima of density over all symplectic lattices in E) by suitable notions of symplectic-perfection and symplectic-eutaxy (see [7], Ch. 11, § 7). Obviously, among symplectic lattices, we have

extreme ⟹ dual-extreme ⟹ symplectic-extreme .

We shall display in Section 5 an infinite series of rational non-perfect, symplectic lattices which are nevertheless dual-extreme, that we shall identify with the symplectic-extreme family F_{n} previously constructed by Bavard in a Riemannian setting ([2]).

The method used is the equivariant version of the Voronoi algorithm established in [5] and outlined in Section 3. Given a finite group G of isometries of E, there are for the G-lattices in E natural notions of G-perfection and of G-neighbourhood. Our examples stem from convenient neighbouring paths connecting G-perfect lattices. The groups G we use are related to the symmetric group S_{n} acting as a permutation group on an orthonormal basis of E: S_{n} itself, or some subgroups stabilizing given subspaces. Root lattices play a key role in our examples.

2. A generalization of some Coxeter lattices.

We describe now certain lattices related to the root lattice 𝔸_{n}, using a basis B_{n} = (e_{1}, e_{2}, …, e_{n}) for its dual 𝔸_{n}^{∗} with Gram matrix

Gram(B_{n}) = \frac{1}{n + 1} ( \begin{matrix}n & −1 & −1 & ··· & −1 \\ −1 & n & −1 & ··· & −1 \\ · & · & · & ··· & · \\ −1 & −1 & −1 & ··· & n \end{matrix} ) = I_{n} − \frac{1}{n + 1}J_{n} ,

where I_{n} and J_{n} denote respectively the unit matrix and he all-ones matrix. The dual basis B_{n}^{∗} = (e_{i}^{∗}) affords for 𝔸_{n} the Gram matrix

A_{n} = I_{n} + J_{n} .

The quotient 𝔸_{n}^{∗}/𝔸_{n} is cyclic of order n + 1, hence for n ≥ 1 and r a divisor of n + 1, there is a unique sublattice of 𝔸_{n}^{∗} which contains 𝔸_{n} to index r; it was called 𝔸_{n}^{r} by Coxeter in 1951. The dual of 𝔸_{n}^{r} is 𝔸n^{\frac{n+1}{r}}. The Coxeter lattices are stable under the group Aut(𝔸_{n}) ≃ {± Id} × S_{n+1}.

In this section we focus on the extreme odd-dimensional lattice Λ = 𝔸_{n}^{(n+1)/2} (n = 5, 7 …). In order to define the densest (resp. the isodual) hyperplane sections of Λ, we look for vectors of norm 2 (resp. \frac{n+1}{4}) in the dual lattice Λ^{∗} = 𝔸_{n}^{2}. They constitute a single orbit under the action of Aut Λ, for which we may choose e_{n}^{∗} (resp. \frac{e_{1}^{∗}−e_{2}^{∗}+···−e_{n−1}^{∗}+e_{n}^{∗}}{2} ) as a representative. Taking the sections of Λ orthogonal to these vectors, we define two canonical series of even-dimensional lattices, isometric only in dimension 6.

Definition 2.1. For n ≥ 4 the n-dimensional generalized Coxeter lattice Cox_{n} is 𝔸_{n}^{(n+1)/2} if n is odd, and the section of 𝔸_{n+1}^{(n+2)/2} orthogonal to e_{n+1}^{∗} if n is even. Moreover, when n is even, we denote by Cox′_{n} the section of 𝔸_{n+1}^{(n+2)/2} orthogonal to \frac{e_{1}^{∗}−e_{2}^{∗}+···+e_{n}^{∗}−e_{n+1}^{∗}}{2} .

More explicitly, for n ≥ 4 even:

Cox_{n} = { ∑_{i=1}^{n} x_{i}e_{i} | (x_{i}) ∈ ℤ^{n} and ∑_{i} x_{i} ≡ 0 mod 2}

Cox′_{n} = { ∑_{i=1}^{n+1} x_{i}e_{i} | (x_{i}) ∈ ℤ^{n+1} and ∑_{i=1}^{n+1}(−1)^{i}x_{i} = 0} ,

where (e_{1}, …, e_{n+1}) is a basis for 𝔸_{n+1}^{∗} whose Gram matrix is I_{n+1} − \frac{1}{n+2}J_{n+1}.

The lattice Cox′_{n} is not the densest section of 𝔸n+1^{\frac{n+2}{2}}, except in the cases n = 4 and n = 6, where Cox′_{4} ≃ 𝔸_{2} ⨂ 𝔸_{2} and Cox′_{6} ≃ Cox_{6} ≃ 𝔻_{6}^{+}. It has nevertheless interesting properties of duality that we shall discuss in Section 5 by inserting Cox′_{n} in an equivariant family of symplectic, n-dimensional lattices.

The remaining of the present section is devoted to the lattice Cox_{n}, n even, which except for n = 4 is the densest section of 𝔸n+1^{\frac{n+2}{2}}. The lattice Cox_{6} ≃ 𝔻_{6}^{+} has exceptional duality and symmetries. For n = 8 onwards, the sequence Cox_{n} acquires a certain regularity.

Proposition 2.2. Let n ≥ 8 be even. The lattice Cox_{n} and its dual have minimal norms \frac{2n}{n+2} and \frac{3}{2} respectively, and spheres

S(Cox_{n}) = {±(e_{i} + e_{j}), 1 ≤ i < j ≤ n; ±(e_{1} + ⋯ + e_{n})} ,
S(Cox_{n}^{∗}) = {±e′_{i}, 1 ≤ i ≤ n} ,

where (e′_{1}, …, e′_{n}) is the basis dual to (e_{1}, …, e_{n}). The automorphism group Aut(Cox_{n}) is isomorphic to {± Id} × S_{n}, acts transitively on S(Cox_{n}^{∗}), and has two orbits on S(Cox_{n}), with one and \frac{n(n−1)}{2}pairs (±x) of vectors respectively.

Proof. The minimal vectors of 𝔸n+1^{\frac{n+2}{2}}, namely ±(e_{i} + e_{j}), 0 ≤ i < j ≤ n + 1 where e_{0} = −(e_{1} + e_{2} + ⋯ + e_{n+1}), belong to ℝ Cox_{n} if and only if 1 ≤ i < j ≤ n or (i, j) = (0, n + 1). The sphere and minimum of its section Cox_{n} follow immediately.

The basis (e′_{i}) has Gram matrix (I_{n} − \frac{1}{n+2}J_{n})^{−1} = I_{n} + \frac{1}{2}J_{n} and associated quadratic form

Q(x_{i}) = ∑ x_{i}^{2} + \frac{1}{2} (∑ x_{i})^{2} .

To determine the minimal norm of

Cox_{n}^{∗} = ⟨e′_{1}, …, e′_{n}, \frac{e′_{1} + ⋯ + e′_{n}}{2} ⟩ ,

we must evaluate Q over half-integers congruent modulo ℤ. It takes values ≥ \frac{3}{2} on ℤ^{n}, with equality only on the canonical basis (up to signs); if the x_{i} are halves of odd integers, we have ∑ x_{i}^{2} ≥ \frac{n}{4} > \frac{3}{2} since n ≥ 8. So, the minimum and sphere of Cox_{n}^{∗} are as stated in 2.2.

Definition 2.1 shows that Cox_{n} is stable under the symmetric group S_{n} permuting {e_{1}, …, e_{n}}. It remains to prove that the group {± Id}× S_{n}, whose action on S(Cox_{n}) is as stated in 2.2, is for n ≥ 8 the full automorphism group of Cox_{n}. Since S(Cox_{n}^{∗}) = {±e′_{i}} has rank n, Aut(Cox_{n}) = Aut(Cox_{n}^{∗}) is generated by some permutations and sign changes of the e′_{i}. Actually, the scalar products e′_{i} · e′_{j} displayed in Gram(e′_{i}) allow all permutations of the e′_{i}, but just the negation of all of them, which completes the proof of 2.2. □

Note that the statements in 2.2 about the minimum of Cox_{n}, its sphere and the action of S_{n} on it still hold in dimension 6. In contrast, the dual lattice Cox_{6}^{∗} has exceptional minimal vectors, namely all permutations of \frac{e′_{1}+e′_{2}+e′_{3}−e′_{4}−e′_{5}−e′_{6}}{2} . We conclude that the group S_{6} fixes one minimal vector of Cox_{6}, but no minimal vector of its dual: the similarities Cox_{6} ≃ Cox_{6}^{∗} do not preserve the action of S_{6}.

We now consider the extremal properties of the lattices Cox_{n}. For the sake of completeness, we state the results for all n, even or odd. However, the results are known for n odd: the lattices Cox_{n} and their duals are strongly eutactic (because their group acts irreducibly), and moreover perfect except for Cox_{5}^{∗}. Let now n ≥ 8 be even. Then, the lattices L = Cox_{n} and L^{∗} are not perfect (because s(L^{∗}) < s(L) < \frac{n(n+1)}{2}), L^{∗} is not eutactic (its n pairs of minimal vectors are not pairwise orthogonal), and the proof of the theorem below will show that L is not strongly eutactic.

Theorem 2.3. For all n ≥ 7, the lattices Cox_{n} are eutactic and dualextreme.

Proof. We only need to consider even dimensions n ≥ 8. Eutaxy and dual-perfection involve the orthogonal projections onto the minimal vectors of L = Cox_{n} and L^{∗}, namely the projections p_{0} onto e_{1}+⋯+e_{n}, p_{i,j} onto e_{i} + e_{j}, 1 ≤ i < j ≤ n, and p′_{i} onto e′_{i}, 1 ≤ i ≤ n respectively.

For the proof of the dual-extremality, we shall establish that there exists up to multiplication by a positive scalar a unique non-trivial linear relation f = 0, where f has the form

a p_{0} + ∑_{1≤i<j≤n} a_{i,j} p_{i,j} − ∑_{1≤h≤n} a′_{h} p′_{h} ,

and that all its coefficients a, a_{i,j}, and a′_{h} are strictly positive. The matrix M = (f(e′_{i}).e′_{j}) has entries

m_{i,j} = \{ \begin{matrix}\frac{1}{μ}(a + a_{i,j}) − \frac{1}{μ^{∗}} (\frac{a′_{i} + a′_{j}}{2} + \frac{∑ a′_{k}}{4} ) if i ≠ j , \\ \frac{1}{μ}(a + ∑_{h} a_{i,h}) − \frac{1}{μ^{∗}} (2a_{i}′ + \frac{∑ a′_{k}}{4} ) if i = j , \end{matrix}

where μ = \frac{2n}{n+2} and μ^{∗} = \frac{3}{2} denote the minimal norms of L and L^{∗}, and where we consider that the a_{i,j} are defined for all i ≠ j, i, j ≤ n, and are symmetric in i, j. For any fixed i ≠ j we have ∑_{h}(m_{i,h} − m_{j,h}) − 2m_{i,i} + 2m_{j,j} = \frac{n−6}{2μ^{∗}}(a′_{j} − a′_{i}); so, if n ≠ 6 the coefficients a′_{i} in any relation f = 0 do not depend on i; it readily follows that there is, up to scale, a unique relation between the projections p_{x}, x ∈ S(L) ∪ S(L^{∗}), which reads

n(n^{2} + 2n − 12) p_{0} + 4n ∑_{i<j} p_{i,j} = 3(n^{2} − 4) ∑_{i} p′_{i} .

Note that taking into account the action of S_{n} on both S(L) and S(L^{∗}), since the space of S_{n}-lattices has dimension 2 – see Section 4 below– the existence of such a relation was evident; for dual-eutaxy the only problem was that of the signs of its coefficients. The same remark applies for the eutaxy relation Id = ap_{0} + b ∑ p_{i,j}. In matrix form, it reads μ(I_{n} + \frac{1}{2}J_{n}) = (n − 2)bI_{n} + (a + b)J_{n}. That leads to the unique system of eutaxy coefficients

(a, b) = (\frac{n(n−4)}{n^{2}−4}, \frac{2n}{n^{2}−4}) ,

which are strictly positive, but distinct for n ≥ 7. □

As mentioned in the introduction, the pair (Cox_{n}, Cox_{n}^{∗}) realizes in all even dimension n ≥ 8 the lower bound s + s^{∗} = \frac{n(n+1)}{2} + 1 for the kissing numbers of a dual-extreme pair.

Note that the eutaxy relation above for Cox_{n} is still valid in dimensions 4 and 6: the lattice Cox_{4} is semi-eutactic (the only one up to dimension 4), and the lattice Cox_{6} ≃ 𝔻_{6}^{+} strongly eutactic (its group W (𝔻_{6}) acts irreducibly). Note also that the proof of 2.3 shows that for any n ≥ 4 the dimension of the span in End^{s}(E) of the projections p_{x}, x ∈ S(L) is equal to s = \frac{n(n−1)}{2} + 1. But for n = 4 or n = 6, the projections onto the minimal vectors of the dual belong to this span, which excludes dual-extremality.

3. Equivariant theory of lattices.

Let G ⊂ O(E) be a finite group of isometries for which E affords a rational representation. The set L_{G} of G-stable lattices in E consists of orbits under the action of the centralizer of G in the linear group GL(E). These orbits are in one-to-one correspondence with the integral equivalence classes of integral representations of G. For the equivariant Voronoi theory we replace End^{s}(E) by the commuting subspace

End_{G}^{s}(E) = {f ∈ End^{s}(E) | ∀σ ∈ G, fσ = σf}

of G, and the orthogonal projections p_{x} by their means

ω_{x} = \frac{1}{|G|} ∑_{σ∈G} p_{σx} .

A lattice Λ ∈ L_{G} is said in [4] to be G-perfect if the ω_{x}, x ∈ S(Λ) span End_{G}^{s}(E). This notion arises in studying the density of G-lattices: a G-lattice Λ is G-extreme, i.e. achieves a local maximum of γ over L_{G}, if and only if it is both G-perfect and eutactic, see [7], Chapters 3 and 11.

We described in the joint paper [5] with Sigrist a Voronoi algorithm with group G, obtaining a connected Voronoi G-graph which classifies all similarity classes of G-perfect lattices affording a given integral representation of G; see also [7], Ch. 13. From now on we rather adopt the point of view of quadratic forms. In the space Sym_{n}(ℝ) of n × n symmetric real matrices equipped with the scalar product ⟨M, N⟩ = Tr(M N), a lattice L ∈ L is represented by one of its Gram matrices A. Its minimal norm is then the minimum min A of the quadratic form A[x] = xA^{t}x (= ⟨A,^{t}xx⟩) for integral vectors x = (x_{1}, x_{2}, …, x_{n}) ≠ 0 ∈ ℤ^{n}, and its sphere is represented in ℤ^{n} by S(A) = {x ≠ 0, x ∈ ℤ^{n} | A[x] = min A}.

Now, given an integral representation ϕ : G → GL_{n}(ℤ) of G, we denote by S_{ϕ} the space of ϕ-invariant matrices, i.e. matrices A ∈ Sym_{n}(ℝ) such that for all σ ∈ G, ^{t}ϕ_{σ}Aϕ_{σ} = A, and by P_{ϕ} the subset of positive definite ϕ-invariant matrices. To a vector x ∈ ℝ^{n} we attach the matrix Ω_{x} := \frac{1}{|G|} ∑_{σ∈G} ϕ_{σ}^{t}xx^{t}ϕ_{σ}, which is invariant under the dual representation of G. By definition, the G-Voronoi domain D_{A} of a matrix A ∈ P_{ϕ} is the convex hull, in the dual space S_{ϕ}^{∗}, of the halflines generated by the Ω_{x}, x ∈ S(A). We can now translate in matrix form the notions above for G-lattices, keeping the same terminology: a eutaxy relation reads A^{−1} = ∑_{x∈G\S(A)} ρ_{x}Ω_{x}, and A is G-perfect if the dimension of its domain is maximal, i.e. equal to

N_{G} = dim(S_{ϕ}) (= dim(End_{G}^{s}(E))) .

We now describe the neighbouring of a G-perfect matrix A ∈ S_{ϕ}. Its domain D_{A} is bounded by a certain set of (N_{G} − 1)-dimensional faces, or facets. (Of course, a G-perfect matrix has at least N_{G} orbits of pairs (±x) of minimal vectors. When it has just N_{G} orbits, we easily obtain each facet of its domain by simply removing one of the orbits.) To each facet there corresponds in S_{ϕ} a uniquely determined Voronoi path orthogonal to it that we now describe. Let F be a facet and F ≠ 0 ∈ S_{ϕ} orthogonal to F and oriented towards the interior of D_{A}. Then the open Voronoi path through F is

c_{F} : t ∈ (0, θ) ↦ A_{t} = A + t F ,

where θ ∈ (0, ∞] is defined by θ = sup {t > 0 | min A_{t} = min A}. We can alternatively define the path c_{F} as the set of matrices M ∈ S_{ϕ} such that min M = min A and S(M) = S_{F} with

S_{F} = {x ∈ S(A) | Ω_{x} ∈ F} .

Following [7], Ch. 13, Th. 3.5., there are two cases:

• If S_{F} spans ℝ^{n}, θ is finite and the matrix A_{θ} is G-perfect again, with sphere S(A_{θ}) ⊋ S_{F}; we say that A and A_{θ} are G-neighbours. Their domains D_{A} and D_{A_{θ}} have just the face F in common.

• If S_{F} spans a strict subspace of ℝ^{n}, θ is infinite. We say that the path is a dead-end.

We summarize now some results of the classification of eutactic lattices applied to equivariant Voronoi paths (for the general setting, see [7] or [1]).

Proposition 3.1. Let c_{F} be an equivariant Voronoi path. Then: (1) There is at most one weakly eutactic matrix in c_{F}.

(2) If c_{F} is a dead-end, it contains no weakly eutactic matrix.

(3) A matrix M ∈ c_{F} is weakly eutactic if and only if it minimizes Hermite’s function γ over c_{F}.

(4) If c_{F} connects two G-extreme matrices, it contains a weakly eutactic matrix.

The dimension N_{G} of Voronoi’s space depends on the canonical decomposition of the representation E (for precise formulae, see [7], Ch. 11, Th. 3.8.). In this paper, we only consider direct sums E = E_{1} ⊥ ⋯ ⊥ E_{k} of k non-isomorphic ℝ-irreducible representations of real type, and then we have N_{G} = k. Note that the irreducible case is trivial for Voronoi’s theory: there are only finitely many similarity classes of G-lattices. In contrast, the dimension N_{G} = 2 is of special interest to effortlessly build 3-designs:

Proposition 3.2. If N_{G} = 2, any weakly eutactic matrix in a G-Voronoi path is strongly eutactic. In particular, there is a strongly eutactic matrix associated with any pair (A, B) of G-extreme neighbours.

Proof. Let S be the (common) sphere of the matrices in the path: the span of the Ω_{x}, x ∈ S, which contains the positive matrix X = ∑_{x∈S} ^{t}x x, also contains the inverse of any weakly eutactic M in the path. From the definition of a Voronoi path, this span has dimension N_{G}−1: under our hypothesis it is a line, hence any eutaxy relation reads M^{−1} = ρX for some ρ necessarily positive: this is strong eutaxy. □

We conclude this section by describing the classification of a family of G-lattices (i.e. an orbit in L_{G} under the action of the centralizer of G in GL(E)) in the case N_{G} = 2. Denote by E_{1} and E_{2} the two irreducible (non isomorphic) components of E.

Theorem 3.3. If N_{G} = 2, the Voronoi graphs have the form

∞_{1} L_{1} ⋯ L_{r} ∞_{2} ,

where L_{1}, …, L_{r} are G-perfect lattices, and where the dead-ends ∞_{1} and ∞_{2} correspond to the orthogonal projections onto the irreducible components E_{1} and E_{2}. Moreover, r = 1 if and only if E_{1} and E_{2} both contain minimal vectors of L_{1}.

Proof. We follow [7], Ch. 13, § 4: in the dual space S_{ϕ}^{∗}, the set of positive matrices is the angular domain with angle π/2 and edges (Ω_{1}) and (Ω_{2}) associated with the projections p_{1} and p_{2} onto E_{1} and E_{2}. The Voronoi domains of the G-perfect matrices form a partition of this right angle into angular domains. We now prove that there are only finitely many such domains. For this proof we adopt the geometric point of view, and evaluate angles in the space End^{s}(E) equipped with the scalar product ⟨f, g⟩ = Tr(f ∘ g). Recall that G-perfect lattices are proportional to integral lattices, and that up to similarity, there are finitely many of them (see [7], Ch. 11, § 9, 9.1 and 9.5). Let E denote the (infinite) set of primitive integral copies of the G-perfect lattices. Their minimal norms take only finitely many values, let m be the biggest. Then the inner products x · y, x, y ∈ S(L), bounded by \frac{m}{2}, take only finitely many integral values when L runs through E. Hence there are finitely many values of ⟨p_{x}, p_{y}⟩ = \frac{(x·y)^{2}}{(x·x)(y·y)}, x, y ∈ S(L), L ∈ E; by averaging over G we obtain the same conclusion for the ⟨ω_{x}, ω_{y}⟩, and thus for the cosine \frac{⟨ω_{x},ω_{y}⟩}{\sqrt{⟨ω_{x},ω_{x}⟩⟨ω_{y},ω_{y}⟩}} of the angle of Ω_{x} and

Ω_{y} in S_{ϕ}^{∗}. So, the partition of the angular domain (Ω_{1}, Ω_{2}) into Voronoi domains is finite, and so is the Voronoi graph (here we do not discard G-lattices isometric to lattices already found). Finally the first and last lattices in the sequence must have minimum vectors in the irreducible subspaces E_{1} and E_{2} respectively, corresponding the edges Ω_{1} and Ω_{2} of their respective domains. □

The proof above shows that either the L_{i} are pairwise non–G-iso-metric, or the graph is symmetric. Examples of both types occur in the next section (the S_{n}-graph for 𝔸_{n} is symmetric if n = 2, but not if n ≥ 3), which also provides graphs with r = 1, 2 or 3.

4. Two sequences of strongly eutactic lattices.

Throughout this section, the group S_{n} is viewed as the permutation group of a given orthonormal basis B = (ε_{1}, ε_{2}, …, ε_{n}) of E.

There are exactly two irreducible stable subspaces of E under this action of S_{n}, namely the line ⟨∑_{i} ε_{i}⟩ and its orthogonal complement: the space of G-invariant symmetric bilinear forms on E has dimension N_{S_{n}} = 2. In matrix form with respect to the basis B, it is the span

S = {aI_{n} + bJ_{n}, (a, b) ∈ ℝ^{2}}

of I_{n} and of the all-ones matrix J_{n}.

Any irreducible root lattice in E admits at least one group action associated with the above permutation representation of S_{n}. Actually, one can prove that, up to ℤ[S_{n}]-isomorphism, there is just one such action, except in the case of 𝔻_{4} and 𝔼_{8}, which have two.

The S_{n}-family of 𝔻_{n}. We represent the lattice

𝔻_{n} = { ∑_{i} x_{i}ε_{i} | (x_{i}) ∈ ℤ^{n}, ∑_{i} x_{i} even }

by the generator matrix G_{n} and the Gram matrix D_{n} = ^{t}G_{n} G_{n}:

G_{n} = ( \begin{matrix}1 & 0 & ··· & 0 & 1 \\ 1 & 1 & ··· & 0 & 0 \\ · & · & ··· & · & · \\ 0 & 0 & ··· & 1 & 0 \\ 0 & 0 & ··· & 1 & (−1)^{n−1} \end{matrix} ) , D_{n} = ( \begin{matrix}2 & 1 & ··· & 0 & 1 \\ 1 & 2 & ··· & 0 & 0 \\ · & · & ··· & · & · \\ 0 & 0 & ··· & 2 & (−1)^{n−1} \\ 1 & 0 & ··· & (−1)^{n−1} & 2 \end{matrix} )

of its basis (ε_{1} + ε_{2}, ε_{2} + ε_{3}, ⋯, ε_{n−1} + ε_{n}, (−1)^{n−1}ε_{n} + ε_{1}). This leads to the following description.

Proposition 4.1. A lattice in E affords the same integral representation of S_{n} as 𝔻_{n} if and only if it is isometric to one with Gram matrix

[a, b] := aD_{n} + bK_{n}, with a > 0 and a + nb > 0 ,

where the matrix K_{n} = ^{t}G_{n} J_{n} G_{n} is equal to 4J_{n} if n is odd and to ( \begin{matrix}4J_{n−1} & 0 0 0 \\ 0 0 0^{t} & 0 \end{matrix} ) if n is even. It has determinant det([a, b]) = 4a^{n−1}(a + nb).

Proof. The Gram matrix M of a basis permuted by S_{n} has constant non-diagonal entries, say b, and constant diagonal entries, say a + b, hence M = aI_{n} + bJ_{n}. The signature and determinant of M follow from its characteristic equation (x − a)^{n−1}(x − (a + nb)). Conjugating by the matrix G_{n} with determinant 2 completes the proof. □

In the following graphs, the S_{n}-perfect lattices are represented by their Gram matrices [a, b] = aD_{n} + bK_{n} scaled to minimum 2. The symbol ∞ indicates a dead-end.

Proposition 4.2. For n ≥ 4, the Voronoi graph associated with the representation ϕ_{𝔻_{n}} of S_{n} is:

n even : ∞ \begin{matrix}𝔻_{n} \\ • \\ [1,0] \end{matrix} \begin{matrix}Cox & n \\ • \\ [ \frac{n+2}{n} , \frac{−1}{n}] \end{matrix} ∞;

n odd : ∞ \begin{matrix}𝔻_{n} \\ • \\ [1,0] \end{matrix}  \begin{matrix}Cox & n \\ • \\ [ \frac{n+1}{n−1} , \frac{−1}{n−1} ] \end{matrix}  \begin{matrix}N_{n} \\ • \\ [ \frac{3n^{2}+2n−1}{2n^{2}−2n} , \frac{−3n−1}{2n^{2}−2n}] \end{matrix} ∞ ,

where N_{n} (n odd) is a S_{n}-perfect lattice with s = n+1 pairs of minimal vectors.

Proof. We have to evaluate the quadratic forms (aI_{n} + bJ_{n})[(x_{i})] on the set

E = {(x_{i}) ∈ ℤ^{n} with ∑ x_{i} ≡ 0 mod 2}

where the group S_{n} acts by permuting the x_{i}. In this point of view, the matrix I_{n} represents the lattice 𝔻_{n}; it has minimum 2 on E attained at two orbits (here, as elsewhere, we waive the distinction between minimal vectors and their negatives): the orbit of (1, −1, 0, …, 0), that we disregard because of Proposition 3.1, and the orbit O of (1, 1, 0, …, 0). The corresponding path from D_{n} is then {^{t}G_{n} M(t) G_{n}}, where

M(t) = (1 + 2t)I_{n} − tJ_{n}, 0 < t < θ_{n}, with θ_{n} = \frac{1}{2⌊\frac{n}{2}⌋} .

Indeed, the matrix 2I_{n}−J_{n} vanishes at O and the matrix^{t}G_{n} M(θ_{n}) G_{n} is S_{n}-extreme, for as we now prove, it is a Gram matrix for the lattice Cox_{n} rescaled to minimal norm 2: when n is odd, \frac{n−1}{n+1}M(θ_{n}) = A_{n}^{−1} is the standard Gram matrix for 𝔸_{n}^{∗}, and thus represents on E the Coxeter lattice 𝔸n^{\frac{n+1}{2}} , of norm \frac{2(n−1)}{n+1}; the case where n is even follows from Definition 2.1. The second orbit O′ of norm 2 vectors of M(θ_{n}) is that of (1, 1, …, 1) when n is even (and by 3.1 the algorithm terminates), and that of (1, …, 1, 0) when n is odd. In this case, it is easy to prove that for (a_{n}, b_{n}) = (\frac{(3n−1)(n+1)}{2n(n−1)}, −\frac{3n+1}{2n(n−1)}), the quadratic form

(a_{n}I_{n} + b_{n}J_{n})[x] = a_{n} ∑(x_{i} − \frac{∑ x_{i}}{n})^{2} + \frac{1}{2}(\frac{∑ x_{i}}{n})^{2} has minimum 2 on the set E with in addition to O′ two minimal vectors ±(2, 2, …, 2). We conclude that the matrix [a_{n}, b_{n}] is S_{n}-perfect, and that the algorithm terminates at it. □

Note that the family defined by 𝔻_{n} always contains the Barnes-Coxeter lattice 𝔸n^{\frac{n+1}{2}} : for n even it lies in the dead-end issuing from Cox_{n}.

We can now enumerate the eutactic lattices in the family.

For n ≥ 7 odd, the path (Cox_{n} −N_{n}) affords no such lattices. [Dimension five is exceptional, for then the representations defined by 𝔻_{5} and its dual are integrally equivalent, and afford two eutactic lattices, namely N_{5} (class e_{6} in [1]) and 𝔻_{5}^{∗}. Following 3.1 (3), we now determine the less dense lattice in the path 𝔻_{n} − Cox_{n}. The typical Gram matrix [1 + 2t, −t], 0 < t < θ_{n} for this path has determinant 4(1 + 2t)^{n−1}(1 + (2 − n)t), whose derivative vanishes at

t_{n} = \frac{1}{2n − 4} ∈ (0, θ_{n})

for n > 4. We can now state the following, where it is to be understood that lattices are defined up to similarity.

Theorem 4.3. For n ≥ 5 the strongly eutactic lattice C_{n} attached to the pair (𝔻_{n}, Cox_{n}) has Gram matrix \frac{n−1}{n−2} D_{n} − \frac{1}{2n−4} K_{n}, s = \frac{n(n−1)}{2} pairs of minimal vectors, and automorphism group {± Id} × S_{n} except for n = 5, where C_{5} ≃ 𝔸_{5}^{2}. Its primitive integral copy has minimal norm n − 2 if n is odd, 2(n − 2) if n is even. □

Examples. We display below Gram matrices for C_{6} and C_{7}:

C_{6} : ( \begin{matrix}8 & 3 & −2 & −2 & −2 & 5 \\ 3 & 8 & 3 & −2 & −2 & 0 \\ −2 & 3 & 8 & 3 & −2 & 0 \\ −2 & −2 & 3 & 8 & 3 & 0 \\ −2 & −2 & −2 & 3 & 8 & −5 \\ 5 & 0 & 0 & 0 & −5 & 10 \end{matrix} ); C_{7} : ( \begin{matrix}5 & 2 & −1 & −1 & −1 & −1 & 2 \\ 2 & 5 & 2 & −1 & −1 & −1 & −1 \\ −1 & 2 & 5 & 2 & −1 & −1 & −1 \\ −1 & −1 & 2 & 5 & 2 & −1 & −1 \\ −1 & −1 & −1 & 2 & 5 & 2 & −1 \\ −1 & −1 & −1 & −1 & 2 & 5 & 2 \\ 2 & −1 & −1 & −1 & −1 & 2 & 5 \end{matrix} ) .

The lattice C_{6} must be added to the list of 19 displayed in [9], whereas C_{7} was discovered by Batut and Sigrist (see [9], p.122) using the cyclic group of order 7 (which does not account for its strong eutaxy).

Other representations of S_{n} related to root lattices. The integral representations of S_{n} defined (up to integral equivalence) by the lattice 𝔸_{n}, n ≥ 2, or by the lattice 𝔻_{n}^{∗}, n ≥ 6, introduce no new 3-design (in each case there is just one pair of S_{n}-extreme lattices, with associated eutactic lattice ℤ^{n} and 𝔻_{n}^{∗} respectively). Note that the 𝔻_{4}^{∗} construction affords the second representation for 𝔻_{4} ≃ 𝔻_{4}^{∗}, for which it has no neighbour (the graph is ∞ 𝔻_{4} ∞).

The representation of S_{6} defined by the root lattice 𝔼_{6} affords the trivial Voronoi graph ∞ 𝔼_{6} ∞.

For n ≥ 8 even, the (maybe not integral) Kneser-neighbour L of the cubic lattice ⨁ ℤε_{i} has (even for n = 8) exactly two ℤ[S_{n}]-structures corresponding to its two constructions 𝔻_{n}^{+} ≃ 𝔻_{n}^{−} consisting in adjoining to 𝔻_{n} the vectors \frac{ε_{1}+···++ε_{n−1}+ε_{n}}{2} and \frac{ε_{1}+···+ε_{n−1}−ε_{n}}{2} respectively. Dimension 8 is of no interest from our point of view: as 𝔻_{8}^{+} the lattice L = 𝔼_{8} has no S_{8}-neighbour, and as 𝔻_{8}^{−} it has just one neighbour, non-eutactic, with s = 9. For n ≥ 10, Voronoi’s graphs have a uniform structure. In particular, they introduce two sequences of strongly eutactic lattices, say C_{n}^{+}, n ≥ 18 and C_{n}^{−}, n ≥ 12, with s = s^{∗} = \frac{n(n−1)}{2}; both lattices contain the lattice C_{n} to index 2, have the same sphere as it, and double density. The S_{n}-neighbour of 𝔻_{n}^{−}, N_{n}^{−} say, is perfect in the classical sense (with just s = \frac{n(n+1)}{2} pairs of minimal vectors), and even extreme for n ≥ 12. It belongs to families of perfect lattices with possible odd minimum of the form

Λ = ⟨e_{1}, …, e_{n}, e = \frac{e_{1} + ⋯ + e_{n}}{d} ⟩, n ≥ 2d ,

constructed by the second author. In the example above, the primitive integral copy of the lattice N_{n}^{−} has odd minimum if and only if n ≡ 2 mod 4. In particular, it affords the first example of a perfect, integral 10-dimensional lattice having an odd minimum (namely 11).

The lattice B_{n}, n odd. The lattices C_{n}, n ≥ 5 described in Theorem 4.3 are not nested. In contrast, in odd dimension, strong eutaxy happens to be preserved by the following section of codimension two. Here we suppose that E has odd dimension 2m + 1, m ≥ 3. We consider the subspace E′ of dimension 2m − 1

E′ = { ∑_{i=1}^{2m+1} x_{i}ε_{i} | x_{1} = 0 and ∑_{k=1}^{m} x_{2k} = ∑_{k=1}^{m} x_{2k+1}}

and its stabilizer G in S_{2m+1}

G ≃ S_{m} × S_{m} ⋊ C_{2} ,

where C_{2} is generated by the product of transpositions

∏_{i≥2,i even}(i, 2m + 3 − i).

The corresponding equivariant families again have dimension N_{G} = 2. In particular, the family of G-lattices defined by the root lattice 𝔸_{2m−1} = 𝔻_{2m+1}∩E′ consists of the sections by E′ of the (2m+1)- dimensional lattices described in Proposition 4.1. The typical invariant matrix can be obtained by removing from the (2m + 1) × (2m + 1) matrices [a, b] the first and last rows and columns; it has the form aA′_{2m−1} + 4bJ_{2m−1}, where

A′_{n} = ( \begin{matrix}2 & 1 & 0 & ··· & 0 & 0 \\ 1 & 2 & 1 & ··· & 0 & 0 \\ 0 & 1 & 2 & ··· & 0 & 0 \\ · & · & · & ··· & ·· \\ 0 & 0 & 0 & ··· & 2 & 1 \\ 0 & 0 & 0 & ··· & 1 & 2 \end{matrix} ) ;

its determinant is det(aA′_{2m−1} + 4bJ_{2m−1}) = 2m a^{2m−2}(a + 2mb).

Proposition 4.4. Let n = 2m − 1 ≥ 5 be an odd integer. The n-dimensional lattice B_{n} = C_{n+2} ∩ E′ with Gram matrix \frac{n+1}{n} A′_{n} − \frac{2}{n} J_{n} is strongly eutactic. Its automorphism group ≃ {± Id} × G acts transitively on its m^{2} pairs of minimal vectors, and its primitive integral copy has minimal norm n.

Proof. It is easy to prove that the family of G-lattices defined by 𝔸_{n} has Voronoi graph

∞  \begin{matrix}𝔸_{n} \\ • \end{matrix}  \begin{matrix}Cox_{n+2} ∩ E′ \\ • \end{matrix}  ∞ ,

where Cox_{n+2} ∩ E′ is G-extreme with two G-orbits of minimal vectors. (Its two eutaxy coefficients \frac{1}{m+1} and \frac{m−1}{2m+2} are distinct except in dimension 5, where 𝔼_{7}^{∗} ∩ E′ ≃ 𝔸_{5}^{2}.) The typical matrix for the Voronoi path is (1 + 2t)A′_{n} − 4tJ_{n}, with 0 ≤ t ≤ θ_{n+2} = \frac{1}{n+1} (see the proof of Theorem 4.3). The derivative of its determinant

(n + 1)(1 + 2t)^{n−1}(1 + (1 − n)t)

again vanishes at t = t_{n+2} = \frac{1}{2n} ∈ (0, \frac{1}{n−1}): the strongly eutactic lattice attached to the pair (𝔸_{n}, Cox_{n+2} ∩E′) coincides with C_{n} ∩ E′. □

The Gram matrix of B_{5} obtained from that of C_{7} by omitting the first and last rows and columns is equivalent to the matrix given by Batut (see [9], p. 121).

5. A family of symplectic lattices in dimension 2m ≥ 6.

Here again E is a Euclidean space of odd dimension 2m+1 (m ≥ 3) equipped with an orthogonal basis (ε_{i}, 1 ≤ i ≤ 2m + 1) acted on by the symmetric group S_{2m+1}. We consider its hyperplane

F = { ∑_{i} x_{i}ε_{i} | ∑_{k=1}^{m} x_{2k} = ∑_{k=0}^{m} x_{2k+1}} ,

and the stabilizer of F in S_{2m+1}

G ≃ S_{m+1} × S_{m} ,

where S_{m+1} acts on {ε_{2k+1}} and S_{m} on {ε_{2k}}.

Regarded as a representation of G, F has three irreducible components, and any family of G-lattices in F has dimension N_{G} = 3. As an example, let us describe the one defined by the root lattice 𝔸_{2m}, leaving the proof to the reader.

Proposition 5.1. A (2m)−dimensional lattice is a G-lattice in the family of 𝔸_{2m} if and only if it is similar to one with Gram matrix

[a, b, c] := ( \begin{matrix}(a−b)I_{m}+bJ_{m} & cI_{m} \\ cI_{m} & cA_{m} \end{matrix} ) ,

0 < c < min{a − b, (m + 1)a + (m^{2} − 1)b} .

In particular, [2, 0, 1] is the Gram matrix of the basis

(ε_{2i} + ε_{2i+1}, ε_{2i+1} − ε_{1}), 1 ≤ i ≤ m

for 𝔸_{2m}. □

Note that the path \{[2, 2 − 2c, c], c ∈ [1, 1 + \frac{1}{m}]\}, which connects the

lattices 𝔻_{2m+1} ∩ F = 𝔸_{2m} and \sqrt{\frac{m+1}{m}} Cox_{2m+1} ∩F = \sqrt{\frac{m+1}{m}} Cox′_{2m}, is not a G-Voronoi path.

From the classification provided by Voronoi’s algorithm we only retain that there are just four G-perfect matrices (all of them G-extreme): apart from [2, 0, 1], there is a not too interesting matrix (minimum 2, s = 2m + 1), and two neighbouring matrices F_{2m} = [2, \frac{−2}{m}, 1] and F′_{2m} = [2, \frac{−2}{m}, 1 + \frac{2}{m}] . The remaining of the section is devoted to the path F_{2m} − F′_{2m}, namely the set of matrices

M(x) := [2, −\frac{2}{m}, \frac{2m + 2}{m}x], x_{0} ≤ x ≤ 1 − x_{0} ,

where x_{0} = \frac{m}{2m+2}, which possess additional symmetries.

Note that M(\frac{1}{2}) = [2, \frac{−2}{m}, 1 + \frac{1}{m}] is a Gram matrix for the lattice

\sqrt{\frac{m+1}{m}} Cox′_{2m}.

To describe group actions, we consider for any x a basis (e_{i}, 1 ≤ i ≤ 2m + 1) of E acted on by G as (ε_{i}) (all permutations of the e_{2k} and all permutations of the e_{2k+1}), with the following inner products: e_{i} · e_{j} is equal to −\frac{1}{2m} for odd i − j, to 2x + \frac{1}{2m} (resp. 2(1 − x) + \frac{1}{2m}) for odd (resp. even) i = j, and to −\frac{2x}{m} + \frac{1}{2m} (resp. −\frac{2(1−x)}{m} + \frac{1}{2m}) for odd (resp. even) i ≠ j. The basis (f_{1}, …, f_{2m}) of F defined by

f_{i} = e_{2i} + e_{2i+1}, f_{i+m} = e_{2i+1} − e_{1}, 1 ≤ i ≤ m

has then Gram matrix M(x).

Now for any x, consider the set

V = {e_{0}, e_{1}, …, e_{2m}, e_{2m+1}} with e_{0} = −(e_{1} + e_{2} + ⋯ e_{2m+1}) .

The inner products e_{i} · e_{j} of these 2m + 2 vectors, for i = j or for i ≠ j, only depend on i and j modulo 2. Thus, in the group of any matrix M(x) there is a subgroup

G′ ≃ S_{m+1} × S_{m+1} ,

where the factors act on the e_{2k} and on the e_{2k+1} respectively. Since F has just two non-isomorphic irreducible G′-subspaces (the spans ⟨e_{i} − e_{j}, i and j odd ⟩ and ⟨e_{i} − e_{j}, i and j even ⟩), any family of G′- invariant matrices has dimension N_{G′} = 2 . Eventually, the product of transpositions Π = ∏_{k=0}^{k=m}(e_{2k}, e_{2k+1}) ∈ S_{2m+2} induces an integral equivalence between the matrices M(x) and M(1 − x), which stabilizes M(\frac{1}{2}) and connects F_{2m} = M(x_{0}) and F′_{2m} = M(1 − x_{0}). Therefore, the stabilizer in GL_{2m}(ℤ) of the path {M(x)} contains a group

G′′ ≃< Π > ×S_{m+1} × S_{m+1} ,

and so does the automorphism group of M(\frac{1}{2}). Since the representation F of G′′ is irreducible, the matrix M(\frac{1}{2}) is strongly eutactic.

We can now state the following proposition.

Proposition 5.2. (1) The matrices M(x) = [2, −\frac{2}{m}, \frac{2m+2}{m}x] are

up to scale symplectic for any x ∈ (0, 1).

(2) For x ∈ [x_{0}, 1 − x_{0}], they have minimum 2.

(3) The matrices F_{2m} = M(x_{0}) and F′_{2m} = M(1 − x_{0}) are G′- extreme, with associated strongly eutactic matrix M(\frac{1}{2}).

(4) For x ∈ [x_{0}, 1 − x_{0}], the automorphism group of M(x) is

Aut(M(x)) = \{ \begin{matrix}\{± Id\} × G′ if x ≠ \frac{1}{2}, \\ \{± Id\} × G′′ if x = \frac{1}{2}, and m ≠ 3. \end{matrix}

Proof. For assertion (1), put y = \sqrt{x(1 − x)}. Then

\frac{m}{2(m + 1)y} M(x) = \frac{1}{y} ( \begin{matrix}A_{m}^{−1} & xI_{m} \\ xI_{m} & (x^{2}+y^{2})A_{m} \end{matrix} )

is the symplectic matrix canonically identified with the point (x+iy)A_{m} in the Siegel space ℍ_{m} = {X + iY | X, Y ∈ Sym_{m}(ℝ), Y > 0} (see [6]). Actually, it lies in the hyperbolic family {zA_{m}, z ∈ ℍ_{1}} attached by Bavard to the lattice 𝔸_{m} ([2]). The determination of the minimum of M(x) is based on the following result of Bavard.

Lemma 5.3. ([2], th. 2.1) The matrix F_{2m} has minimum 2 and \frac{(m+1)(3m+2)}{2} pairs of minimal vectors. □

These vectors are divided under G′-action into two orbits that we describe (up to signs) with respect to the set V = {e_{0}, e_{1}, …, e_{2m+1}} corresponding to x = x_{0} :

O_{1} = {e_{i} + e_{j}, i − j odd }, O_{2} = {e_{i} − e_{j}, i < j, i and j odd } ,

which satisfy the unique eutaxy relation

(m + 2)(m + 1) Id = 2m ∑_{x∈O_{1}} p_{x} + 4 ∑_{x∈O_{2}} p_{x} .

Clearly, Lemma 5.3 is also valid for F′_{2m} ∼ F_{2m}, but we have to replace O_{2} by the orbit O′_{2} = {e_{i} − e_{j}, i < j, i and j even}. From Voronoi’s theory it follows that F_{2m} and F′_{2m} are G′-extreme and that, for x_{0} < x < 1 − x_{0}, M(x) has minimum 2 and sphere S(M(x)) = O_{1}. The strong eutaxy of M(\frac{1}{2}) completes the proof of assertions (2) and (3). Note that under G-action, O_{2} is stable, but O_{1} is divided into two G-orbits, and so is O′_{2}. We conclude that F_{2m} and F′_{2m} are both G-extreme, but that their equivalence does not preserve group action.

For assertion (4), note that any automorphism of M(x) must stabilize the orbit O_{1}, and thus the matrix M(\frac{1}{2}). Taking into account the inclusions we already know, it remains to establish that {± Id} × G′′ is the full automorphism group of M(\frac{1}{2}). We leave the somewhat tedious proof to the reader. □

In geometric language, the properties of the matrix M(\frac{1}{2}) read:

Corollary 5.4. For m ≥ 2 the lattice Cox′_{2m} is symplectic, strongly eutactic, with minimum \frac{2m}{m+1} and s = (m + 1)^{2} pairs of minimal vectors. Its automorphism group is {± Id} × G′′ except for m = 3 where Aut(𝔻_{6}^{+}) = W (𝔻_{6}) contains {± Id} × G′′ to index 10. □

Now let us discuss the extremal properties of the isodual lattice F F F_{2m}

with Gram matrix F_{2m} = ( \begin{matrix}\frac{2(m+1)}{m} A_{m}^{−1} & I_{m} \\ I_{m} & A_{m} \end{matrix} ).

Theorem 5.5. For m ≥ 2 the lattice F F F_{2m} is dual-extreme, and nonperfect except in the case m = 2 where it is isometric to 𝔻_{4}.

Proof. Since Λ = F F F_{2m} is eutactic and isodual, it is dual-eutactic. Let σ : Λ ↦ Λ^{∗} be the symplectic similarity corresponding to the matrix

( \begin{matrix}0 & I_{m} \\ −I_{m} & 0 \end{matrix} ).

To prove that Λ is dual-perfect, we extract from the spheres S(Λ) and S(Λ^{∗}) = σ(S(Λ)) the following set Σ with |Σ| = \frac{2m(2m+1)}{2} elements:

• e_{i} + e_{j}, i − j odd , (i, j) ≠ (0, 1);

• e_{i} − e_{j}, i and j ≥ 3 odd ;

• σ(e_{2k+1} − e_{2k′+1}) = \frac{m}{m+2}(e_{2k′} − e_{2k}).

In matrix form with respect to the basis (e_{k}, 1 ≤ k ≤ 2m + 1) of E, it is easy to prove that the orthogonal projections p_{x}, x ∈ Σ are linearly independent. Similarly, one can prove that a relation ∑_{x∈S(Λ)} λ_{x}p_{x} = 0 has infinitely many solutions depending on m parameters. The rank of the s = \frac{(3m+2)(m+1)}{2} projections is then s − m < \frac{2m(2m+1)}{2} except when m = 2. □

6. Tables.

Tables 6.1 and 6.2 display the main invariants of the n-dimensional lattices constructed in this paper. The symbols μ, γ, γ′, s, s^{∗}, and g denote respectively the minimum of the primitive integral copy of the lattice L, its Hermite and dual-Hermite invariants, the half-kissing-number of L, of L^{∗}, and the order of the automorphism group. The markers (∗) and (∗∗) point out the usual exceptions in low dimension: (∗) corresponds to n = 5 when C_{5} ≃ A_{5}^{2}, and (∗∗) to n = 6 when Cox_{6} ≃ Cox′_{6} ≃ 𝔻_{6}^{+}.

Table 6.1 Strongly eutactic lattices of dimension n ≥ 5

Lattice  μ  γ  γ′  s, s^{∗}  g

C_{n}

n−2  if n is odd
2(n−2) if n is even (\frac{2n−4}{n−1})^{\frac{n−1}{n}} \sqrt{2} \frac{n^{2}−n}{2}, ^{n}^{(∗)} 2 n! ^{(∗)}
\begin{matrix}B_{n} \\ n odd \end{matrix}  ^{n}  ^{n} (\frac{2}{n+1})^{\frac{n+1}{n}} \sqrt{\frac{2n}{n+1}} ^{( \frac{n+1}{2} )^{2}} , ^{n+1}  ^{(2 ( \frac{n+1}{2} )!)^{2}}

\begin{matrix}Cox & ′n \\ n even \end{matrix}

n if n≡2 mod 4
2n if n≡0 mod 4

\frac{2n}{n+2} γ′=γ s=s^{∗}=( \frac{n}{2} +1)^{2} 4 (( \frac{n}{2} +1)!)^{2}^{(∗∗)}

Table 6.2. Dual-extreme lattices of even dimension n ≥ 6

Lattice or matrix  μ  γ  γ′  s, s^{∗}  g

\begin{matrix}F_{n} \\ n even ≥6 \end{matrix}

\frac{n}{2} if n≡0 mod 4
n if n≡2 mod 4 \sqrt{\frac{4n}{n+4}} γ′=γ s=s^{∗}= \frac{3n2+10n+8}{8} 2(( \frac{n}{2} +1)!)^{2}

\begin{matrix}Cox & n \\ n even ≥6 \end{matrix}

n if n≡0 mod 4
\frac{n}{2} if n≡2 mod 4

\frac{n 2^{\frac{n−3}{n}}}{(n+2)^{\frac{n−1}{n}}} \sqrt{\frac{3n}{n+2}} \frac{n^{2}−n+2}{2}, ^{n}^{(∗∗)} 2 n!^{(∗∗)}

References

[1] Batut C (2000) Classification of quintic eutactic forms. Math. Comp. 70: 395– 417

[2] Bavard C (2001) Familles hyperboliques de réseaux symplectiques. Math. Ann. 320: 799–833

[3] Bergé A-M (2000) Une forme dual-extrême irrationnelle. J. Théorie des Nombres de Bordeaux 12: 281–291

[4] Bergé A-M, Martinet J (1992) Réseaux extrêmes pour un groupe d’automorphismes. Astérisque 198-200: 41–66

[5] Bergé A-M, Martinet J , Sigrist F (1992) Une généralisation de l’algorithme de Voronoï pour les formes quadratiques. Astérisque 209: 137–158

[6] Buser P, Sarnak P (1994) On the period matrix of a Riemann surface of large genus (with an Appendix by Conway JH and Sloane NJA). Invent. Math. 117: 27–56

[7] Martinet J (2003) Perfect Lattices in Euclidean Spaces. Grundlehren 327. Berlin Heidelberg New-York: Springer

[8] Martinet J (1997) Une famille de réseaux dual-extrêmes. J. Théorie des Nombres de Bordeaux l9: 169–181

[9] Martinet J, Venkov B (2001) Les réseaux fortement eutactiques. In: Martinet J (ed) Réseaux euclidiens, designs sphériques et formes modulaires Monographie 37 de l’Enseignement Mathématique, pp 112–134. Genève

[10] Venkov B (2001) Réseaux et designs sphériques. In: Martinet J (ed) Réseaux euclidiens, designs sphériques et formes modulaires Monographie 37 de l’Enseignement Mathématique, pp 10–86. Genève

A.-M. Bergé, J. Martinet A2X, Institut de Mathématiques Université Bordeaux 1 351, cours de la Libération F–33405 TALENCE cedex

E-mail : berge@math.u-bordeaux.fr, martinet@math.u-bordeaux.fr