THE PERFECTION DEFAULT MODULE

OF A EUCLIDEAN LATTICE

(AFTER A TALK IN LUMINY, SEPTEMBER 2000)

by J. Martinet^{1}

Abstract. Following an idea of Roland Bacher, we discuss a notion of perfection over ℤ for a Euclidean lattice. Résumé. Suivant une idée de Roland Bacher, nous discutons une notion de perfection sur ℤ pour les réseaux euclidiens. Titre français : Le module de défaut de perfection d’un réseau euclidien.

§ 1. Introduction. Some time ago, Bacher ([Bc]) suggested that the property of perfection for a Euclidean lattice L, which can be defined using symmetric tensors of degree 2 (one requires that the tensors x ⊗ x for x minimal in L span the Symmetric square of the Euclidean space), could be considered over ℤ, not only as usual over ℚ or ℝ. We can define in this way new invariants, which contain more information on the lattice that the mere perfection rank, as defined in [M], chapter III, section 2. To this end, we construct a finitely generated ℤ-module (the perfection default module, see below, %) inside a convenient symmetric square, whose rank is the perfection co-rank (i.e., dim End^{s}(E)-perfection rank of Λ). The study of its torsion submodule is the object of this note.

The notion of a perfect lattice goes back to Korkine & Zolotareff and Voronoï (1873, 1908). Perfect lattices are precisely those which possess a finite perfection default module. They are the central object of our study.

Several years ago, for algorithmic reasons, Batut and myself considered perfection modulo a prime p: if the linear forms which define the perfection of a lattice have maximal rank modulo some prime p, then the lattice is certainly perfect. However, we were not able to assert conversely that a lattice which is not perfect modulo some conveniently chosen prime p is indeed not perfect in the usual sense, since there may be a phenomenon of “bad reduction”. Bacher’s notion of a perfection default module might be an important tool to solve the question above: “good primes” are those which do not divide the order of its torsion submodule.

§ 2. Symmetric squares. Let R be a commutative ring and let M be an R-module. Usually (e.g. in [Bou1], III, pp. 67 and 76), the symmetric algebra (as well as the exterior algebra) of M is defined as a quotient of the tensor algebra ⨂ M. In what follows, we need modify the definition of the symmetric algebra. We only need to consider the degree 2 part of the symmetric algebra.

Key words and phrases. Perfect lattices, symmetric squares. October, 2000.

1laboratoire A2X, UMR 5465 associée au C.N.R.S. – Université Bordeaux 1

2.1. Definition. The symmetric square of M denoted by Sym^{2}(M) (or Sym_{R}^{2}(M)

if we need the ring R in the notation) is the submodule of ⨂^{2}M generated by the symmetric tensors x ⊗ x, x ∈ M.

[Note that the inclusion Sym^{2}(M) ↪ ⨂^{2}M induces an isomorphism of Sym^{2}(M) onto the

classical ⋁^{2}M anytime the map x ↦ 2x is bijective, compare [Bou1], III, p. 68.]

In the applications to lattices, M will be either a vector space over ℚ or ℝ, in which case the above map is bijective, or a finitely generated torsion free ℤ-module, in which case it is not, except for M = {0}.

Let T be a generating subset of M, endowed with an arbitrary total ordering. Then, the identity

(2.2)  (∑_{x∈T} λ_{x}x) ⊗ (∑_{y∈T} λ_{y}y) = ∑_{x∈T} λ_{x}λ_{x} x ⊗ x + ∑_{x<y} λ_{x}λ_{y}(x ⊗ y + y ⊗ x)

shows that Sym^{2}(M) is generated by the split symmetric tensors x ⊗ x, x ∈ T on the one hand, and the tensors (x ⊗ y + y ⊗ x), x, y ∈ T on the other hand. The identities

(2.3)  (x ± y) ⊗ (x ± y) − x ⊗ x − y ⊗ y = ±(x ⊗ y + y ⊗ x)

shows that we may replace for any pair (x, y) ∈ T × T the tensor x ⊗ y + y ⊗ x by one of the tensors (x + y) ⊗ (x + y) or (x − y) ⊗ (x − y).

2.4. Definition. For S ⊂ M, let S^{(2)} = {x ⊗ x | x ∈ S}, and let Perf_{S}(M) be the submodule of Sym^{2}(S) generated by S^{(2)}. We call Perf_{S}(M) the perfection module of M ( with respect to S). We often write Perf(M) rather than Perf_{S}(M) when the reference to S is clear from the context.

Since we are interested in the submodule of Sym^{2}(M) generated by S^{(2)}, we may assume that 0 does not belong to S and that S = −S (just replace if necessary S by S ∪ −S). Denote now by S′ a system of representatives of S/{±1}. We then have S = S′ ∪ (−S′), S ∩ S′ = ∅, and of course S^{(2)} = S′^{(2)}.

Let N be a submodule of M and let P = M/N. The embedding N ↪ M induces maps N ⨂ N → N ⨂ M → M ⨂ M, which are injective whenever Tor_{1}^{R}(P, N) = Tor_{1}^{R}(P, M) = 0, whose product N ⨂ N → M ⨂ M is then also injective.

Take now N = ⟨S⟩. The quotient Sym^{2}(M)/ Perf(M) is an extension of the two modules Sym^{2}(N)/ Perf(M) = Sym^{2}(N)/ Perf(N) and Sym^{2}(M)/ Sym^{2}(N). For this reason, we shall most of the time restrict ourselves to pairs (M, S) where S generates M. This hypothesis does not imply that S^{(2)} generates Sym^{2}(M) ! In applications to lattices, such a circumstance is indeed rather scarce, see below, section %. Note for further use the following result, which easily follows from formulae 2.2 and 2.3:

2.5. Proposition. If S contains a generating system {e_{1}, …, e_{r}} of M together with all differences e_{j} − e_{k}, j ≠ k, then Perf(M) = Sym^{(2)}(M). □

2.6. Definition. The quotient

Dft_{S}(M) = Sym^{2}(M)/ Perf_{S}(M)

is called the perfection default module of (M, S). We shall forget the index S when the reference to S is clear from the context.

§ 3. Dedekind domains. In this section, we suppose that R is a Dedekind domain. We denote by K is fraction field, by M a finitely generated torsion free R-module, and by V be the K-vector space M ⨂_{R} K. We still consider a symmetric subset S of M which does not contain 0.

Since R is Dedekind, the torsion product Tor_{1}^{R}(M_{1}, M_{2}) is zero anytime at least one of the modules M_{1}, M_{2} is torsion free (reduce by localization to the case when R is a principal ideal domain, and apply [Bou1], X, p. 29, prop. 3.) Thus, for any submodule N of M, Sym^{2}(N) can be identified with a submodule of Sym^{2}(M). In particular, the canonical map x ↦ x ⊗ 1 : M ≃ M ⨂ R → V is injective. We use it to embed M inside V. By making use of the composition of the maps M ⨂ M → M ⨂ V → V ⨂ V, we obtain a canonical embedding

Sym^{2}(M) ↪ Sym^{2}(V) .

3.1. Definition. The rank r of S^{(2)} in Sym^{2}(V) is called the perfection rank of S, and the difference \frac{n(n+1)}{2} − r = dim Sym^{2}(V) − r the perfection co-rank of S. We say that M is perfect (with respect to S) if r is maximum, i.e. if S^{(2)} spans Sym^{2}(V).

Clearly,

M is perfect ⟺ V is perfect ⟺ Dft_{S}(M) is torsion.

3.2. Proposition. If (M, S) is perfect, then S spans V.

Proof. Let W be the span of S in V. There exists a direct sum decomposition V = W ⨁ W′, which shows that Sym^{2}(W′) is a direct summand of Sym^{2}(V). Because of the inclusion S^{(2)} ⊂ Sym^{2}(W), M cannot be perfect unless Sym^{2}(W′) = 0, which implies W′ = 0, hence W = V. □

The rank of the perfection default module is the perfection co-rank of M. Its torsion submodule is a new invariant, that we shall investigate later, at least for perfect modules. In the applications, K will be a number field. This implies that the torsion submodule of the perfection default module is finite, since all residue fields of R are then finite.

3.3. Proposition. If Dft_{S}(M) = 0, then S generates M.

Proof. If M = {0}, there is nothing to prove. Otherwise, let N be the submodule of M generated by S, and let P = M/N. Since M is perfect, we have dim N = dim M = n ≠ 0. The commutative diagram

0 ⟶ N ⨂ N ⟶ N ⨂ M ⟶ N ⨂ P ⟶ 0
↓  ↓  ↓
0 ⟶ M ⨂ M ⟶ M ⨂ M ⟶ 0 ⟶ 0

yields via the snake lemma an injective map N ⨂ P ↪ M ⨂ M/N ⨂ N. Now, we have the inclusions ⟨S ⨂ S⟩ ⊂ N ⨂ N ⊂ M ⨂ M. Since Dft(M) = 0, we have ⟨S ⨂ S⟩ = M ⨂ M, hence N ⨂ N = M ⨂ M, whence N ⨂ P = 0. Since N is projective and non-zero, this implies P = 0, i.e. N = M. □

Recall that given a finitely generated torsion free R-module P of rank p and a submodule Q of P of rank q (q ≤ p), there exist a K-basis B = (e_{1}, …, e_{p}) of W = K ⨂ P (we identify P to a submodule of W, so that W = KP), fractional ideals 𝔟_{1}, …, 𝔟_{p} of R and integral ideals 𝔞_{1}, …, 𝔞_{q} of R such that

(1) P = 𝔟_{1}e_{1} ⨁ … ⨁ 𝔟_{p}e_{p};

(2) Q = 𝔞_{1}𝔟_{1}e_{1} ⨁ … ⨁ 𝔞_{q}𝔟_{q}e_{q};

(3) 𝔞_{1} | 𝔞_{2}, …, 𝔞_{q} | 𝔞_{q−1} . Moreover, the integers p, q and the sequence 𝔞_{1}, …, 𝔞_{q} are invariants of the isomorphism class of the pair (P, Q).

3.4. Definition. The ideals 𝔞_{1}, …, 𝔞_{q} are called the elementary divisors of the pair (P, Q ⊂ P). The ordered set (𝔞_{1}, …, 𝔞_{q}), denoted Smith(P, Q), is called the Smith invariant of (P, Q).

The torsion submodule of P/Q is P/(P ∩ KQ), and we have a direct sum decomposition P/Q = P/(P ∩ KQ) ⨁ P′ where P′ is a projective R-module of rank p − q. The first term 𝔞_{1} is the annihilator of P/Q. The last one 𝔞_{q} is a “measure of imprimitivity” for the pair (P, Q); it becomes trivial when we replace Q by Q′ = 𝔞_{q}^{−1}Q; note that when 𝔞_{q} is principal, Q′ is isomorphic to Q. The product of the ideals 𝔞_{i} is an “R-index”: over ℤ, it is simply the index [P ∩ KQ : Q]. If K is more generally a number field and if R = ℤ_{K} (the ring of integers of K), we have [P ∩ KQ : Q] = N_{K/ℚ}(𝔞_{1}) … N_{K/ℚ}(𝔞_{q}).

We now apply the definition above to P = Sym^{2}(M) and Q = Perf_{S}(M) (i.e., Q = ⟨Sym_{2}(S)⟩ ⊂ Sym_{2}(M)).

3.5. Definition. We denote by Smith_{S}^{(2)}(M), or simply by Smith^{(2)}(M), the Smith invariant of (Sym^{2}(M), Perf(M)).

If 𝔞_{1}, …, 𝔞_{q} are the elementary divisors of the pair (Sym^{2}(M), Perf(M)), then Dft(M) is isometric to the direct sum R/𝔞_{1} ⨁ … ⨁ R/𝔞_{q} ⨁ R^{q−p}. When ℝ = ℤ and Dft(M) is a finite group, then 𝔞_{i} = (a_{i}) for a well defined integer a_{i} ≥ 1. We shall use the notation Smith^{(2)}(M) = (b_{1}^{β_{1}}, …, b_{r}^{β_{r}}) to say that a_{1} = ⋯ = a_{β_{1}} = b_{1}, a_{β_{1}+1} = ⋯ = a_{β_{1}+β_{2}} = b_{2}, …, ⋯ = a_{p} = b_{β_{r}}.

3.6. Remark. When W is endowed with a symmetric bilinear form taking values in R on M (we then say that M is integral), M is contained in its dual M^{∗} = {x ∈ V | ∀ y ∈ M, x.y ∈ R}, so that we can define the Smith invariant Smith(M) of M, namely the Smith invariant of the pair (M^{∗}, M). Experimental data indicate that Smith^{(2)}(M) has essentially nothing to do with Smith(M).

§ 4. Symmetric endomorphisms and matrices. Let W be a vector space of finite dimension n over some field C. The map

(e, φ) ↦ (x ↦ φ(x) e) : W × W^{∗} → End(W)

(W^{∗} is the dual L(W, C) of W) induces a canonical isomorphism W ⨂ W^{∗} ≃ End(W). (This homomorphism is easily seen to be surjective, and both sides have the same dimension). Suppose now that W is endowed with a non-degenerate symmetric bilinear form (x, y) ↦ x · y. We can then identify W and W^{∗} (by f ↦ φ = (x ↦ f · x)), obtaining in this way an isomorphism W ⨂ W ≃ End(W), which is characterized by the condition

∀x ∈ W , (e ⊗ f) (x) = (f · x) e .

Recall that u ∈ End(W) is symmetric if it satisfies the identity u(x) · y = x·u(y) ; We denote by End^{s}(W) the vector space of symmetric endomorphisms of W. Among symmetric endomorphisms, orthogonal projections onto non-isotropic lines D play a crucial rôle. Given a non-zero e ∈ D, this projection is defined by the formula p_{D}(x) ( = p_{e}(x)) = \frac{x · e}{e · e} e. The endomorphism x ↦ (f · x) e is symmetric if and only if f and e are collinear, and if e · e ≠ 0, the image of e ⊗ e in End^{s}(W) is (e · e) p_{e}.

Let B = (e_{1}, …, e_{n}) be a basis of L. For x ∈ W, let X be the column vector of the components of x on B, and let e ∈ W. For the endomorphism u_{e} : x ↦ (e · x) e, we have the matrix representation Mat(u_{e}, B^{∗}, B) = X^{t}X. We can thus pass from symmetric squares to matrix rings by the correspondence

x ⊗ x ∈ Sym^{2}(W) ⟷ X^{t}X ∈ Sym_{n}(C) .

We can now recover the usual definition of perfection for Euclidean lattices. The traditional definition of perfection (in terms of quadratic forms) is that q is perfect if the rank-1 n×n matrices X^{t}X, X ∈ S(q) span the space Sym_{n}(ℝ) of real symmetric matrices; we pass from lattices to quadratic forms by taking for B a basis of the lattice and considering the quadratic form q(x) = ^{t}XAX where A = (e_{i} · e_{j}) is the Gram matrix Gram(B) of B.

Applying the definitions of the previous sections with R = ℤ, K = ℚ, and C = ℝ, choosing for W a Euclidean space E, for M a lattice Λ ⊂ E, and for S the set S(Λ) of minimal vectors of Λ, we see immediately that Λ is perfect in the sense of definition 3.1 if and only if the projections p_{e}, e ∈ S span End^{s}(E), and this is precisely one of the possible definitions for the perfection of Λ as a Euclidean lattice, see [M], chapter III, definitions 2.2 and 2.7. We have thus proved:

4.1. Proposition. A Euclidean lattice Λ is perfect if and only if the pair (Λ, S(Λ)) is perfect in the sense of definition 2.6. □

Let us come back to the C-vector space W. We define the norm of x ∈ W by the formula N(x) = x · x. (This extends the definition currently used in the theory of lattices, where this norm is the square of the Euclidean norm x ↦ ∥x∥.) The bilinear map

(x, y) ↦ x · y : W × W → C

induces a linear map from W ⨂ W onto C which is well defined by the condition that it takes the value x · y on the split tensors x ⊗ y. By restriction, we obtain a linear map N : Sym^{2}(W) → C which is well defined by the condition

(4.2)  ∀x ∈ W, N(x ⊗ x) = N(x) .

We have the formula

N(x ⊗ y + y ⊗ x) = N((x + y) ⊗ (x + y)) − N(x ⊗ x) − N(y ⊗ y) = 2 x · y .

4.3. Definition. The map N from Sym^{2}(W) onto C defined above is called the norm map.

Let S = S′ ∪ −S′ (S′ ∩ −S′ = ∅) be a symmetric subset of W and let t be a symmetric tensor belonging to the span of S^{(2)} in Sym^{2}(W). Choose a representation

t = ∑_{x∈S′} λ_{x}(t) x ⊗ x

of t on S^{(2)} and let

λ(t) = ∑_{x∈S′} λ_{x}(t) .

4.4. Proposition. Suppose that all the vectors of S have the same non-zero norm, that we denote by N(S). Then, we have

λ(t) = \frac{N(t)}{N(S)} .

In particular, λ(t) does not depend on the representation of t on S^{(2)}.

Proof. By the linearity of the norm map, we have N(t) = ∑ λ_{x}(t) N(x ⊗ x). □

4.5. Corollary. If t = y ⊗ y, then λ(t) = \frac{N(y)}{N(S)}. □

§ 5. Euclidean lattices. We again apply the previous sections taking R = ℤ, K = ℚ, C = ℝ, M, now denoted Λ, being a lattice in W, now denoted E, endowed with a Euclidean structure. We denote by S the sets of its minimal vectors, and by S′ a half system of minimal vectors. The norm of x ∈ E is N(x) = x · x, and the norm (or minimum) of Λ is the common norm of its minimal vectors.

For any similarity u of E, we have S(u(Λ)) = u(S(Λ)). Hence, the perfection default module of Λ only depends on the similarity class of Λ. Now, by an old result of Korkine and Zolotareff (see [M], chapter III, proposition 2.11), a perfect lattice is proportional to an integral lattice, so that its similarity class contains an integral primitive lattice, well defined up to isometry.

5.1. Theorem (A.-M. Bergé). Let Λ be an integral and primitive perfect lattice, of norm m. Then, the annihilator of the torsion submodule of the perfection default module of Λ is divisible by \frac{m}{2} if m is even, and by m if m is odd.

Proof. Let α ∈ ℤ be the annihilator of the perfection default module of Λ (which is a finite group because Λ is perfect). It is the smallest integer such that, for all y ∈ Λ, there is a relation

α y ⊗ y = ∑_{x∈S′} λ_{x}(y) x ⊗ x

with λ_{x}(y) ∈ ℤ. By Corollary 3.5, we have

α N(y) = ( ∑_{x∈S′} λ_{x}(y)) m .

This proves that m divides the product α × g.c.d.(N(y), y ∈ Λ), whence the result since the g.c.d. of the norms on Λ is 2 when Λ is even and 1 when Λ is odd. □

We shall now use this theorem to characterize lattices whose perfection default module is trivial. Recall that a root in a lattice Λ is a non-zero primitive vector e of Λ such that Λ is invariant under the reflection σ_{e} with respect to the hyperplane orthogonal to e. A lattice generated by its roots is called a root lattice. It is easy to see that the roots of a lattice constitute a root system in the space they span, and the classification of root systems (see [Bou2], VI) shows that a root lattice is isometric to an orthogonal sum of irreducible root lattices, each of which is similar to one of the integral primitive lattices ℤ, 𝔸_{n}, n ≥ 2, 𝔻_{n}, n ≥ 4, 𝔼_{6}, 𝔼_{7}, or 𝔼_{8} (see [M], chapter IV,sections 1–5 for the definitions). Finally, we see (“Witt’s theorem”) that integral lattices generated by vectors of norm 1 or 2 are root lattices and are isometric to direct sums of copies of ℤ, 𝔸_{n}, n ≥ 1, 𝔻_{n}, n ≥ 4, or 𝔼_{n}, n = 6, 7, 8, which are pairwise non-isometric.

5.2. Theorem. A lattice Λ has a trivial perfection default module if and only if it is similar to one of the irreducible root lattices ℤ, 𝔸_{n}, n ≥ 2, 𝔻_{n}, n ≥ 4, 𝔼_{6}, 𝔼_{7}, or 𝔼_{8}.

Proof. If Dft(Λ) = {0}, Λ is perfect, hence irreducible and proportional to an integral primitive lattice, which is unique up to isometry. Let m = N(Λ). By Theorem 5.1, m divides 2. By proposition 3.3, S generates Λ. By Witt’s theorem, Λ is one of the lattices which are listed in the theorem we want to prove,

Conversely, we must show that irreducible integral primitive root lattices have trivial perfection default modules, a result which was known to Bacher. This is clear if m = 1, where Λ = ℤ. We may thus restrict ourselves to lattices with m = 2 and n ≥ 2. For such a lattice, we make use of bases of type “Korkine-Zolotareff” in the sense of [M], chapter IV. These are bases whose Gram matrix A = (a_{i,j}) fulfils the conditions a_{i,i} = 2 and a_{i,j} = 1 for i ≠ j except for {i, j} = {1, 2} in the case of 𝔻_{n} and {i, j} = {1, 2} or {1, 3} if the case of 𝔼_{n}, where a_{i,j} = 0. Clearly, S^{(2)} contains all tensors e_{i} ⊗ e_{i}. We shall show that it also contains all tensors e_{i} ⊗ e_{j} + e_{j} ⊗ e_{i} for 1 ≤ i < j ≤ n,; By formula 2.3, we may replace any such tensor by (e_{i} − e_{j}) ⊗ (e_{i} − e_{j}).

The case of 𝔸_{n} is now a consequence of proposition 2.5.

For 𝔻_{n}, we must show that e_{1} ⊗ e_{2} + e_{2} ⊗ e_{1} is a linear combination of vectors belonging to S ⨂ S. This is achieved by making use of the minimal vector e_{1} + e_{2} − e_{3}.

A similar argument applies to 𝔼_{6}, 𝔼_{7}, 𝔼_{8}, using the minimal vectors e_{1} + e_{2} − e_{4} and e_{1} + e_{3} − e_{4}. □

§ 6. Numerical results. We account here for some calculations of Smith^{(2)} which have been done using the PARI package together with some more specific programs written by Batut.

We have first considered the 48 perfect lattices of dimension n ≤ 7 and the 10916 known perfect lattices (at the date this paper is written, [B-M]) with n = 8. All these lattices are even, except the root lattice ℤ and the lattice P_{7}^{2} ≃ 𝔼_{7}^{∗}, of norm 3. For the 10962 even known perfect lattices of dimension up to 8, the following facts have been observed: 6.1. Facts.

(1) The components of Smith^{(2)} of rank r > n are trivial;

(2) The first component of Smith^{(2)} (i.e., the annihilator of the perfection default module) is equal to \frac{m}{2} or to m.

(3) For m = 4, Smith^{(2)} reduces to its first component, which is always equal to \frac{m}{2} = 2.

Note that all these lattices possess a basis of minimal vectors. This is no more true in dimensions n ≤ 10, and the remark we have made in section 2 about lattices which are not generated by their minimal vectors shows that the facts 2 and 3 above cannot be true when S does not generate Λ.

For instance, the lattices 𝔻_{n}^{+}, when scaled to a primitive integral lattice, have norm 2 if n ≡ 0 mod 4 and 4 if n ≡ 2 mod 4. For all (even) n, we have Smith^{(2)}(𝔻_{n}^{+}) = 4 · 2^{n−1}.

Consider now the Coxeter lattices 𝔸_{n}^{r}, r | n + 1, the sublattices of 𝔸_{n}^{∗} which contain 𝔸_{n} to the index r. Results similar to that we obtained for 𝔻_{n}^{+} hold for 2 ≤ r ≤ \frac{n+1}{3} (exactly the same one for r = 2). However, the “exotic” perfect lattices 𝔸_{n}^{(n+1)/2} produce original examples. When rescaled to an integral primitive lattice, 𝔸_{n}^{(n+1)/2} is even with minimum n − 1 if n ≡ 1 mod 4 and odd with minimum \frac{n−1}{2} if n ≡ 3 mod 4, in all cases generated by their minimal vectors. However, for all n ≥ 5 odd, we likely have

Smith^{(2)}(𝔸_{n}^{(n+1)/2}) = (\frac{(n − 1) (n − 3)}{4}, (\frac{n − 3}{2})^{n−1}) .

The ratio \frac{a_{1}}{m}, equal to \frac{n − 3}{4} or \frac{n − 3}{2}, tends to infinity with n. This shows that the second experimental fact verified for known perfect lattices up to dimension 8 is not general. However, no bad prime p > n occur.

I have no counter-examples to facts 1 and 3. However, since the value of Smith^{(2)} depends on subtle linear relations over ℤ between minimal vectors, it does not seem reasonable to conjecture that fact 3 holds for all norm 4 lattices generated by their minimal vectors.

§ 7. Relative perfection over Z. We still denote by Λ a lattice in the ndimensional Euclidean space E. Let F be a hyperplane of E such that M = Λ ∩ F is a lattice in F. It was proved by Barnes (see [M], chapter XII, theorem 3.5) that if M is perfect, then Λ is perfect whenever there exists a set of n independent minimal vectors in L ∖ M. The following statement is a kind of enlargement of Barnes’s theorem:

7.1. Theorem. Suppose that there exists e ∈ Λ ∖ M and generators e_{1}, …, e_{r} of M such that

(1) e, e + e_{1}, …, e + e_{r} ∈ S(Λ).

(2) e, e_{1}, …, e_{r} generate Λ.

Then, the canonical map Dft(M) → Dft(Λ) is surjective.

Proof. Set e_{0} = 0 and e′_{i} = e + e_{i} for 0 ≤ i ≤ n. Then, {e′_{0}, e′_{1}, …, e′_{n}} is a generating set for Λ. Hence, Sym^{(2)}(Λ) is generated by the symmetric tensors e′_{i} ⊗ e′_{i}, 0 ≤ i ≤ n, which belong to Perf_{S(Λ)}(Λ), and e′_{i} ⊗ e′_{j} + e′_{j} ⊗ e′_{i}, 0 ≤ i < j ≤ n, for which we write

e′_{i} ⊗ e′_{j} + e′_{j} ⊗ e′_{i} = e′_{i} ⊗ e′_{i} + e′_{j} ⊗ e′_{j} − (e′_{j} − e′_{i}) ⊗ (e′_{j} − e′_{i})
= e′_{i} ⊗ e′_{i} + e′_{j} ⊗ e′_{j} − (e_{j} − e_{i}) ⊗ (e_{j} − e_{i})
∈ Perf_{S(Λ)}(Λ) + Sym^{(2)}(M) . □

The theorem above could have been used to prove that irreducible root lattices have trivial perfection default modules, using the inclusions 𝔸_{n−1} ⊂ 𝔸_{n} (n ≥ 2), 𝔻_{n−1} ⊂ 𝔻_{n} (n ≥ 4), and 𝔼_{n−1} ⊂ 𝔼_{n} (n ≥ 6), together with the standard isometries 𝔻_{3} ≃ 𝔸_{3} and 𝔼_{5} ≃ 𝔻_{5}. Consider now the ascending sequence of laminated lattices

Λ_{0} = {0} ⊂ Λ_{1} ⊂ ⋯ ⊂ Λ_{n} ⊂ …

These are norm 4 lattices, integral for n ≤ 24, uniquely defined up to isometry in the ranges 0 ≤ n ≤ 10 and 14 ≤ n ≤ 24, which are scaled copies of root lattices for n ≤ 8 ([C-S], chapter 6). Hence, the corresponding perfection default modules are trivial up to n = 8. For n ≥ 9, they are not, because primitive integral scaled copies have a norm N > 2 (N = 4 if 9 ≤ n ≤ 24). We have actually Dft(Λ_{9}) ≃ ℤ/2ℤ. (Proof: note that Λ_{9} is a scaled copy of 𝔼_{8} + 𝔻_{9}, see [M], chapter 5, section 5.) I have verified that Theorem 7.1 applies up to n = 16 (Λ_{16} is the Barnes-Wall lattice BW_{16}). I have also verified in the same way that Dft(K_{n}) is also cyclic of order 2 for n = 9, 10, 11, 12.

Theorem 7.1 also allows to prove that Dft(L) ≃ ℤ/2ℤ for lattices L belonging to some classical infinite series. This is for instance easily verified for the Barnes lattices L_{n}^{r}, n ≥ 5, r ≥ 2, on the definition given in [M], chapter V, section 4.

Acknowledgments. I thank Anne-Marie Bergé, with whom I had numerous helpful discussions. I also thank the authors of the PARI system, and in particular Christian Batut, whose various programs on lattices have been very useful in many instances.

References

[Bc] R. Bacher, Private communication, 1998. [Bt-M] C. Batut, J. Martinet, Martinet’s WEB page, http://www.math.u-bordeaux.fr/˜ martinet.

[Bou1] N. Bourbaki, Algèbre, ch. I–X, Hermann, Masson, Paris, 1970, 1980.

[Bou2] N. Bourbaki, Groupes et algèbres de Lie, chapitres IV, V, VI, Masson, Paris, 1981. [C-S] J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag,

Grundlehren nu. 290, Heidelberg, 1988.

[M] J. Martinet, Les réseaux parfaits des espaces euclidiens, Masson, Paris, 1996.

Jacques Martinet A2X, Institut de Mathématiques Université Bordeaux 1 351, cours de la Libération F–33405 TALENCE cedex E-mail : martinet@math.u-bordeaux.fr