Journal de Théorie des Nombres de Bordeaux ?? (200?), 000–000

Sublattices of certain Coxeter lattices

par Anne-Marie BERGÉ et Jacques MARTINET

À Georges Gras, pour ses soixante ans

Résumé. Dans cet article, nous décrivons les sous-réseaux de certains réseaux de Coxeter, prolongeant les résultats de [Ber]. Notre description utilise des graphes.

Abstract. In this paper, we describe the sublattices of some lattices, extending previous results of [Ber]. Our description makes intensive use of graphs.

1. Introduction

In his 1951 paper Extreme forms ([Cox]), the late Coxeter studied lattices Λ which satisfy inclusions Λ_{0} ⊂ Λ ⊂ Λ_{0}^{∗} for some root lattice Λ_{0}. (As usual, L^{∗} stands for the dual of a lattice L, and a root lattice is an integral lattice generated by norm 2 vectors.) We shall more specially consider the root lattice Λ_{0} = 𝔸_{n}, the section of ℤ^{n+1} endowed with its canonical basis B_{0} = (ε_{0}, ε_{1}, …, ε_{n}) by the hyperplane x_{0} + x_{1} + ⋯ + x_{n} = 0. Then Λ_{0}^{∗}/Λ_{0} is cyclic of order n + 1, so that there exists for every divisor r of n + 1 a well defined lattice 𝔸_{n}^{r} such that 𝔸_{n} ⊂ 𝔸_{n}^{r} ⊂ 𝔸_{n}^{∗} and [𝔸_{n}^{r} : 𝔸_{n}] = r. We denote by Cox_{n} (n odd) the lattice 𝔸_{n}^{(n+1)/2} scaled to the minimum which makes it integral and primitive. The lattices Cox_{n} are perfect lattices which have various curious properties. In particular, they are hollow in the sense of [Ber], and their minimum is odd whenever n ≡ 3 mod 4. This paper is a continuation of [Ber], which was devoted to the classification of crosssections of Cox_{n}, but this time we consider sublattices of finite index (or more generally, of finite index in a cross-section). As [Ber], our paper makes intensive use of graphs to describe the possible sublattices of Cox_{n}.

As in [Ber], Euclidean properties play only a modest rôle: the minimal vectors of 𝔸_{n}^{∗} are vectors ±v_{0}, ±v_{1}, …, ±v_{n} where the v_{i} satisfy the unique (up to proportionality) dependence relation v_{0} + v_{1} + ⋯ + v_{n} = 0. We set

S = {v_{i} + v_{j}, 0 ≤ i < j ≤ n} .

Manuscrit reçu le xx xx 2004.

Key words and phrases. eutactic lattices, spherical designs, kissing number, bipartite graphs.

The lattice Cox_{n} is generated by S, which is actually the set of its minimal vectors (up to sign). We then essentially deal with submodules of the ℤ-module generated by the v_{i}: the Euclidean structure is only used in the last section where we consider some spherical 2-designs related to Cox_{n} (this is the property called strong eutaxy in [Ven]).

Notation. We denote by E the span of the v_{i}, assuming that n = dim E ≥ 5, and of course that n is odd (for the notation Cox_{n} to make sense). We denote by L (resp. M) the ℤ-module generated in E by v_{0}, …, v_{n} (resp. by S). Note that [L : M] = 2.

In Section 1, we characterize in terms of graphs the submodules N of L generated by subsets of S. We then determine in Section 2 the structure of the quotient modules L/N and M/N. Finally, Section 3 is “the Euclidean section”. We construct here a new infinite sequence of strongly eutactic sublattices of Cox_{n} having the same minimum.

2. Graph of subsets of S

Definition 2.1. With a subset B of S we associate the following graph Γ_{B}. Its vertex set is V = {0, 1, ⋯, n}, and ij is an edge of Γ_{B} if and only if v_{i} + v_{j} lies in B.

In order to study such a graph, some more notation is needed.

Notation. For every graph G with vertex set V_{G} ⊂ V, put

e_{G} = ∑_{i∈V_{G}} v_{i} ,

B_{G} = {e_{ij} = v_{i} + v_{j}, ij edge of G} ,

and

L_{G} = ⟨v_{i}, i ∈ V_{G}⟩ and M_{G} = ⟨B_{G}⟩ ⊂ L_{G} ,

where for a subset X of E, the notation ⟨X⟩ stands for the ℤ-module generated by X.

If G is a tree with bipartition (V_{0}, V_{1}), we define the excess of G to be

d_{G} = | |V_{1}| − |V_{0}| | .

In the case of an isolated vertex G = {i}, d_{G} = 1.

[Recall that a tree is a connected graph without cycle; two vertices i and j of the tree are in the same vertex set V_{k}, (k = 0, 1), if the length of the path from i to j is even; see [Bo].]

Eventually, we say that two graphs G and G′ with the same vertex set are equivalent if the corresponding modules M_{G} and M_{G′} coincide.

Theorem 2.2. Let Γ be a graph with vertex set V = {0, 1, …, n} and p ≤ n edges, and let

Γ = ⋃_{C∈C} C

be its partition into connected components. Then the p vectors e_{ij} = v_{i} + v_{j}, ij edge of Γ, are linearly independent if and only if the following conditions hold:

(1) every C ∈ C contains at most one odd cycle, and no even cycle;

(2) there exists exactly n + 1 − p trees in C;

(3) at least one of the trees has a strictly positive excess.

Before proving the theorem, let us discuss the modules associated with the type of subgraphs occurring in its statement. It may be convenient to use some canonical graphs. The shapes we have in mind (see Figure 1) are those of a kite (triangle linked to a path), of a double star (two stars with adjacent centres) or of a shooting star (a star linked to an even path). (A star may reduce to a single vertex.)

\underbrace{[figure: •; •; •; •; •]}_{kite} [figure: •; •; •; •; •; •; i_{0}; i_{1}; •; •; •; •; •; •]

double star

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •]

shooting star

Figure 1

Our basic tool is the following, where we keep the notation e_{ij} = v_{i} + v_{j}.

Lemma 2.3. Let i_{0} i_{1} i_{2} … i_{m−1} i_{m} be a walk of length m in Γ (a cycle if i_{m} = i_{0}, a path otherwise). Then

v_{i_{0}} + (−1)^{m−1}v_{i_{m}} = e_{i_{0} i_{1}} − e_{i_{1} i_{2}} + ⋯ + (−1)^{m−1}e_{i_{m−1} i_{m}} .

Proof. Clear. □

Proposition 2.4. Let C be a connected graph with vertex set V_{C} strictly included in V, containing one odd cycle and no even cycle. Then the following holds for the set B_{C} = {e_{ij}, ij edge of C}, its ℤ-span M_{C}, and the vector e_{C} = ∑_{i∈V_{C}} v_{i}.

(1) For all vertices i, j of C, the vectors v_{i} ± v_{j} belong to M_{C}.

(2) C is equivalent to any kite with vertex set V_{C}.

(3) M_{C} admits B_{C} as a ℤ-basis, and rank (M_{C}) = |C|.

(4) For all i ∈ V_{C}, the vector 2v_{i} is an indivisible element of M_{C}.

(5) The vector e_{C} belongs to M_{C} if and only if |C| is even.

Proof. Since C contains an odd cycle, two vertices i and j are joined by a path of odd length and by a path of even length (one of them including the cycle). Assertion 1 follows by applying Lemma 2.3 to these paths. This implies that M_{C} = ⟨e_{ij}, (i, j) ∈ V_{C} × V_{C}⟩ only depends on its vertex set; assertion 2 follows immediately. In particular, M_{C} contains the vectors 2v_{i}, i ∈ V_{C}, and from 2L_{C} ⊂ M_{C} ⊂ L_{C} it follows that rank(M_{C}) = rank(L_{C}) = |C| (because |C| < n + 1). Since C contains a unique cycle, B_{C} has |C| elements, and therefore is a ℤ-basis for its span. Let i_{C} be a vertex of the cycle γ contained in C. By Lemma 2.3, we obtain 2v_{i_{C}} = ∑_{jk edges of γ} ±e_{jk}, where the e_{jk} belong to the basis B_{C} for M_{C}; thus 2v_{i_{C}} is a primitive element of M_{C}, and by assertion 1, this extends to all vertices of C. Eventually, from assertion 1 again it follows that the vector e_{C} is congruent to |C|v_{i_{C}} modulo M_{C}, and the fifth assertion results from the fourth one. □

Proposition 2.5. Let C be a non-trivial tree with vertex set V_{C} ⊂ V, bipartition (V_{0}, V_{1}) (|V_{1}| ≥ |V_{0}|), and excess d_{C} = |V_{1}| − |V_{0}|. For i ∈ V_{C}, define σ(i) ∈ {0, 1} by i ∈ V_{σ(i)}.

(1) For all i, j in V_{C} we have (−1)^{σ(j)}v_{j} ≡ (−1)^{σ(i)}v_{i} mod M_{C}.

(2) C is equivalent to a double star with the same bipartition.

(3) B_{C} is a basis for M_{C} if and only if (|C|, d_{C}) ≠ (n + 1, 0).

(4) If (|C|, d_{C}) ≠ (n + 1, 0), for all i ∈ V_{C} there exists an indivisible element w_{i} ∈ M_{C} such that d_{C} v_{i} = w_{i} − (−1)^{σ(i)}e_{C} .

[In the case of an isolated vertex C = {i}, the relation of assertion 4 is still valid with w_{i} = 0 (and d_{C} = 1).]

Proof. Let i, j ∈ V_{C}. Lemma 2.3 applied to the path i ↔ j, whose length is congruent to σ(i) − σ(j) modulo 2, proves assertion 1. Choose a pair (i_{0}, i_{1}) ∈ V_{0} × V_{1} of neighbours in C, and let μ_{1} (resp. μ_{0}) be the maximal length of the paths i_{0} ↔ j, j ∈ V_{1} (resp. i_{1} ↔ i, i ∈ V_{0}). We reduce first μ_{1}, and then μ_{0}, to be equal to 1, by successive applications of the following lemma:

Lemma 2.6. Let i j k ℓ ⊂ C be a path of length 3. Then replacing the edge k ℓ by the edge i ℓ we obtain a tree equivalent to C.

Proof of Lemma 2.6. By substituting v_{i} + v_{ℓ} = e_{ij} − e_{jk} + e_{kℓ} to e_{kℓ} in B_{C} we obtain a new generator system for M_{C} = ⟨B_{C}⟩. □

For the remaining assertions, only depending on the bipartition and on the module M_{C}, we may suppose that C is a double star with edges i_{0}j and i_{1}i, (i, j) ∈ V_{0} × V_{1}. The discussion of the linear independence of the |C|−1 vectors of B_{C} is then straightforward. For assertion 4 we may restrict ourselves to the case i = i_{0} (see assertion 1). We then have e_{C} +d_{C}v_{i_{0}} = w_{i_{0}} with

w_{i_{0}} = ∑_{x∈B_{C}} x + (1 − |V_{0}|)e_{i_{0}i_{1}} ∈ M_{C} .

Note that at least one coefficient of w_{i_{0}} on B_{C} is equal to 1. Thus, if B_{C} is a basis for M_{C}, w_{i_{0}} is primitive as required. This completes the proof of Proposition 2.5. □

Proof of Theorem 2.2. In the case when Γ is a tree, this results from Proposition 2.5. From now on we discard this case. The partition of Γ gives rise to a partition B_{Γ} = ∪ B_{C} and to a sum M_{Γ} = ∑ M_{C} (for every C ∈ C, B_{C} is the set of vectors v_{i} + v_{j}, ij edge of C, and M_{C} is its span).

First note that if the vectors of B_{Γ} are linearly independent, Γ contains no even cycle (otherwise, Lemma 2.3 applied to this cycle would provide a dependence relation between them). In the following we suppose that this condition holds. We now discuss the number k_{C} of cycles contained in a given component C ∈ C. Such a component has |B_{C}| = |C| + k_{C} − 1 edges; on the other hand, the inclusion M_{C} ⊂ L_{C} implies rk M_{C} ≤ |C|. We then have the equivalences

B_{C} is a basis for M_{C} ⟺ rk M_{C} = |B_{C}| ⟺ k_{C} = 0 or 1

(the last equivalence makes use of Propositions 2.4 and 2.5). From now on, we assume k_{C} = 0 or 1 for each C ∈ C, i.e. that condition (1) of Theorem 2.2 holds. Then ∑_{C} (1−k_{C}) = −|B_{Γ}|+|V | is the number of trees in C, as asserted in (2).

Now, B_{Γ} will be a basis for its span M_{Γ} = ∑ M_{C} if and only if this sum is direct. Since any dependence relation between the v_{i} is proportional to v_{0} + v_{1} + ⋯ + v_{n} = 0, we have, for (x_{C}) ∈ ∏_{C} M_{C}, the equivalence

∑_{C} x_{C} = 0 ⟺ ∃ a ∈ ℝ such that ∀ C ∈ C, x_{C} = a e_{C} .

So, the sum M_{Γ} = ∑ M_{C} is direct if and only if there exists C ∈ C such that e_{C} does not belong to the space ℝM_{C}. Using the propositions above, one sees that this condition is equivalent to condition (3) of Theorem 2.2. This completes the proof of Theorem 2.2. □

Remark 2.7. It can be convenient to use, as a reference tree, a shooting star instead of a double star. The degree of the centre of the star is then equal to d + 1, where d is the excess of the tree. In particular, the shooting star is a path if and only if d = 0 or 1.

3. Quotient structures.

In this section we assume that B ⊂ S is a basis for E and we will describe the additive structures of the quotient groups L/M_{B} and M/M_{B}, using the partition Γ_{B} = ⋃_{C∈C} C into components. By Theorem 2.2, there is a unique tree T ∈ C, whose excess d ≥ 1 is called excess of B. The indices [L : M_{B}] and [M : M_{B}] only depend on the number of the components and on the excess of B, but the actual structures also depends on the type of the graph:

Definition 3.1. The basis B is of odd type if its graph Γ_{B} contains a component C with |C| odd, and of even type if there is no such component.

Remark 3.2. Note that the type, as well as the number of components and the excess of the basis, may depend of the choice of the basis B for a given submodule N of M. In particular, certain odd types can be reduced to even types: a graph Γ_{B} = T ∪ C_{0} ∪ G where G is a union of kites of even order and T a tree of excess 1 is equivalent to the graph obtained by replacing T ∪ C_{0} by a tree of excess 2 (see Figure 2).

[figure: •; •; •; •; •; •; •; •; ; T ∪ C_{0}; •; •; •; •; •; •; •; •; ; T′] Figure 2

Theorem 3.3. Let B be a basis for E contained in S, let d ≥ 1 be its excess and let c = |C| be the number of components of its graph. Put A = ℤ/2dℤ (resp. ℤ/dℤ × ℤ/2ℤ) if B is of odd (resp. even) type. Then

(1) the group L/M_{B} is isomorphic to A × (ℤ/2ℤ)^{c−2};

(2) the group M/M_{B} has order d 2^{c−2} and is isomorphic to ℤ/(d/2)ℤ if c = 1, and to A × (ℤ/2ℤ)^{c−3} if c ≥ 2.

Proof. To lighten the notation, we assume B of odd type, leaving to the reader the few changes needed for the even case. Let C = {T } ∪ O ∪ E be the set of components of the graph of B, where T denotes the tree, and E (resp. O) denotes the set of components C ≠ T of even (resp. odd) order |C|. Let us fix a component C_{0} ∈ O, and vertices i_{T} ∈ V_{1} (the biggest vertex class of the tree) and i_{C} ∈ V_{C} for each C ≠ T.

Let x = ∑_{i=0}^{i=n} a_{i}v_{i} be a vector of L, with (a_{i}) ∈ ℤ^{n+1} well defined up to translation. Put A_{T} = − ∑_{i∈V_{T}} (−1)^{σ(i)}a_{i} (with σ as in Proposition 2.5), and A_{C} = ∑_{i∈V_{C}} a_{i} for C ≠ T. We consider the following integers (b_{C})_{C≠C_{0}} :

b_{T} = A_{T} + dA_{C_{0}}, b_{C} = A_{C} for C ∈ E, b_{C} = A_{C} − A_{C_{0}} for C ∈ O .

[Note that, under the translation a_{i} ↦ a_{i} + 1, b_{C} is invariant modulo 2d (resp. 2) for C = T (resp. C ≠ T).] By Propositions 2.4 and 2.5, and using in particular the relation d v_{iT} = w_{T} − ∑_{C≠T} e_{C} for some w_{T} ∈ M_{T}, we obtain:

x ≡ A_{T} v_{i_{T}} + ∑_{C≠T} A_{C} v_{i_{C}} ≡ y mod M_{B} ,

with y = \frac{b_{T}}{2d} (2w_{T} − 2 ∑_{C≠T} e_{C}) + ∑_{C≠T,C_{0}} \frac{b_{C}}{2} 2v_{i_{C}}. Since the vectors 2w_{T} − 2e_{C_{0}} ∈ M_{T} ⊕ M_{C_{0}} and 2v_{i_{C}} ∈ M_{C} (C ≠ T, C_{0}) are indivisible, y belongs to the module M_{B} = (M_{T} ⊕ M_{C_{0}}) ⊕_{C≠T,C_{0}} M_{C} if and only if \frac{b_{T}}{2d} and, for C ≠ T, C_{0}, \frac{b_{C}}{2} are integers. Let us denote by ã (resp. \overline{a}) the residue class modulo 2d (resp. 2) of an integer a. Then, the map

ϕ : x ↦ (b̃_{T}, (\overline{b_{C}})_{C≠T,C_{0}})

from L to ℤ/2dℤ × (ℤ/2ℤ)^{c−2} induces an additive isomorphism L/M_{B} ≃ ℤ/2dℤ × (ℤ/2ℤ)^{c−2} as required for assertion 1.

To prove assertion 2 we only need to describe the image ϕ(M) in ℤ/2dℤ × (ℤ/2ℤ)^{c−2} of the submodule

M = \{ ∑_{i} a_{i}v_{i} ∈ L such that ∑_{i} \overline{a_{i}} = \overline{0}\} .

With the notation of the beginning of this proof, we have

\begin{aligned}\sqrt{a_{i}} & = ∑_{C∈C} \overline{A_{C}} = (\overline{b_{T}} − d\overline{A_{C_{0}}}) + (∑_{C∈E} \overline{b_{C}}) + \overline{A_{C_{0}}} + ∑_{C∈O∖C_{0}}(\overline{b_{C}} + \overline{A_{C_{0}}}) \\ & = ∑_{C≠C_{0}} \overline{b_{C}} , \end{aligned}

because, since ∑_{C} |C| = (2|V_{1}| − d) + ∑_{C≠T} |C| = n + 1 is even, so is |O| + d. The canonical homomorphism f : ℤ/2dℤ → ℤ/2ℤ gives rise to a homomorphism τ : (u_{C})_{C≠C_{0}} ↦ f(u_{T})+∑_{C≠C_{0},T} u_{C} of ℤ/2dℤ×(ℤ/2ℤ)^{c−2} onto ℤ/2ℤ. We can now state that the isomorphism L/M_{B} ≃ ℤ/2dℤ × (ℤ/2ℤ)^{c−2} gives rise to an exact sequence

0 ⟶ M/M_{B}⟶ℤ/2dℤ × (ℤ/2ℤ)^{c−2} ⟶^{τ} ℤ/2ℤ ⟶ 0 ,

from which assertion 2 is a straightforward consequence. This completes the proof of Theorem 3.3. □

By the actual construction of graphs Γ_{B} one can determine the possible quotient structures M/M_{B}. For instance, the possible orders of cyclic quotients are the integers a ≤ max(\frac{n−1}{2}, n − 4) and moreover the even integers b ≤ 2(n − 7). Apart from the usual exceptions in low dimensions, the cyclic structure does not provide the maximal index [M : M_{B}], which is equal to 5 × 2^{\frac{n−9}{3}} , 6 × 2^{\frac{n−10}{3}} or 7 × 2^{\frac{n−11}{3}} according as n is congruent to 0, 1 or 2 modulo 3 (and equal to 2 for n = 5).

A finer invariant for the module M_{B} is the set of vectors v_{i} + v_{j}, i ≠ j it contains.

Proposition 3.4. Let B ⊂ S be a basis for E with graph Γ = T ∪_{C≠T} C reduced in the sense of Remark 2, and excess d. Then the ℤ-module M_{B} with basis B contains the vectors v_{i} + v_{j} where i and j are distinct vertices in a same component and moreover at odd distance if this component is the tree. There are no other vectors v_{i} + v_{j} in M_{B} except if d = 1 or d = 2 and Γ is of even type, and then M_{B} also contains the v_{i} + v_{j} for all vertices i, j in the tree.

Proof. This follows directly from Propositions 2.4 and 2.5; we leave the details to the reader. □

In particular, one can determine the maximal values of s_{B} = |M_{B} ∩ S| for a given dimension n. Apart from the case M_{B} = M, i.e. when the reduced graph Γ is a tree of excess 2 (and then s_{B} = \frac{n(n+1)}{2}), the maximum value s_{n} of s_{B} in a given dimension n is attained when [M : M_{B}] = 2: for n = 5, 7, 9, s_{n} = \frac{n^{2}+2n−15}{4} (Γ is a tree of excess 4); for n ≥ 7, s_{n} = \frac{n^{2}−7n+24}{2} (Γ = T ∪ C, where T is a tree of excess 2 and of order 4 or n − 3).

4. Application to Coxeter lattices.

In this section, E is an n-dimensional Euclidean space of odd dimension n ≥ 5. Recall that Cox_{n} is a scaled copy of 𝔸_{n}^{(n+1)/2} ⊂ E. As in the previous sections, 𝔸_{n}^{∗} is generated by n + 1 vectors v_{0}, v_{1}, …, v_{n} which add to zero. For a non-zero x ∈ E, let p_{x} ∈ End^{s}(E) be the orthogonal projection to the line ℝ x. Recall that a lattice Λ with set of minimal vectors S(Λ) is weakly eutactic if there exists in End^{s}(E) a linear relation

Id = ∑_{x∈S(Λ)/±1} ρ_{x} p_{x} ,

eutactic if there exists such a relation with strictly positive eutaxy coefficients ρ_{x}, and strongly eutactic if there exists such a relation with equal ρ_{x} (which are then equal to \frac{n}{s} > 0). This last condition amounts to saying that the set S(Λ) is a spherical 2-design (or 3-design); see [Mar], Sections 3.1 and 16.1, and [Ven], Section 6.

An easy averaging argument shows that if Λ is weakly eutactic, there exist systems of eutaxy coefficients which are constant on the orbits of the automorphism group of Λ. In particular, if the automorphism group of Λ acts transitively on S(Λ), weak eutaxy implies strong eutaxy.

The lattice Cox_{n} is strongly eutactic. Various strongly eutactic lattices related to Cox_{n} have been constructed in [B-M], using the equivariant Voronoi algorithm for the symmetric group 𝔖_{n} or for its subgroup (𝔖_{m} × 𝔖_{m}) ⋊ C_{2}, m = \frac{n−1}{2}. Using such a deformation of the Euclidean structure, we obtained lattices denoted there by C_{n} (n ≥ 5) and B_{n} (n ≥ 5 odd). In the scale which make them integral and primitive, C_{n} has minimum n − 2 (n odd) or 2(n − 2) (n even), and B_{n}, a section of C_{n+2}, has minimum n.

The two theorems below will allow us to recover the lattices above as cross-sections of Coxeter lattices having a much larger dimension.

Theorem 4.1. Let n, p be integers with n ≥ 7 odd and 3 ≤ p ≤ n − 2. Let F_{n,p} be the span in E of v_{1}, …, v_{p} and set Cox_{n,p} = Cox_{n} ∩F_{n,p}. The following conditions are equivalent:

(1) Cox_{n,p} is weakly eutactic.

(2) Cox_{n,p} is strongly eutactic.

(3) p = \frac{n+3}{2}.

[It results from [Ber] that for p ≥ 6, the p-dimensional sections of Cox_{n} generated by minimal vectors whose kissing number is maximal (indeed, equal to \frac{p(p−1)}{2} ) are the images of Cox_{n,p} under an automorphism of Cox_{n}.]

Theorem 4.2. Let n, p be integers with n ≥ 7 odd and 3 ≤ p ≤ n − 2 odd. Let F′_{n,p} be the span in E of the vectors v_{i} + v_{j}, 1 ≤ i, j ≤ p, i odd, j even, and set Cox′_{n,p} = Cox_{n} ∩F′_{n,p}. The following conditions are equivalent:

(1) Cox′_{n,p} is weakly eutactic.

(2) Cox′_{n,p} is strongly eutactic.

(3) p = \frac{n−1}{2}.

Figure 3 below displays the graphs corresponding to the lattices Cox_{11,7} and Cox′_{11,5}:

\underbrace{[figure: •; •; •; •; •; •; •; •; •; •; •; •]}_{Cox_{11,7}}  \underbrace{• • • • • • • • • • •  •}_{Cox′_{11,5}} Figure 3

Proof. We shall only prove Theorem 4.1, leaving to the reader the proof of Theorem 4.2, which follows the same pattern.

For the proof, we work with 𝔸_{n}^{(n+1)/2} (recall that Cox_{n} is a scaled copy of 𝔸_{n}^{(n+1)/2}). Its minimal vectors are the ±(v_{i}+v_{j}), i < j, and it has minimum m = \frac{2n−2}{n+1}. Due to the transitive action of 𝔖_{p} on S(Cox_{n,p}), the assertions (1) and (2) are equivalent. To prove that (1) and (3) are equivalent, we evaluate ∑_{1≤i<j≤p} p_{v_{i}+v_{j}}. Taking into account the action of 𝔖_{p}, it suffices to evaluate this sum on v_{1}.

Using a Gram matrix for 𝔸_{n}^{∗}, it is not difficult to check that

\begin{matrix}∑_{i<j} p_{v_{i}+v_{j}}(v_{1}) = & ∑_{1≤i<j≤p} \frac{(v_{i} + v_{j}) · v_{1}}{m} (v_{i} + v_{j}) \\ = & ∑_{2≤i<j≤p} \frac{−2}{m(n + 1)} (v_{i} + v_{j}) + ∑_{2≤j≤p} \frac{n − 1}{m(n + 1)} (v_{1} + v_{j}) . \end{matrix}

Writing this sum as ∑_{1≤k≤p} a_{i}v_{i} and replacing m by its value, we obtain

∑_{i<j} p_{v_{i}+v_{j}}(v_{1}) = \frac{p − 1}{2} v_{1} + \frac{n − 2p + 3}{2(n − 1)} ∑_{2≤j≤p} v_{j} .

This formula clearly shows that ∑_{i<j} p_{v_{i}+v_{j}} is proportional to the identity if and only if n − 2p + 3 = 0. □

Proposition 4.3. The lattice defined in Theorem 4.1 (resp. Theorem 4.2) is isometric to C_{p} (resp. B_{p}).

Proof. We make use of the notion of a minimal class and its equivariant version as defined in [Mar], Sections 9.1 and 11.9.

The 𝔖_{p}-equivariant minimal class of the lattice C′_{p} = Cox 2p − 3, p defined in Theorem 4.1 has dimension 1, which shows that it is an equivariant Voronoi path, whose minimal vectors can be extracted from those of Cox_{n}. In Theorem 4.3 of [B-M] it is proved that C_{p} lies on an equivariant Voronoi path connecting Cox_{p} and the root lattice 𝔻_{p}. This allows us to identify the 𝔖_{p}-equivariant minimal classes of C_{p} and C′_{p}. Since a minimal class contains at most one weakly eutactic lattice (up to similarity), the (strongly) eutactic lattices C_{p} and C′_{p} are similar, hence isometric since they have the same minimum. (They are even 𝔖_{p}-isometric, i.e. isometric under an isometry which commutes with the action of 𝔖_{p}.)

The case of B_{p} is dealt with by a similar argument. □

Remark 4.4. In Theorem 4.1, when n = 5, we have p = 4 and the lattice is only semieutactic (its eutaxy coefficients are non-negative, but some of them are zero). However, the 4-dimensional section having the largest possible kissing number is again a strongly eutactic lattice (similar to 𝔸_{2} ⨂ 𝔸_{2}, with s = 9).

Remark 4.5. The lattice B_{p} can be also defined for any even p ≥ 6. Then it is only semi-eutactic, but it nevertheless has only one non-zero eutaxy coefficient. In such a situation, the subset of its minimal vectors whose corresponding eutaxy coefficient is non-zero constitutes a spherical 2-design.

We could show that for n ≥ 7 odd, the lattices B_{(n−1)/2} and C_{(n+3)/2} are the only weakly eutactic cross-sections of Cox_{n} having the same minimum as Cox_{n}.

References

[Ber] A.-M. Bergé, On certain Coxeter lattices without perfect sections, J. Algebraic Combinatorics 20 (2004), 5–16.

[B-M] A.-M. Bergé and J. Martinet, Symmetric Groups and Lattices, Monatsh. Math. 140

(2003), 179–195.

[Bo] B. Bollobás, Modern graph theory, Graduate texts in Mathematics 184, Springer-Verlag, Heidelberg, 1998.

[Cox] H.S.M. Coxeter, Extreme forms, Canad. J. Math. 3 (1951), 391–441.

Sublattices of certain Coxeter lattices 11

[Mar] J. Martinet, Perfect Lattices in Euclidean Spaces, Grundlehren 327, Springer-Verlag, Heidelberg, 2003.

[Ven] B. Venkov, Réseaux et designs sphériques (notes by J. Martinet), in Réseaux euclidiens, designs sphériques et groupes, L’Ens. Math., Monographie 37, J. Martinet ed., Genève (2001), 10–86.

Anne-Marie Bergé Université Bordeaux 1 351, cours de la Libération 33451 Talence Cedex, France

E-mail : berge@math.u-bordeaux1.fr
URL: http://www.math.u-bordeaux.fr/ berge/

Jacques Martinet Université Bordeaux 1 351, cours de la Libération 33451 Talence Cedex, France

E-mail : martinet@math.u-bordeaux1.fr
URL: http://www.math.u-bordeaux.fr/ martinet/