A GENERALIZATION OF SOME LATTICES OF COXETER

ANNE-MARIE BERGÉ AND JACQUES MARTINET

Abstract. This paper introduces a wide generalization of a family of integral lattices defined by Coxeter, which share with the Coxeter lattices the following properties: they are perfect, with often an odd minimum, and have no nontrivial perfect sections with the same minimum.

Let E be an n-dimensional Euclidean space with scalar product x · y and norm N(x) = x · x. A lattice in E is a discrete subgroup Λ of E of rank n. We set m = min Λ = min N(x), x ∈ Λ∖{0} (the minimum of Λ), S = S(Λ) = {x ∈ Λ | N(x) = m\} (the sphere of Λ), and s = \frac{1}{2}|S| (its (half ) kissing number).

In his 1951 paper [Cox], Coxeter considered the lattices L defined for odd n ≥ 5 by the conditions 𝔸_{n} ⊂ L ⊂ 𝔸_{n}^{∗} and [𝔸_{n}^{∗} : L] = 2, where 𝔸_{n} stands as usual for the root lattice generated by the root system of type A_{n}. He determined their minimum and proved that they are perfect, with s = \frac{n(n+1)}{2}, the smallest possible kissing number for a perfect lattice. In the sequel, we denote by Cox_{n} their scaled copy to the smallest minimum m which makes them integral (2(n − 1) if n ≡ 1 mod 4, \frac{n−1}{2} if n ≡ 3 mod 4; see [M], Section 5.2).

In her paper [Be], the first author proved that these lattices are hollow for all n ≥ 7 in the sense that they do not have any perfect r-dimensional sections with the same minimum in the range 1 < r < n. (Cox_{5}, which has hexagonal sections of minimum m, is an exception.) Her method relies on the fact that S(𝔸_{n}^{∗}) is of the form {±v_{0}, ±v_{1}, …, ±v_{n}} with vectors v_{i} which add to zero, and that the minimal vectors of Cox_{n} (up to sign) are then the sums v_{i} + v_{j} for 0 ≤ i < j ≤ n.

In relation with his recent joint paper [M-V] with Boris Venkov, the second author tried to construct in various dimensions integral perfect lattices Λ having an odd minimum, using a construction going back to Watson ([W]; see also [M1]): one starts with a lattice Λ′ having a basis (e_{1}, …, e_{n}) of minimal vectors, and then considers lattices of the form Λ = ⟨Λ′, e⟩ for a vector e = \frac{e′}{d}, e′ ∈ Λ′, d ≥ 2. chosen in such a way that the minimal vectors of Λ lie in the set S_{0} = {±e_{i}}∪{±(e−e_{i} −e_{j})}. It appeared that S_{0} could be viewed as a set {±(v_{i}+v_{j})} where the v_{i}, i = 0, 1, …, n belong to a lattice M containing Λ to index 2 and are related by a single relation c_{0}v_{0} + c_{1}v_{1} + ⋯ + c_{n}v_{n} = 0 with at most one null coefficient c_{i}, an observation which throws new light on Watson’s constructions. Using this device, we are able to prove:

Theorem 0.1. For every n ≥ 10, there exist perfect, hollow n-dimensional lattices having an odd minimum.

The mere existence of perfect lattices having an odd minimum (hollow or not) for all large enough dimensions is new (the Coxeter family solves the problem only for dimensions congruent to 3 modulo 4). The proof of Theorem 0.1 will be given in Section 3. That these lattices are actually hollow lattices is proved following closely the arguments of [Be].

To guess systems (c_{i}) which ensure that all vectors in S_{0} have the same odd norm is rather easy. As usual, the difficult problem is to show that these vectors are minimal. The theorem below (chosen because the corresponding lattices are highly symmetric; their automorphism group contains the symmetric group on n letters) is an instance of such a result. To state it, we introduce some more notation. We consider positive integers a_{1}, …, a_{n}, d and a lattice Λ = ⟨Λ′, e⟩ for which e = (a_{1}e_{1} + ⋯ + a_{n}e_{n})/d. We shall prove that there is a unique system of scalar products (e_{i}·e_{j}) such that all vectors in S_{0} have a given common norm m > 0. Set A = a_{1} + ⋯ + a_{n} and q = A − 2d. For the system (v_{i}), we then have c_{0} = q and c_{i} = a_{i} if i ≥ 1, and thus C := ∑_{i=0}^{n} c_{i} = 2A − 2d is even, as well as A − q, and the transition between Coxeter-like and Watson-like constructions is given by the formulae e_{i} = v_{0} + v_{i} and e = 2v_{0}, whence e − e_{i} − e_{j} = −(v_{i} + v_{j}), and in the other direction, v_{0} = \frac{e}{2} and v_{i} = −(\frac{e}{2} − e_{i}).

Theorem 0.2. If n ≥ 10 and a_{1} = ⋯ = a_{n} = 1, the vectors of S_{0} are minimal if and only if 0 ≤ q ≤ \sqrt{n + 1} − 1, and S(Λ) reduces to S_{0} if and only if these

inequalities are strict. (If q = 0, S(Λ) = S_{0} ∪ \{±(e − e_{i})\}; if q = \sqrt{n + 1} − 1, S(Λ) = S_{0} ∪ {±(e_{i} − e_{j})}.)

The proof will be given in Section 2. This suffices to prove Theorem 0.1 for all dimensions n ≥ 21 not divisible by 8 (and indeed for all n outside a set of density zero). The missing dimensions will be dealt with using systems (a_{i}) with a_{1} = ⋯ = a_{n−2} = 1, a_{n−1} = 1 or 2, and a_{n} = 2, see Section 3 and the appendix.

1. A perfect system

In this section, we assume that E has dimension n ≥ 5. We denote by End^{s}(E) the space of symmetric endomorphisms of E, and for any x ∈ E, by p_{x} ∈ End the orthogonal projection onto the line spanned by x (and set p_{0} = 0). The perfection rank of a subset S of E is the rank of the system {p_{x}, x ∈ S}, and we say that S is perfect if this perfection rank is equal to dim End^{s}(E) = \frac{n(n+1)}{2}. As usual, a lattice Λ in E is perfect if the set of its minimal vectors is perfect.

We now give a method of construction of perfect lattices, starting from a system of n + 1 vectors of E.

Theorem 1.1. Let V = {v_{i}, 0 ≤ i ≤ n} ⊂ E be a set of n + 1 distinct, non-zero vectors of E, of rank n, satisfying a non-trivial relation with integral coefficients. If the set

S_{V} = {v_{i} + v_{j}, 0 ≤ i < j ≤ n} ⊂ E

consists of minimal vectors of the lattice Λ_{V} = ⟨S_{V}⟩ it spans, then the set S_{V} and the lattice Λ_{V} are perfect.

We first establish a more general property of perfection for the set S_{V}.

Lemma 1.2. Let ∑_{0≤i≤n} c_{i} v_{i} = 0, c_{i} ∈ ℝ be a non-trivial relation between the v_{i} ∈ V. Set C = ∑_{0≤i≤n} c_{i}. Then, if the coefficients c_{i} satisfy the hypotheses

c_{i} ≠ \frac{C}{4} for 0 ≤ i ≤ n and ∑_{0≤i≤n} \frac{C}{C − 4c_{i}} − (n − 3) ≠ 0 ,

the set S_{V} is perfect. This is true in particular when c_{i} < \frac{C}{4} for all i, which implies

∑_{0≤i≤n} \frac{C}{C − 4c_{i}} − (n − 3) > 8 .

Proof. Since |\{(i, j), 0 ≤ i < j ≤ n\}| = \frac{n(n+1)}{2}, it suffices to prove that the p_{x}, x ∈ S_{V} are linearly independent in End^{s}(E). Consider in End^{s}(E) an endomorphism

∑_{0≤i<j≤n} N(v_{i} + v_{j}) λ_{ij} p_{v_{i}+v_{j}} with (λ_{ij}) ∈ ℝ^{\frac{n(n+1)}{2}} .

We suppose for example c_{0} ≠ 0, and use the basis B = (v_{1}, …, v_{n}) and its dual basis B^{∗}. For convenience, we set

∀ i ≥ 1, ∀ j < i, λ_{ij} = λ_{ji}, μ_{i} = \frac{λ_{0i}}{c_{0}^{2}} and μ = ∑_{1≤i≤n} μ_{i} .

The n × n matrix (a_{hk}) of the endomorphism ∑ λ_{ij}N(v_{i} + v_{j}) p_{v_{i}+v_{j}} in the bases (B^{∗}, B) is given by

a_{hk} = \{ \begin{matrix}λ_{hk} + c_{h}c_{k}μ − c_{0}(c_{k}μ_{h} + c_{h}μ_{k}) if h ≠ k \\ ∑_{l≠k} λ_{lk} + c_{k}^{2} μ − c_{0}(2c_{k} − c_{0})μ_{k} if h = k. \end{matrix}  (1.1)

This implies that ∑_{h≠k} a_{hk} − a_{kk} = c_{k}(C − 2c_{0} − 2c_{k}) μ − c_{0}(C − 4c_{k}) μ_{k} for all k ≥ 1. Using now the hypothesis C − 4c_{k} ≠ 0 for all k, we obtain

\frac{∑_{h≠k} a_{hk} − a_{kk}}{C − 4c_{k}} = \frac{c_{k}(C − 2c_{0} − 2c_{k})}{C − 4c_{k}} μ − c_{0} μ_{k} ,  (1.2)

and thus

∑_{1≤k≤n} \frac{∑_{h≠k} a_{hk} − a_{kk}}{C − 4c_{k}} = ( ∑_{1≤k≤n} \frac{c_{k}(C − 2c_{0} − 2c_{k})}{C − 4c_{k}} − c_{0}) μ ,  (1.3)

where

∑_{1≤k≤n} \frac{c_{k}(C − 2c_{0} − 2c_{k})}{C − 4c_{k}} − c_{0} = \frac{C − 4c_{0}}{8} ( ∑_{0≤i≤n} \frac{C}{C − 4c_{i}} − (n − 3)) ≠ 0 .

Now, suppose that ∑ N(v_{i} + v_{j}) λ_{ij} p_{v_{i}+v_{j}} is zero, i.e. that a_{hk} = 0 for all h, k. It follows from (1.3) that μ = 0, hence by (1.2) that λ_{0k} = μ_{k} = 0 for all k ≥ 1, and finally by (1.1) that λ_{hk} = 0 for all 1 ≤ h < k ≤ n. Hence S_{V} is perfect.

Suppose now that C − 4c_{i} > 0 for all i ≥ 0, which implies C > 0, and thus

u_{i} = \frac{C}{C − 4c_{i}} > 0. We have ∑ \frac{1}{u_{i}} = n − 3, hence

∑_{0≤i≤n} \frac{C}{C − 4c_{i}} − (n − 3) = ∑_{0≤i≤n} u_{i} − ∑_{0≤i≤n} \frac{1}{u_{i}} .

Multiplying side by side the arithmetic-geometric inequalities

∑_{i} u_{i} ≥ (n + 1)(∏_{i} u_{i})^{\frac{1}{n+1}} and ∑_{i} \frac{1}{u_{i}} ≥ (n + 1)(∏_{i} \frac{1}{u_{i}})^{\frac{1}{n+1}} ,

we obtain ( ∑_{i} u_{i})( ∑_{j} \frac{1}{u_{j}} ) ≥ (n + 1)^{2} , where equality holds if and only if all u_{i} are equal. In particular,

∑ u_{i} − ∑ \frac{1}{u_{i}} ≥ \frac{(n + 1)^{2} − (∑ \frac{1}{u_{i}})^{2}}{∑ \frac{1}{u_{i}}} ,

i.e. ∑_{0≤i≤n} \frac{C}{C−4c_{i}} − (n − 3) ≥ \frac{8(n−1)}{n−3} > 8 . □

Proof of Theorem 1.1. We prove that the coefficients of the relation ∑_{0≤i≤n} c_{i}v_{i} = 0, where without loss of generality we may suppose C = ∑ c_{i} ≥ 0, satisfy the hypotheses of Lemma 1.2, under the mere assumptions

\{ \begin{matrix}for 0 ≤ i ≤ n : N(2v_{i}) ≥ m \\ for 0 ≤ i < j ≤ n : N(v_{i} − v_{j}) ≥ N(v_{i} + v_{j}) = m > 0 . \end{matrix}  (1.4)

[Indeed the lattice Λ_{V} contains the non-zero vectors 2v_{i} = (v_{i} + v_{j}) + (v_{i} + v_{k}) − (v_{j} + v_{k}) and v_{i} − v_{j} = v_{i} + v_{j} − 2v_{j}, and the hypothesis min Λ_{V} = m implies conditions (1.4).]

For i = 0, …, n, let

α_{i} = v_{i} · v_{i} .

It results from the hypotheses (1.4) that for i ≠ j, we have

α_{i} + α_{j} − m = −2v_{i} · v_{j} = \frac{N(v_{i} − v_{j}) − N(v_{i} + v_{j})}{2} ≥ 0 .  (1.5)

We now use the relation ∑_{j≠i} c_{j}(v_{i} · v_{j}) = v_{i} · ∑_{j≠i} c_{j}v_{j} = −c_{i}α_{i}. By performing linear combinations on the equalities in (1.5), we obtain the n + 1 Watson-like identities

∑_{j≠i} c_{j} \underbrace{(N(v_{i} − v_{j}) − N(v_{i} + v_{j}))}_{≥0} = c_{i} N(2v_{i}) ,  (W(i))

and the relations ∑_{j≠i}(α_{i} − m) + ∑_{j≠i} c_{j}α_{j} = 2α_{i}c_{i}, which also read

(C − 4c_{i})(α_{i} − \frac{m}{4}) = K where K = \frac{m C}{2} − ∑_{j} c_{j}(α_{j} − \frac{m}{4}) .  (1.6)

We now prove that K is strictly positive. Since 2v_{i} ∈ Λ_{V} is non-zero and 4α_{i} = N(2v_{i}), we have α_{i} ≥ \frac{m}{4} for all i, and the inequalities in (1.5), namely α_{i} + α_{j} ≥ m for all i ≠ j, show that there exists at most one index i_{0} such that α_{i_{0}} ≤ \frac{m}{4}, and then α_{i_{0}} = \frac{m}{4} by condition (1.4). Thus K ≤ 0 would imply by (1.6) c_{i} ≥ \frac{C}{4} ≥ 0 for all i ≠ i_{0}, and Watson’s identity (W(i_{0})) then implies c_{i_{0}} ≥ 0. Since the c_{i} are non-negative and not all zero, their sum C is strictly positive and satisfies the inequality C = ∑_{i≠i_{0}} c_{i} + c_{i_{0}} ≥ n \frac{C}{4} , a contradiction for n ≥ 5. From (1.6) and K > 0, it follows that C > 4c_{i} (and α_{i} > \frac{m}{4}) for all i, and then, by Lemma 1.2 the set S_{V} is perfect, and so is the lattice Λ_{V} if m is its minimal norm. (Actually, this last condition is fulfilled under the hypotheses (1.4) at least in the cases studied in Section 2.) This completes the proof of the theorem. □

From Formula (1.6) we can now make explicit K and the Gram matrix for the set V. For further use, we state these results as a lemma.

Lemma 1.3. Under the hypotheses of Theorem 1.1, one has

K = \frac{2m C}{∑i \frac{C}{C−4c_{i}}} − (n − 3) ,

and

∀ i, j, v_{i} · v_{j} = \{ \begin{matrix}\frac{m}{4} + \frac{K}{C − 4c_{i}} (= α_{i}) if j = i, \\ \frac{m − α_{i} − α_{j}}{2} if j ≠ i. \end{matrix}

[As above, ∑_{i=0}^{n} c_{i}v_{i} = 0 and C = ∑_{i=0}^{n} c_{i}.]

Proof. By (1.6) we have α_{i} − \frac{m}{4} = \frac{K}{C−4c_{i}} for all i, hence K = \frac{m C}{2} − K ∑_{i} \frac{c_{i}}{C−4c_{i}}, where ∑_{i} \frac{c_{i}}{C−4c_{i}} = \frac{1}{4}( ∑_{i} \frac{C}{C−4c_{i}} − (n + 1)); this shows that a K = 2m C where a = ∑_{i} \frac{C}{C−4c_{i}} − (n − 3) is > 8 by Lemma 1.2. This gives K and the α_{i}, and the other scalar products stem from Formula (1.5). □

Incidentally, we have proved the upper bound K < \frac{mC}{4} , from which one can derive some other properties of the c_{i}, namely:

(i) ∀ i, c_{i} ≥ 0, and c_{i} = 0 for at most one index i_{0} (note that from (W(i_{0})), c_{i_{0}} = 0 ⇔

N(v_{i} − v_{i_{0}}) = m for all i ≠ i_{0}).

(ii) C = ∑ c_{i} is even; see the introduction.

2. Calculation of minima

In this section, we prove Theorem 0.2 and then sketch the proof of an analogous result that we shall also need in the next section to prove Theorem 0.1. The basic idea is to use a “decomposition into many squares” of the quadratic form N(x).

Proof of Theorem 0.2. The notation is that of Lemma 1.2, with c_{0} = q and c_{i} = 1 for i ≥ 1, where 0 ≤ q ≤ \sqrt{n + 1} − 1. Thus C = n + q satisfies the conditions C > 4c_{i}. Lemma 1.3 shows that the scalar products v_{i} · v_{j} take at most four values, namely (with 1 ≤ i < j ≤ n)

α_{0} = v_{0} · v_{0}, α = v_{i} · v_{i}, β_{0} = −v_{0} · v_{i}, β = −v_{i} · v_{j} ,

uniquely determined by the value of m = N(v_{i} + v_{j}). For convenience, we give m the value

m = \frac{n^{2} − 2n(q + 1) + 4q − q^{2}}{2} ,

a strictly positive (and even) integer (note that A = n ≡ q mod 2). We then have

\begin{aligned}α_{0} & = \frac{1}{4}(n^{2} − 3n), α = \frac{1}{4}(n^{2} − n(2q + 1) + 2q − 2q^{2}), \\ β_{0} & = \frac{1}{4}(n − 3)q, β = \frac{1}{4}(n − 2q − q^{2}) , \end{aligned}

and the bounds we assumed for q are equivalent to conditions (1.5): for all i ≠ j, v_{i} · v_{j} ≤ 0 ⟺ N(v_{i} − v_{j}) ≥ m.

We first suppose q ≥ 1, and use the basis B = (v_{1}, …, v_{n}) for E. The vector x = ∑_{1≤i≤n} x_{i}v_{i} belongs to the lattice Λ_{V} = ⟨v_{i} + v_{j}⟩ if and only if its components have the form x_{i} = \frac{a_{i}}{q} where the integers a_{i} are pairwise congruent modulo q and have an even sum. Its norm is

N(x) = (α − (n − 1)β) ∑_{i} x_{i}^{2} + β ∑_{i<j}(x_{i} − x_{j})^{2},

where α−(n−1)β = \frac{1}{4} (n−3)q^{2}. Note that m = 2α−2β = 2(α−(n−1)β)−2(n−2)β, and hence

N(x) − m = (α − (n − 1)β)(Q_{1} − 2) + β(Q_{2} − 2(n − 2)) ,  (2.1)

where Q_{1} = ∑_{i} x_{i}^{2} and Q_{2} = ∑_{i<j}(x_{i} − x_{j})^{2} . We consider three cases according to whether max|x_{i} − x_{j}| is 0, 1 or ≥ 2.

(i) One value case. Then x = −a(v_{1} + ⋯ + v_{n})/q = −av_{0}, a ∈ ℤ, a ≠ 0, lies in the lattice if and only if a is even, which implies N(x) ≥ 4α_{0} > m.

(ii) Two values case, say, x_{1} = ⋯ = x_{k}, x_{k+1} = ⋯ = x_{n} = x_{1} +1. We then have Q_{2} = k(n−k) ≥ 2(n−2) except for k = 1 or n−1, and Q_{1} = nx_{1}^{2}+(n−k)(2x_{1}+1) ≥ k(n − k)/n. If k(n − k) ≥ 3(n − 3), then Q_{1} > 2 (because n ≥ 10), hence N(x) > m by (2.1). It remains to discuss the cases k(n − k) = (n − 1) or 2(n − 2).

Suppose first that (x_{i}) ∈ ℤ^{n}. Then we have Q_{1} ≥ 2, with equality on the sequence (1^{2}, 0^{n−2}) (up to sign and permutations), for which Q_{2} = 2(n−2), N(x) = m, and x is one of the vectors ±(v_{i} + v_{j}), 1 ≤ i < j ≤ n. When Q_{1} > 2 and k(n − k) ≥ 2(n − 2), we have N(x) > m by (2.1). Next suppose that k(n − k) = (n − 1). We then have Q_{1} ≥ n − 1, with equality attained on (1^{n−1}, 0^{1}), and (2.1) shows that N(x) − m ≥ (n − 3)(α − nβ), where α − nβ = (q − 1)(nq + n − 2q) ≥ 0. This implies N(x) > m except when q = 1 and Q_{1} = n − 1, in which case we find the vectors x = ±(v_{i} + v_{0}) of a Coxeter lattice.

Consider now the case where x_{1} = \frac{a}{q}, with a ≢ 0 mod q (which excludes the

Coxeter lattices, for which q = 1). We then have Q_{1} = n(x_{1} + 1 − \frac{k}{n})^{2} + \frac{k(n−k)}{n}, where we may assume (exchanging x_{1} and x_{1} + 1 if need be) that k = 1 or k = 2. From our hypotheses we have \frac{q}{n} < \frac{1}{4}; this shows that Q_{1} has minimum

\frac{n − 2kq + kq^{2}}{q^{2}} = 2 + \frac{n − 2kq + (k − 2)q^{2}}{q^{2}} ,

attained only on x_{1} = \frac{1}{q} − 1. For k = 2, we have Q_{1} > 2, hence N(x) > m by (2.1). For k = 1, formula (2.1) gives

\begin{aligned}N(x) − m & ≥ (α − (n − 1)β) \frac{n − 2q − q^{2}}{q^{2}} − ((n − 3)β) \\ & = \frac{n − 2q − q^{2}}{q^{2}} (α − (n − 1)β − \frac{n − 3}{q^{2}}) = 0 , \end{aligned}

and N(x) = m occurs only on ((\frac{1}{q} − 1)^{1}, \frac{1}{q}^{n−1}), which corresponds to the vectors ±(v_{0} + v_{i}).

(iii) Case when there is a difference |x_{i} − x_{j}| ≥ 2 with, say, x_{1} ≤ x_{2} ≤ ⋯ ≤ x_{n}. We have

Q_{1} ≥ x_{1}^{2} + x_{n}^{2} = \frac{(x_{1} − x_{n})^{2} + (x_{1} + x_{n})^{2}}{2} ≥ 2 ,

and equality holds only on the sequence (−1^{1}, 0^{n−2}, 1^{1}).

Now from the identity Q_{2} = nQ_{1} − (∑ x_{i})^{2}, it follows Q_{2} ≥ 2n: indeed Q_{2} is invariant under translation, and we may assume ∑ x_{i} = 0, and then Q_{2} = nQ_{1} ≥ 2n. Formula (2.1) shows that N(x) − m ≥ 4β ≥ 0. Hence we have N(x) > m, except if β = 0 (i.e. n = 2q +q^{2}) and Q_{1} = 2, i.e. for x = ±(v_{i} −v_{j}), 1 ≤ i < j ≤ n.

So far, we left aside the case where q = 0, because v_{1}, …, v_{n} then no longer constitute a basis for E. To deal with this special case, we considered the less symmetric basis (v_{0}, v_{1}, …, v_{n−1}). The theorem was proved using calculations analogous to the previous ones. Since we do not need this special case in the remaining of the paper, we leave the details to the reader. □

The theorem we have just proved does not suffice to prove Theorem 0.1. For this reason, we give below the analogous statement for systems (a_{i}) = (1^{n−1}, 2), for which A = n + 1, hence q = n + 1 − 2d. The proof follows the same pattern, and we shall restrict ourselves to the case where q = 3 and n ≡ 0 mod 4, which suffices for our applications.

Theorem 2.1. If n ≥ 10 and (a_{1}, …, a_{n}) = (1, …, 1, 2), the vectors of S_{0} are minimal if and only if 1 ≤ q ≤ \sqrt{n} − 1, and S_{0} is then the set of all minimal vectors of Λ.

[Recall that S_{0} = {±e_{i}} ∪ {±(e − e_{i} − e_{j})} = ± S_{V}.]

Proof (sketch for q = 3 and n ≡ 0 mod 4). We work in the basis (v_{0}, v_{1}, …, v_{n−1}) for E. Then x = ∑_{i=0}^{n−1} x_{i}v_{i} belongs to Λ if and only if the x_{i} are either all integral or all halves of integers, and add to an even number. As above, for 1 ≤ i < j ≤ n−1, we set α_{0} = v_{0} · v_{0}, α = v_{i} · v_{i}, β_{0} = −v_{0} · v_{i}, and β = −v_{i} · v_{j}, and give m the value

m = \frac{n^{3} − 10n^{2} + 24n − 16}{16} .

This gives the scalar products v_{i} · v_{j} the values

\begin{aligned}α_{0} & = \frac{1}{32}(n^{3} − 5n^{2} + 4n − 8), α = \frac{1}{32}(n^{3} − 9n^{2} + 4n + 56), \\ β_{0} & = \frac{1}{32}(3n^{2} − 20n + 40), β = \frac{1}{32}(n^{2} − 20n + 72) , \end{aligned}

We write the norm in the form

α_{0}N(x) = (α_{0}x_{0} − β_{0} ∑_{i=1}^{n−1} x_{i})^{2} + B ∑_{i=1}^{n−1} x_{i}^{2} + C ∑_{1≤i<j≤n−1}(x_{i} − x_{j})^{2}

where B = α_{0}(α−(n−2)β)−(n−2)β_{0}^{2} and C = (α_{0}β+β_{0}^{2}). Using arguments similar to those we used in the proof of Theorem 0.2, we show that min Λ = m, attained on the sequences (up to permutation and sign) (\frac{1}{2}, …, \frac{1}{2}), (\frac{3}{2}, −\frac{1}{2}, \frac{1}{2}, …, \frac{1}{2}) and (0, 1, 1, 0, …, 0), which correspond to v_{0} + v_{n}, v_{1} + v_{n} and v_{1} + v_{2} respectively. □

3. Hollow lattices having an odd minimum

In this section, we consider as in the previous sections lattices Λ = ⟨Λ′, e⟩ with e = \frac{a_{1}e_{1}+···+a_{n}e_{n}}{d}, such that S(Λ) = S_{0}.

Theorem 3.1. A lattice of the form Λ = ⟨Λ′, \frac{a_{1}e_{1} + ⋯ + a_{n}e_{n}}{d} ⟩ such that S(Λ) = S_{0} and dim Λ ≥ 7 is hollow.

Proof. We refer the reader to [Be], whose combinatorial methods used in the case where c_{i} = 1 for all i (that of the Coxeter lattices) extend in a straightforward way to non-necessarily equal strictly positive coefficients c_{i}. These methods indeed show that for any r-dimensional subspace F of E with 1 < r < n, we have |S_{0} ∩ F | ≤ \frac{r(r−1)}{2} + 1 < \frac{r(r+1)}{2} (and even s(Λ ∩ F) ≤ \frac{r(r−1)}{2} if r ≤ n − 2). □

We now prove two lemmas on lattices Λ constructed with systems (a_{i}) = (1^{n}) and (1^{n−1}, 2) respectively, from which Theorem 0.1 will follow easily.

Lemma 3.2. If one of the following three conditions holds:

(1) n ≡ 1 mod 4 and n ≥ 17;

(2) n ≡ 2 mod 4 and n ≥ 10;
(3) n ≡ 3 mod 4 and n ≥ 7,

there exists an integral lattice of the form Λ = ⟨Λ′, \frac{e_{1} + ⋯ + e_{n}}{d} ⟩ with an odd minimum and S(Λ) = S_{0}.

Proof. We use the notation we introduced in the proof of Theorem 0.2 and start with the scale defined there, for which the vectors of S_{0} have norm

m = \frac{n^{2} − 2n(q + 1) + 4q − q^{2}}{2} .

(Here, A = n hence q = n−2d.) We shall first choose q ≡ n mod 2 in [1, \sqrt{n + 1}−1) which ensures by Theorem 0.2 the existence of a lattice Λ with min Λ = m and S(Λ) = S_{0}, then divide out m by the highest power a of 2 it contains, and check that the scalar products on a^{−1/2}Λ remain integers. Since Λ is generated by the vectors e_{i} = v_{0} + v_{i} and e = 2v_{0}, this last condition holds if and only if the three scalar products (v_{0} + v_{1}) · (v_{0} + v_{2}), 2v_{0} · 2v_{0} and 2v_{0} · (v_{0} + v_{i}) are divisible by a. These are α_{0} − 2β_{0} − β, 4α_{0} and 2(α_{0} − β_{0}) respectively.

If n ≡ 1 (resp. 2) mod 4, we take q = 3 (resp. q = 2), and n ≥ 17 (resp. n ≥ 10). In both cases, we have m ≡ 2 mod 4. Theorem 0.2 ensures that min Λ = m and S(Λ) = S_{0}, and it is easily checked using the explicit values for α_{0}, β_{0}, β given in the proof of Theorem 0.2 that the three scalar products above are even.

If n ≡ 3 mod 4, we take q = 1. Then Λ is proportional to Cox_{n}, which is known to be integral when scaled to minimum \frac{n−1}{2} ; see [M], Proposition 2.5.3.

[If n ≡ 2^{k} mod 2^{k+1}, there exists n(k) such that some Λ can be rescaled to an integral lattice having an odd minimum if and only if n ≥ n(k), for instance, n(4) = 28, n(8) = 88. But dimensions in an infinite set of density zero are excluded, so that we cannot handle all large enough n ≡ 0 mod 4.] □

Lemma 3.3. If n ≡ 0 mod 4 and n ≥ 16, there exists an integral lattice of the

form Λ = ⟨Λ′, \frac{e_{1} + ⋯ + e_{n−1} + 2e_{n}}{d} ⟩ with an odd minimum and S(Λ) = S_{0}.

Proof. We use the data given for q = 3 in the proof of Theorem 2.1. This time, it is easily verified that the value chosen there for m is odd, an that all scalar products (v_{0} +v_{i})·(v_{0} +v_{j}), 2v_{0} ·2v_{0} and 2v_{0} ·(v_{0} +v_{i}) are integral (and incidentally odd). □

Proof of Theorem 0.1. If n ≠ 12, 13, Theorem 0.1 is an immediate consequence of Lemma 3.3 if n is divisible by 4 and of Lemma 3.2 otherwise. If n = 12 or n = 13, we consider explicit Gram matrices relative to systems (1^{n−2}, 2^{2}); see the appendix below. □

Appendix 1: Numerical data

We first list for each dimension n ∈ [10, 25] one perfect, hollow lattice which can be used to prove Theorem 0.1, giving the minimum m, the index d = [Λ : Λ′], and the vector A = (a_{1}, …, a_{n}) in the usual symbolic form (1^{x}, 2^{y}, …). We have considered vectors (1^{n−k}, a_{k+1}, …, a_{n}) for small values of k and of the a_{i}, i > k. (When the sum a_{k+1} + ⋯ + a_{n} becomes too large, low-dimensional examples no more exist.) We have listed below the smallest minimum m we could find and then for given m, the smallest denominator d. For n ≡ 3 mod 4, our lattice is Cox_{n}, for which m = d = \frac{n−1}{2} and a_{1} = ⋯ = a_{n} = 1.

n = 10 m = 11 d = 4 (1^{10}) n = 18 m = 11 d = 8 (1^{18}) n = 11 m = 5 d = 5 (1^{11}) n = 19 m = 9 d = 9 (1^{19}) n = 12 m = 31 d = 6 (1^{10}, 2^{2}) n = 20 m = 91 d = 10 (1^{18}, 2^{2}) n = 13 m = 15 d = 7 (1^{11}, 2^{2}) n = 21 m = 15 d = 10 (1^{20}, 2) n = 14 m = 29 d = 6 (1^{14}) n = 22 m = 89 d = 10 (1^{22}) n = 15 m = 7 d = 7 (1^{15}) n = 23 m = 11 d = 11 (1^{23}) n = 16 m = 57 d = 8 (1^{14}, 2^{2}) n = 24 m = 19 d = 12 (1^{22}, 2^{2}) n = 17 m = 39 d = 7 (1^{17}) n = 25 m = 107 d = 11 (1^{25})

We list below experimental results for lattices with A ≤ n + 2, giving an interval I such that S(Λ) = S_{0} ⟺ q ∈ I except for a few special values of n. The data below have been checked for all dimensions n ≤ 100. Of course, q must moreover satisfy the congruence q ≡ A mod 2.

\underline{A = n},  A = (1^{n})  : 1 ≤ q < \sqrt{n + 1} − 1 ; n ≥ 7, n ≠ 8.

\underline{A = n + 1}, A = (1^{n−1}, 2) : 1 ≤ q ≤ \sqrt{n} − 1 ; n ≥ 10. \underline{A = n + 2}, A = (1^{n−1}, 3) : 1 ≤ q ≤ \sqrt{n − 3} − 1 ; n ≥ 16, n ≠ 19. \underline{A = n + 2}, A = (1^{n−2}, 2^{2}) : 1 ≤ q ≤ \sqrt{n − 1} − 1 ; n ≥ 11.

The minima we found are sometimes very large: 107 (see above) for n = 25, but also 241 for n = 32, 791 for n = 65, 1121 for n = 70, or 2257 for n = 96. It would be interesting to construct integral perfect lattices (maybe, non-hollow) having a much smaller minimum, and in particular minimum m = 3, for which our knowledge is very poor.

[Besides Cox_{7}, we know examples for 15 ≤ n ≤ 23, using cross-sections of the unimodular lattice O_{23} listed in [Bt-M]). Using the classification of unimodular lattices of minimum 3 up to n = 28 (due to Bacher and Venkov in [Bc-V], extending previous works of Borcherds and of Conway), we see that for n = 24 to 28, perfect unimodular lattices of minimum 3 exist for n = 27 (1 lattice out of 3) and n = 28 (28 lattices out of 38). Finally, two perfect lattices (in dimensions 29 and 31) can be found in Nebe and Sloane’s catalogue [N-S]. Note that it is proved in [M-V] that integral perfect lattices of minimum 3 do not exist in dimensions 8 and 9.]

Appendix 2: Extreme Lattices

Recall that a eutaxy relation among vectors of a finite set T ⊂ E is an equality

Id = ∑_{x∈T} λ_{x} p_{x} ,

that T is eutactic if there exists a eutaxy relation with strictly positive coefficients λ_{x}, and that a lattice Λ is eutactic if S(Λ) is. Note that eutaxy relations exist if T is perfect, that such a relation is then unique if and only if |T | = \frac{n(n+1)}{2}, and that a lattice is extreme (i.e., the density its corresponding canonical sphere packing is a local maximum) if and only if it is both perfect and eutactic.

We have considered our set S_{0} in the case where Λ is of the form

⟨Λ′, \frac{e_{1} + ⋯ + e_{n}}{d} ⟩

and S(Λ) contains S_{0} (i.e., setting as usual n = 2d + q, when 0 ≤ q ≤ \sqrt{n + 1} − 1). Due to the action of the symmetric group S_{n}, the eutaxy relation has the form

Id = λ ∑_{1≤i<j≤n} p_{v_{i}+v_{j}} + μ ∑_{1≤k≤n} p_{v_{0}+v_{k}} .

Calculating the trace and the image of v_{0}, we obtain

λ= \frac{n^{2} − 2(q + 2)n + q^{2} + 8q}{(n − 3)(n + q)(n − 3q)} and μ= \frac{n^{2} − 2(q^{2} + q + 1)n + 5q^{2} + 8q}{(n − 3)(n + q)(n − 3q)} ,

with which easily characterize the eutaxy property. Using Theorem 3.6.2 of [M] when q = 0 and a direct calculation when q = \sqrt{n + 1} − 1, we have proved:

Theorem. Let Λ be of the form ⟨Λ′, \frac{e_{1}+···+e_{n}}{d} ⟩, with S(Λ) ⊃ S_{0}. Then Λ is extreme if and only if q ≤ \frac{\sqrt{2n+1}−1}{2} or q = \sqrt{n + 1} − 1. □

Note that λ and μ are equal if and only if q = 1. This proves that for 1 < q < \sqrt{n + 1} − 1, there are exactly two orbits of minimal vectors in Λ (those of v_{0} + v_{1} and of v_{1} + v_{2}), then that Aut(Λ) ≃ {± Id} × S_{n}. If q = 1, Λ = Cox_{n}, there is a unique orbit, and Aut(Λ) ≃ \{± Id\} × S_{n+1}. (If q = 0 or q = \sqrt{n + 1} − 1, there are three orbits.)

Appendix 3: The Watson Identity

A perfection relation on a finite subset S of E is a dependence relation

∑_{x∈S} a_{x} p_{x} = 0

in End^{s}(E) between the orthogonal projections p_{x} onto the vectors x ∈ S.

Given a lattice Λ written in the form Λ = ⟨Λ′, e⟩ for a lattice Λ′ having a basis (e_{1}, …, e_{n}) and a vector

e = \frac{a_{1}e_{1} + ⋯ + a_{n}e_{n}}{d} , a_{i}, d ∈ ℤ , d ≥ 2 ,

Watson’s identity is

∑_{i=1}^{n} a_{i} (N(e − e_{i}) − N(e_{i})) = (A − 2d)N(e)  (∗) (∗) (∗)

(or a slightly modified form involving the absolute values of the a_{i}). In the language of Section 1, there are n+1 “Watson’s identities”, one for each choice of a c_{i}, exactly one of which is equivalent to the identity displayed above.

With this identity is associated a unique perfection relation between the 2n + 1 vectors e_{i}, e′_{i} = e − e_{i}, and e. Setting m_{i} = N(e_{i}) and m′_{i} = N(e′_{i}), and denoting by p_{i}, p′_{i} and p the projections onto e_{i}, e′_{i} and e respectively, it reads

∑_{i=1}^{n} a_{i} (m_{i} p_{i} − m′_{i} p′_{i}) = −(A − 2d) N(e) p .  (∗ ∗) (∗ ∗) (∗ ∗)

The proof, whose details will be left to the reader, consists in writing a perfection relation of the type above with indeterminate coefficients, then evaluating both sides on two vectors e_{i}, e_{j} and on a basis of the space orthogonal to the span of e_{i}, e_{j}.

As a consequence, we obtain:

Proposition. The perfection rank of the system {p_{i}, p′_{i}} is equal to 2n if A ≠ 2d, but only to 2n − 1 when “Watson’s condition” A = 2d holds. □

In the applications, it is generally assumed that e_{1}, …, e_{n} are minimal. Then (∗) shows that the vectors e′_{i} = e − e_{i} also are minimal, and (∗ ∗) then shows that the perfection rank of Λ is strictly smaller that its kissing number s. We could show that in dimensions n ≤ 6, all perfection relations stam from a relation of type (∗ ∗), with a denominator d equal to 2 or 3. This is no longer true if n ≥ 8, where for instance perfection relations related to an identity of Zahareva (see[M1], proof of Proposition 9.1) occur. It would be interesting to deal with the case of dimension 7, where denominators d = 2, 3, 4 must be considered.

Acknowledgements. This research was carried out in the “Laboratoire A2X”, UMR 5465, FR 2254, C.N.R.S & Université Bordeaux 1. The authors would like to thank the authors of the PARI package, and in particular C. Batut who allowed us to use some specific programs of his.

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A.-M. Bergé, J. Martinet 11H55: NUMBER THEORY ; Inst. Math., Université Bordeaux 1 Euclidean Lattices, Perfect Lattices 351, cours de la Libération 33405 Talence cedex, France e-mail: berge@math.u-bordeaux.fr

martinet@math.u-bordeaux.fr