FAMILIES OF EQUIANGULAR LINES AND LATTICES

JACQUES MARTINET

Abstract. We construct Euclidean lattices whose sets of minimal vectors support some large equiangular families of lines, using notably reduction modulo 2 of lattices. We also consider some related problems, and answer a question raised by Greaves ([G], Subsection 1.3.2).

Introduction

In this note we consider constructions of sets of equiangular lines in a Euclidean space E derived from Euclidean lattices. We postpone to Section 1 the necessary prerequisites on Euclidean lattices together with the statement of various general results, stating here only the following theorem:

Theorem 0.1. Let Λ be an even (integral) lattice in a Euclidean space E, of minimum m, and let x_{0} ∈ Λ of norm (the square of the length) 2m−2. Then the set vectors of norm 2m + 2 in Λ orthogonal to x_{0} and congruent to x_{0} modulo 2 supports an equiangular family of lines of common angle arccos \frac{1}{m+1}.

In the rest of the introduction we solely recall some basic facts and questions concerning families of equiangular lines. Since the publication in 1973 of Lehmens-Seidel seminal paper [L-S], there has been a large literature on the subject, among which I would like to quote the collective work [JTYZZ] (Ann. Math., 2021). We refer to Greaves’ survey [G] for the results mentioned below without a reference. We shall in particular note that for a set of a given rank n, the numbers of lines t = n, n + 1, 2n and \frac{n(n+1)}{2} play somewhat special rôles.

Consider a set X of t lines of rank n with common angle θ ∈ [0, \frac{π}{2}], with which we associate the parameter α = arccos θ, and assume that t > n. We then have θ ≠ \frac{π}{2}, thus 0 < α < 1. Equip each line with a norm-1 vector, and consider their Gram matrix G(X), with entries 1 on its diagonal and ±α off its diagonal. The matrix

2000 Mathematics Subject Classification. 11H55, 11H71.

Key words and phrases. Euclidean lattices, equiangular families, reduction modulo 2.

S(X) := \frac{1}{α}(I_{n} − G(X))

is a Seidel matrix for X. It depends on the ordering and orientations of the lines in X, but its spectrum solely depends on X, and in particular its smallest eigenvalue λ, equal to −\frac{1}{α} < −1. It is immediate that if all ±α are +α (resp. −α), we have λ = −1, which must be excluded (resp, λ = −(t − 1), t = n + 1 and α = \frac{1}{n}): in other words, for an acute (resp. obtuse) equiangular family of vectors, we have t ≤ n (resp. t ≤ n + 1, with α = \frac{1}{n} if equality holds). Also we see that t ≥ n + 1 implies that the angle may take only finitely many values.

The absolute bound (Gerzen’s theorem) is the inequality t ≤ \frac{n(n+1)}{2}, which holds because the orthogonal projections to the line of X are independent symmetric automorphisms.

The importance of the value 2n (or 2n + 1) comes from Neuman’s theorem: if t > 2n, \frac{1}{α} is an odd integer; compare Theorem 0.1.

In general one considers the maximum number of lines in an equiangular set without taking into account their rank: in terms of the definition below, one considers values or estimations for t_{n}, not for t′_{n}.

Definition 0.2. Let t_{n} be the maximal cardinality of an equiangular set contained in an n-dimensional space (resp. spanning an ndimensional space). [Thus we have t′_{n} ≤ t_{n} = max_{k≤n} t_{k}.]

The exact values of t_{n} are known up to n = 17. As we shall see, we have t′_{n} = t_{n} for n ≤ 7 and 14 ≤ n ≤ 17, but in the range 7 < n < 14, we have the strict inequality t′_{n} < t_{n}. It is also known (cf. Neuman’s theorem above) that t_{n} is strictly larger than 2n except if n = 2, 3 4, 5 or 14. This suggests the following double question:

Question 0.3. For which dimensions n is it true that t′_{n} = t_{n}? that t′_{n} > 2n?

A neighbour question closer to the subject of this note is:

Question 0.4. For which dimensions is the bound t′_{n} attained on the set of minimal vectors of a Euclidean lattice ?

I have put emphasis on t′_{n} rather than on t_{n} because I believe that t′_{n} presents irregularities as a function of n which throw some light on individual properties of dimensions. In this respect it is worth considering the absolute bound \frac{n(n+1)}{2}. This is known to be attained for n = 2, 3, 7, 23, but on no other dimensions. These dimensions are somewhat special, related to the existence of the regular hexagon and icosahedron for n = 2, 3, and to the lattices 𝔼_{8} and Leech’s Λ_{24} of dimension n + 1 for n = 7, 23.

[This can be related to the double transitivity of the groups S_{3}, A_{5}, O_{7}(2) ∼ S_{6}(2) and Co_{3} in degrees 3, 6, 28 and 276, respectively. For n = 3, one identifies the action of 𝔸_{5} on the diagonals of a regular icosahedron as its action on its 5-Sylow subgroups; lattices for n = 2, 7, 23 will be described later.]

Recent works have shown the special behaviour of the “magic” dimensions 8 and 24 for kissing number and sphere packing problems. This suggest that the absolute bound could be strict in all other dimensions. The following question of maximality could be considered in place of it. Consider a configuration X of equiangular lines which is maximal in the n-dimensional space its spans. Say that X is universally maximal if X remains maximal when embedded in dimension n+1 (or any larger dimension). The four dimensions above are universally maximal: trivially for n = 2, because t′_{n} = t_{n} = t_{n+1} for n = 3, 7, 23.

Problem 0.5. For which dimensions n does there exist a universally maximal configuration of t′_{n} lines?

Here are two more results we shall need later. First the relative bound (see [G], Th. 1.9): if t ≤ \frac{1}{a^{2}} , then t ≤ \frac{n(1−α^{2})}{1−nα^{2}}.

Next ([JTYZZ]): if α = \frac{1}{2k−1} (k ≥ 2), then we have t′_{n} = ⌊\frac{k(n−1)}{k−1}⌋ for n large enough.

In Section 1 we explain how to apply the theory of lattices modulo 2 as developed in [Ma1] and [Ma2] to the construction of equiangular systems of lines. In Section 2 we consider root lattices (certain lattices of minimum 2) and their duals, and lattices of minimum 3 derived from them, and in Section 3, lattices of minimum 4 and 5.

Acknowledgments. The author thanks Bill Allombert for his help in working with PARI.GP.

1. Basic Results

Let E be a Euclidean space of dimension n, and Λ be a (full) lattice in E. We denote by m its minimum, by S the set of its minimal vectors and set s = \frac{1}{2}|S|. More generally, given r > 0, let

S_{r} = \{x ∈ Λ | x · x = r\} and let s_{r} = \frac{1}{2}|S_{r}|;

thus, S = S_{m} and s = s_{m}. We also set N(x) = x · x (the norm of x, the square of the usual ∥x∥). We say that Λ is integral if all scalar products on Λ are integral, and then even if all norms are even, odd otherwise. Note that Λ is integral if and only if it is contained in its dual Λ^{∗} = {x ∈ E | ∀ y ∈ Λ, x · y ∈ ℤ}.

Consider vectors x, y ∈ Λ, with x ≠ 0, y ≠ 0 and y ≠ ±x, and such that y ≡ x mod 2. Set

e = \frac{y−x}{2} and f = \frac{y+x}{2} .

Then e, f are nonzero, the set {±e, ±f} only depends on {±x, ±y}, and we have

x = −e + f and y = e + f .

Moreover, in an equality ±y = ±x + 2u, u is one of ±e or ±f.

Proposition 1.1. ([Ma1], § 1) With the notation above, we have (1) N(x) + N(y) ≥ 4m.

(2) If Λ is integral, N(y) ≡ N(x) mod 4.

(3) If (N(x) + N(y) = 4m, then e and f are minimal and x · y = 0.

Proof. (1) We have

N(x) + N(y) = \frac{1}{2}(N(y − x) + N(y + x)) = 2(N(e) + N(f)) ≥ 4m since e and f are nonzero.

(2) We then have N(y) − N(x) = N(e + f) − N(−e + f) = 4e · f ≡ 0 mod 4 .

(3) We then have N(e) + N(f) = 2m, hence N(e) = N(f) = m and x · y = N(f) − N(e) = 0. □

Proposition 1.1 shows that a list of representatives of classes modulo 2 of smallest possible norms must include vectors of norm N ≤ 2m, and that the corresponding classes then include a unique pair ±x if N < 2m, and at most n such pairs if N = 2m. Among known examples are the celebrated lattices 𝔼_{8} and Λ_{24}.

Proposition 1.2. Let m′ ≥ m and m′′ ≥ m′, and let x, y, y′ ∈ Λ such that N(x) = m′, N(y) = N(y′) = m′′, y ≡ y′ ≡ x mod 2, and y′ ≠ ±y, Then |y · y′| ≤ m′′ − 2m.

Proof. Write as in Proposition 1.1

x = −e + f, y = e + f and x = −e′ + f′, y′ = e′ + f′. Calculating N(y − y′) in two ways we obtain the equalities

2m′′ − 2y · y′ = 4N(e − e′) ≥ 4m

(since e′ = e implies y′ = y), hence

y · y′ = m′′ − 2N(e − e′) ≤ m′′ − 2m,

since N(e ± e′) ≥ m.

Calculating similarly N(y + y′), we obtain

2m′′ − 2y · y′ = 4N(−2x + f + f′) ≥ 4m

(since x = f + f′ implies y′ = −y), hence

y · y′ = m′′ − 2N(e − e′) ≤ m′′ − 2m.

□

Theorem 1.3. Assume that Λ is integral of even minimum. Let x_{0} ∈ Λ of norm 2m − 2. Consider the set E(x_{0}, 2m + 2) = {±y_{1}, …, ±y_{k}} of those vectors of norm 2m + 2 congruent to x_{0} modulo 2. Then the y_{i} support an equiangular family of lines of angle arccos \frac{1}{m+1} and rank

≤ n − 1. [By an abuse of language, we accept sets of less than three lines.]

Proof. We have |y_{i} ·y_{j}| ≤ 2 by Proposition 1.2. Next by the calculation done in the proof of this proposition we have y_{i} · y_{j} = m′′ − N(e − e′) with m′′ = 2m + 2, and since N(e) = N(e′) = m, we have N(e − e′) = 2m − 2e · e′, which implies

y_{i} · y_{j} = −2 m + 2 + 4 e · e′ ≡ 2 mod 4 ,

hence y_{i} · y_{j} = ±2.

The bound for the rank comes from Proposition 1.1, (4). □

Remark 1.4. (1) The restriction to even minima is essential. Indeed, if m is odd, we may apply Proposition 1.1 to the even sublattice Λ_{ev} of Λ, of minimum m_{ev} ≥ m + 1. Since m′ + m′′ is strictly smaller than 4(m + 1), E(x_{0}, 2m + 2) is then empty.

(2) We may also clearly restrict ourselves to irreducible lattices.

The proposition below, extracted from Section 5 of [Ma2], shows that the set of equiangular lines constructed in Theorem 1.3 can be realized as the set of minimal vectors of an integral (relative) lattice of minimum m + 1.

Proposition 1.5. Let x_{0} ∈ Λ of norm m′ < 2m, let C be its class modulo 2, let L_{0} = C ∪2Λ and let L = L_{0} ∩x_{0}^{⊥}. Set m′′ = 4m−m′, and assume that E(x_{0}, m′′) is not empty. Then L is a lattice with invariants

dim L = n − 1, min L = m′′, and S(L) = E(x_{0}, m′′) .

Proof. We have C = x_{0} + 2Λ and 2x_{0} ∈ 2Λ, hence L_{0} is a lattice (containing 2Λ to index 2). By Proposition 1.1, the first minimum of the norm on C is m′, attained uniquely at ±x_{0}, and since E is not empty, its second minimum is m”, attained exactly on E. Since min 2Λ = 4m > m′′, these are the first two minima of L_{0}, and since E is orthogonal to x_{0}, we have min L = m′′ and S(L) = E.

[The calculation of det(L) is carried out in [Ma2], Section 5.] □

Remark 1.6. The vectors in Λ′ are sums of minimal vectors. This shows that its even part that Λ′_{ev} is generated by sums e + e′, e, e′ ∈ S(Λ′). Easy calculations then show that on Λ′_{ev}, all scalar products are even and all norms are divisible by 4. Hence \frac{1}{\sqrt{2}}Λ′_{ev} is an even (integral) lattice.

From an algorithmic viewpoint, listing the y_{i} in Theorem 1.3 from vectors of norm 2m + 2 can need lengthy computations. The following proposition allows us to complete this list using only minimal vectors. This also makes easy the calculation of the rank of E. In the following proposition we keep the notation of 1.3, setting moreover

{S_{0} = {x ∈ S | x_{0} · x = m − 1 .}

Proposition 1.7. The map

x ↦ x_{0} − 2x : S_{0} → Λ

induces a bijection of S_{0} onto E, and we have rk E = rk S_{0} − 1.

Proof. The first assertion follows from the calculation of x_{0} · \frac{y−x}{2} and N(\frac{y−x}{2}) for y ∈ S_{2m+2}.

Let r = rk E, let y_{1}, …, y_{r} be r independent vectors in E, and let x_{i} = \frac{x_{0}−y_{i}}{2} . We have

⟨y_{1}, ..., y_{r}⟩ = ⟨y_{1}, y_{2} − y_{1}, …, y_{r} − y_{1}⟩ = ⟨y_{1}, x_{2} − x_{1}, …, x_{r} − x_{1}⟩. Now ⟨S_{0}⟩ contains the x_{0} − x_{i} (because −y_{i} = x_{0} − 2(x_{0} − x_{i})) and x_{0} − 2x_{1} (= y_{1}). Hence ⟨S_{0}⟩ = ⟨x_{0}, y_{1}, …, y_{r}⟩, and since the y_{i} but not x_{0} are orthogonal to x_{0}, we have rk S_{0} = r + 1. □

Let v be a nonzero vector in E and let H := (ℝv)^{⊥}. The orthogonal projection to H (or along v) of x ∈ E is

p(x) = x − \frac{v·x}{v·v} v.

Scalar products and norms of projections are given by the formulae

p(x) · p(y) = x · y − \frac{(v · x)(v · y)}{N(v)} and N(p(x)) = N(x) − \frac{(v · x)^{2}}{N(v)} . (∗∗∗)

Note that for x ∈ S and x ≠ ±v, we have |v · x| ≤ \frac{N(v)}{2}, since we have N(v ± x) ≥ m, hence ∓2(v · x) ≤ N(v).

Consider a lattice Λ. Then p(Λ) is a lattice if and only if v ∈ Λ, and we may assume that v is primitive (because p only depends on the line ℝv). We can then construct bases (v_{1}, …, v_{n}) for Λ with v_{1} = v, so that (p(v_{2}), …, p(v_{n}) is a basis for p(Λ).

In the setting of Theorem 1.3 projections along x_{0} preserve the norms on E, and shall be used to construct lattices whose minimal vectors support equiangular families of lines. However I cannot give a priori a general procedure to find minimal vectors in projections. Note that if |v · x|,takes values α_{1} < ⋯ < α_{t} on S(Λ) ∖ {±v}, the corresponding norms of the projections occur in the reverse order, and α_{t} = \frac{N(v)}{2} needs N(v) < 4m by (∗∗∗). We shall consider the projections of the x ∈ S(Λ) such that v · x = ±α_{t}, in particular when N(v) = 2m − 2 and α_{t} = \frac{N(v)}{2}, since we then obtain equiangular families of line by applying Theorem 1.3. It will often happen that S(p(Λ)) is the subset of p(S(Λ)) of the p(x) with v · x = \frac{N(v)}{2}. This will be checked for all lattices we shall construct in the forthcoming sections.

We state below a proposition which gives us some precisions on projections. Give v we consider a partition S^{+} ∪ S^{−} of S for which x ∈ S^{+} needs v · x ≥ 0 (the choice of x ∈ S^{+} among ±x is well-defined only when v · x ≠ 0). Using formulae (∗), we easily prove:

Proposition 1.8. Consider a lattice Λ of minimum m, a vector v ∈ Λ of norm 2m − 2 and the orthogonal projection p along v, and assume that S(p(Λ) is contained in p(S(Λ)). Let x and y ≠ ±x in S^{+}, not colinear with v. Then we have

\frac{m−3}{4} ≤ x · y ≤ \frac{3m−1}{4} .

In particular if m = 2 (resp. m = 4) we have x · y ∈ {0, 1} (resp. x · y ∈ {1, 2}. □

2. Root Lattices and their duals

Root lattices are integral lattices generated by norm-2 vectors. Theses are orthogonal sums of irreducible lattices, isometric to exactly one of 𝔸_{n} (n ≥ 1), 𝔻_{n} (n ≥ 4) or 𝔼_{n} (n = 6, 7, 8, the definition of which we recall below; see [Ma], Chapter 4 for details.

Inside the lattice ℝ^{n+1} (resp. ℝ^{n}), equipped with its canonical basis (ε_{0}, …, ε_{n}) (resp. (ε_{1}, …, ε_{n}), Let

𝔸_{n} = {x ∈ ℤ^{n+1} | ∑_{i=0}^{i=n} x_{i} = 0} and 𝔻_{n} = {x ∈ ℤ^{n} | ∑_{i=1}^{i=n} x_{i} ≡ 0 mod 0}.

In ℝ^{8} let e = \frac{ε_{1}+···+ε_{8}}{2} , set 𝔼_{8} = 𝔻_{8} ∪ (𝔻_{8} + e), and define 𝔼_{7} and 𝔼_{8} by successive sections orthogonal to ε_{7} − ε_{8} and ε_{6} − ε_{7}.

For a lattice Λ in the list above the automorphism group acts transitively on its set of minimal vectors, among which we may choose arbitrarily x_{0} when applying Theorem 1.3. Let us choose x_{0} = ε_{0} − ε_{1}, ε_{1} − ε_{2} and e when Λ = 𝔸_{n}, 𝔻_{n} and 𝔼_{n}, respectively.

For Λ = 𝔸_{n}, we have

S_{0} = {−ε_{0} + ε_{i}, ε_{1} − ε_{i}, i ≥ 2} and ± E = {ε_{0} + ε_{1} − 2ε_{i}, i ≥ 2}.

For Λ = 𝔻_{n}, we have

S_{0} = {−ε_{1} ± ε_{i}, ε_{2} ± ε_{i}, i ≥ 3} and ± E = {ε_{1} + ε_{2} ± ε_{i}, i ≥ 3}.

For Λ = 𝔼_{8}, S_{0} consists of 28 × 2 = 56 vectors, the ε_{i} + ε_{j}, 1 ≤ i < j ≤ 8 and the 28 vectors obtained by negating 6 basis vectors in e. The 28 vectors in E (up to sign) are then obtain by permutations of

\frac{3ε_{1}+3ε_{2}−ε_{3}−ε_{4}−ε_{5}−ε_{6}−ε_{7}−ε_{8}}{2} .

The results for 𝔼_{7} and 𝔼_{6} are obtained by sections of 𝔼_{8}. Alternatively we could also have used the sets of minimal vectors in projections of 𝔸_{n}, 𝔻_{n}, 𝔼_{n}.

Summarizing, we obtain:

Proposition 2.1. Let Λ be an irreducible root lattice of dimension n. Then the equiangular family of Theorem 1.3 is of rank r = n − 1 and contains t lines according to the following data:

Λ = 𝔸_{n}: t = r; Λ = 𝔻_{n}: t = 2r − 2 ;

Λ = 𝔼_{8}: t = 28 ; Λ = 𝔼_{7} : t = 16 ; Λ = 𝔼_{6}: t = 10.

[𝔻_{n+1} provides the asymptotic bound for α = \frac{1}{3}, valid for all n ≥ 15.] □

As for the duals, we may discard 𝔻_{4}^{∗} ∼ 𝔻_{4}, E_{8}^{∗} = E_{8} and 𝔻_{n}^{∗}, n ≥ 5 (because S(D_{n}^{∗}) = S(ℤ^{n})). The set S(𝔸_{n}^{∗}) supports the equiangular set of vectors with common angle − arccos \frac{1}{n}, and S(𝔼_{7}^{∗}) is is the projection of E attached to 𝔼_{8}. The lattice 𝔼_{6}^{∗} does not apply directly to equiangular systems of lines.

[The function |x · y| takes two values on non-proportional x, y ∈ S(𝔼_{6}^{∗}). We shall return to such lattices in [Ma3] in connection with the theory of strongly regular graphs (with S(𝔼_{6}^{∗}) we can recover the Schläfli graph, attached to the system of lines on a non-singular cubic surface).]

The values of t_{n} are known up to n = 17 (see [G]). For n = 2 to 7, t_{n} is equal to 3, 6, 6, 10 and 16, respectively, and we have t_{n} = 28 for 7 ≤ n ≤ 14.

Proposition 2.2. (1) We have t′_{n} = t_{n} for 2 ≤ n ≤ 7, and except for n = 3, t′_{n} is attained on the set of minimal vectors of a lattice.

(2) For n = 3, the maximum number of lines defined by a lattice is 4, attained uniquely on the configuration of S(𝔸_{3}^{∗}).

(3) For 8 ≤ n ≤ 13, t′_{n} is strictly smaller than t_{n}.

Proof. (1) This is clear for n = 2 and 3. For n = 4, 5, 6, 7, consider the projections of 𝔻_{5}, 𝔼_{6}, 𝔼_{7} and 𝔼_{8} (Proposition 2.2).

(2) This results from the classification of minimal classes for n = 3 (see [Ma], Theorem 9.2): one checks that a lattice with s ≥ 4 must have a system of minimal vectors containing e_{1}, e_{2}, e_{3}, e_{1} + e_{2} + e_{3} with equal scalar products e_{i} · e_{j}.

(3) Let n as above, Assume that there exists an n-dimensional system of s = 28 equiangular lines. Let θ be their common angle and let α = arccos θ. We then have s > 2n, so that by Neumann’s theorem ([G], Theorem 1.16), we have α = \frac{1}{m} for some odd integer m ≥ 3. If m ≥ 5, the relative bound s ≤ \frac{1−α^{2}}{1−sα^{2}} (see [G], Theorem 1.9) implies s ≤ 2n, a contradiction. Hence we have m = 3, and a theorem of Kao-Yu’s ([K-Y]) implies n ≤ 27. (If n ≤ 14, a system of 28 equiangular lines with angle arccos \frac{1}{3} comes from dimension 7.) □

An other way to construct lattices of minimum 3 consists in viewing them as lattices containing to index 2 their even sublattice. For further use we consider more generally lattices of minimum m ≥ 3 odd. To state the results below we introduce some notation. Given a lattice L and a > 0, we denote by ^{a}L the group L equipped with the scalar product a (x · y). (Thus we have a (x · y) ≃ \sqrt{a} L.) Given an integral lattice L, L_{ev} denotes the even part of L, i.e., the set of x ∈ L having an even norm. If L is odd, we have [L : L_{ev}] = 2 and L = ⟨L_{ev}, e⟩ where e ∈ L is any vector of odd norm.

Proposition 2.3. Let Λ be a lattice of dimension n ≥ 2 and of odd minimum m ≥ 3, generated by its minimal vectors. Then L := ^{1/2}Λ_{ev} is an even lattice of minimum m′ ≥ \frac{m}{2}, and either we have s(Λ) ≤ n, or L has a basis of vectors of norm m − 1.

Proof. First note that Λ_{ev} is generated by the sums x + y, x, y ∈ S(Λ), with y ≠ x since 2x = (x+y)+(x−y). We have N(x+y) = 2(m±1) ≡ 0 mod 4, which shows that L is even.

Choose a half-system e_{1}, …, e_{s} of minimal vectors of Λ such that e_{1} · e_{i} = +1 for i ≥ 2 and that e_{1}, …, e_{n} are independent, and set e′_{i} = e_{1} − e_{i}, i ≥ 2. Suppose first that all e_{i} · e_{j}, i < j are equal to +1 and that s > n. Then we may write x := e_{n+1} as a ℚ-linear combination of e_{1}, …, e_{n}, say, x = ∑ λ_{k} e_{k}. Calculating x·e_{i} for i ≤ n, we obtain

1 = x · e_{i} = mλ_{i} + ∑_{i≠j} λ_{j} = (m − 1)λ_{i} + ∑_{i} λ_{j},

which implies that the λ_{i} have the common value λ = \frac{1}{n+m−1}.

From x = λ ∑ e_{i} we deduce that m = x · x = λ n, which implies

\frac{1}{n+m−1} = \frac{m}{n} ⇔ (m + n) (n − 1) = 0 ,

a contradiction.

[Lattices with e_{i} · e_{j} = +1 and s = n exist, and are unique up to isometry. They can be represented by the Gram matrix M with entities M_{i,i} = m − 1 and M_{i,j} = \frac{m−1}{2} if j ≠ i, except M_{1,1} = m + 1 and M_{1,2} = M_{2,1} = 0.]

Otherwise, let i > 1 and j > i such that e_{i} · e_{j} = −1. If i > n, write e_{i} as a Q=linear combination of e_{k}, k ≤ n. There must be at least three nonzero components, so that exchanging i with some i′ ≤ n, me may assume that i ∈ [2, n], that me may then exchange with 2. We thus reduce ourselves to the case when i = 2, and then we may similarly assume that j = 3. Finally we check that e_{2} + e_{3}, e′_{2}, …, e′_{n} is a basis for Λ_{ev}, which completes the proof of the dichotomy between the two cases above. □

Note that given L we can reconstruct Λ from L by the formula

Λ = ^{2}L_{1} with L_{1} = ⟨L, \frac{v}{2}⟩ and v ∈ L of norm 2m.

Using Proposition 2.3 together with the classification of root lattices we now prove:

Theorem 2.4. Let Λ be an n-dimensional lattice of minimum 3 whose set S of minimal vectors supports an equiangular family of lines of rank n. Then except if n = 5, 6 or 7, the maximal number s_{n} of lines is equal to 2(n − 1), and except if n = 2, attained uniquely on the projection of 𝔻_{n+1} described in Proposition 2.1.

Proof. The first case of Proposition 2.3 needs n ≥ 2(n−1), hence n = 2.

Let now n ≥ 3. Then Λ is of the form ⟨L, \frac{v}{2}⟩ where Λ is a root lattice and v a norm-6 vector which is not congruent modulo 2 to a shorter vector. This condition eliminates 𝔼_{6} and 𝔼_{8}, and implies n ≥ 5 (resp. n ≥ 6) if Λ = 𝔸_{n} (resp. 𝔻_{n}), and because S must be of rank n, n = 5 (resp. 6). Finally we are left with 𝔸_{5}, 𝔻_{6} and 𝔼_{7}, which accounts for the exceptional dimensions 5, 6, 7.

From now on we assume that L = L_{1} ⊥ L_{2} and v = \frac{x+y}{2}, x ∈ L_{1}, y ∈ L_{2}. The vectors congruent to \frac{v}{2} modulo l are of the form \frac{x+2z,y+2t}{2} with z,t in L. We have

N(x + 2z) = N(x) ⇔ x · z + N(z) = 0 ,

and if N(z) ≥ N(x) (a condition which is automatic if N(x) = 2), this needs z = −x.

We may assume that N(x) = 2 and N(y) = 4, and since x′ = x + 2z reduces to x′ = ±x, dim S = n needs dim L_{1} = 1, i.e., L_{1} = 𝔸_{1}, and then either L ≃ 𝔸_{1} ⊥ 𝔸_{1} ⊥ 𝔸_{1} or L2 is irreducible. The former case accounts for the theorem in dimension 3. We are now left with L = 𝔸_{1} + 𝔸_{k}, k ≥ 3, L = 𝔸_{1} + 𝔻_{k}, k ≥ 4 and L = 𝔸_{1} + 𝔼_{k}, k = 6, 7, 8 (n = k +1), the convenient y +2t not ±y being obtained with N(t) = 2 and y · t = −2.

If L = 𝔸_{1} ⊥ 𝔸_{k} we may take y = ε_{0}+ε_{1}−ε_{2}−ε_{3} and must take k = 3. We obtain s = 6, which accounts for the theorem in dimension 4.

If L = 𝔸_{1} + 𝔻_{k} we may take (a) y = 2ε_{1} or (b) y = ε_{1} + ε_{2} + ε_{3} + ε_{4} (in the same orbit if n = 4). Then y + 2t is one of ±ε_{i} (2(n − 1) solutions, rk S = n) or ±ε_{1} ± … with an even number of minus signs (8 solutions, rk S = 5).

If L = 𝔸_{1} ⊥ 𝔼_{n−1}, we perform direct calculations. The maximal value of rk S is then 6, 7 and 9 for n = 7, 8 and 9, respectively. In the latter cases S generates a sublattice of index 2 in Λ, isometric to the lattice constructed with 𝔸_{1} ⊥ 𝔻_{8}¿

We have proved that fir n ≠ 2, 5, 6, 7, the maximum of s is 2(n − 1), attained on a unique lattice (up to isometry) generated by its minimal vectors. This completes the proof of Theorem 2.4. □

[As a Gram matrix for the projection of 𝔻_{n+1} we may choose A = (a_{i,j}) with entries 3 on the diagonal and 1 off the diagonal, except a_{1,2} = a_{2,1} = −1.]

It might well be that the methods of [K-Y] could prove better upper bounds than t′_{n} ≤ 27 for some n. In the other direction, we have the lower bounds t′_{n} ≥ 2(n − 1) coming from 𝔻_{n+1}, and we shall see in Section 3 that we have t′_{13} ≥ 26, attained with a system of angle arccos \frac{1}{5}. Experimentation based on Proposition 2.3 applied to various “classical” lattices of minimum 4 in dimensions 8 to 12 always produced lattices with s < 2(n − 1). This supports the following conjecture:

Conjecture 2.5. For 8 ≤ n ≤ 13 we have t′_{n} ≤ 2n, and t′_{13} = 26.

The upper bound 2n allows that some t′_{n} may be attained on angles not of the form arccos \frac{1}{m}, m odd, like in dimension 3.

3. Lattices of minimum 4 and 5

In this section we concentrate on lattices of minimum 5 in which pairs of non-proportional minimal vectors have scalar product ±1. In dimensions n ∈ [15–23] (and in some higher dimension) examples show that we have the strict inequality t′_{n} > 2n,so that the highest values of t′_{n} are attains on sets of angle arccos \frac{1}{5}, whence the special interest of these lattices.

Most of the lattices we shall consider (though not all) have been constructed using projections of a lattice of minimum 4 along a norm-6 vector, and using descending chains of cross-sections. We first consider such an exception.

3.1. A 13-Dimensional Lattice. This is the lattice L denoted by C2 × PSL(2, 25) : C2 in [WebN-S] (after its automorphism group). It has s = 26, and defines the equiangular family referred to in Conjecture 2.5. It has a unique (up to isometry) 14-dimensional extension \overline{L} with s = 28, the value of t_{14}, which extends uniquely wxactly up to dimension 19 under the condition that s be as large as possible. We recover these lattices as the unique sequence of densest sections, for which Gram matrices denoted by Qan can be read in [WebM], file Min5.GP.

[The even sublattice L_{ev} of L deserves a remark: this is the best known example (found by Conway and Sloane, [C-S]) of a lattice in its dimension having a large “Bergé-Martinet invariant” γ′, defined by γ′(Λ) = (γ(Λ)γ(Λ^{∗}))^{1/2}.]

3.2. Lattices from the Leech Lattice. There is a unique orbit of norm-6 vectors in the Leech lattice Λ_{24}, so that Theorem 1.3 defines a unique lattice (up to isomorphism). Its set of minimal vectors consists in 276 pairs ±x, which are the projections along the corresponding norm-6 vector of vectors x ∈ S(Λ_{24} such that v · x = ±3. This configuration is known as the Witt design. Its automorphism group is 2 × Co_{3}.

Consequently, for any pair (L, v) of a relative lattice L ⊂ Λ_{24} of minimum 4 and of a norm-6 vector v ∈ L, provided that there exists an x ∈ S(L) with v · x = ±3, projection along v defines a lattice whose set of minimal vectors supports an equiangular family contained in the Witt design.

We have considered projections of various lattices contained in the Leech lattice in dimensions 14−24 taken from [WebM], file Lambda.gp. The best results were obtained using Conway-Sloane’s laminated lattices Λ_{n} or successive sections of such lattices. The file Min5.gp contains series of Gram matrices Qbn, 16 ≥ n ≥ 8 and Qcn, 23 ≥ n ≥ 8, starting with Gram matrices for projections of Λ_{17} and Λ_{24}, respectively.

The table below displays the known values of t_{n} (taken from [GSY] and the largest values found on lattices for equiangular families, using Qan and Qbn for n = 14, Qbn for n = 15, 16, and Qcn for 17 ≤ n ≤ 23.

Table for dimensions 14 – 23

\begin{matrix}n & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 \\ t_{n} & 28 & 36 & 40 & 48 & 57 − 59 & 72 − 74 & 90 − 94 & 126 & 176 & 276 \\ lat ≥ & 28 & 36 & 38 & 48 & 56 & 72 & 90 & 126 & 176 & 276 \end{matrix}

Inspection of the second line of this table clearly shows that we have t′_{n} = t_{n} for all n ∈ [14, 23], and inspection of the third line shows that this values are attained on minimaL vectors of lattices for n = 14, 15, 17 and 21–23, As for the remaining values of n I conjecture:

Conjecture 3.1. For n = 16, 18, 19 and 20, the largest number of lines produced by the minimal vectors of a lattice is 38, 56, 72 and 90, respectively.

We now prove two complements, first for n = 14, then for n = 18, the latter answering a question raised in [G], Subsection 1.3.2.

Proposition 3.2. The sets of equiangular lines afforded by S(Qa_{14}) and S(Qb_{14}) are not isometric.

Proof. The construction of S(Qa_{14}) from S(Qa_{13}) shows that removing two convenient lines from S(Qa_{14}) produces a set of rank 13 (indeed, in a unique way), whereas one must remove at least four lines from S(Qb 14)) to obtain a set of rank ≤ 13. □

In [GSY] the authors construct four sets of 57 vectors which, when rescaled to norm 5 have pairwise scalar products ±1. I have checked that in all examples the sublattice they generate in ℤ^{18} has a minimum m ≤ 4 (and contains norm 5 vectors with pairwise scalar products not ±1). Since any subset of norm 5 vectors in Qa_{23} generate a lattice of minimum 5 in its span, we have:

Proposition 3.3. The four equiangular systems above are not contained in the Witt design. □

3.3. Beyond dimension 23. The arguments used to prove Proposition 2.2, (3) (relying on results of [K-Y] together with the relative bound) prove:

Proposition 3.4. In the range 24 ≤ n ≤ 41, t′_{n} is strictly smaller than t_{n} (equal to 276). □

In analogy with what was observed in Section 2 for angle arccos \frac{1}{3}, we expect that arccos \frac{1}{5} should play a major rôle in dimensions, say, 24 to 50. However, whereas lattices contained in the Leech lattice constitute a rich source of even lattices of minimum 4, our knowledge of such lattices in larger dimensions is poor. I have carried out some experimentation (far from being exhaustive) on the even sublattices of unimodular lattices of minimum 3, using the Bacher-Venkov classification in dimensions 27 and 28 (see the file unimod23to28.gp.gp in[WebM]). Using projections of the even part of the lattice denoted there by o27b1 we found lattices with (n, s) = (26, 82) and (25, 108), and then (n, s) = (24, 100) by cross-sections. Gram matrices can be downloaded from Part 4 of Min5.gp.

Also no infinite series of lattices having a fixed minimum m ≥ 4 are known (by contrast with minima m = 2 or 3 for which we can use root lattices). This leaves wide open the following question:

Question 3.5. Do there exist for each odd m ≥ 5 infinite series L_{n} of lattices with minimum m and pairwise scalar products ±1 on S(L_{n})

such that s ∼ \frac{m+1}{m−1} n for n → ∞? (If not the exact bound ⌊\frac{m+1}{m−1} (n − 1)⌋ of [JTYZZ].)

I have also considered angles arccos \frac{1}{7}, using projections of lattices of minimum 6. The only example which deserves to be mentioned is due to G. Nebe ([Ne], answering a question of mine), the proof of which relies on the theory of modular forms: applying Theorem 1.3 to an even, unimodular lattice of minimum 6 yields an equiangular family of 100 lines.

References

[C-S] J.H. Conway, N.J.A. Sloane, On Lattices Equivalent to Their!la Duals,

J. Number Theory 48 (1994), 373–382.

[G] G. Greaves, Equiangular lines, University of Montreal (August 2022).

[GSY] G. Greaves, J. Syatriadi, P. Yatsyna, Equiangular lines in Euclidean spaces: dimensions 17 and 18 , Combinatorica 41 (2021), 839-–872; preprint at arXiv:2104.04330v1 [math.CO], 9 Apr 2021.

[JTYZZ] Z.Jiang, J. Tidor, Y. Yao, S. Zhang, Y. Zhao, Equiangular Lines with a Fixed Angle, Ann. Math. 194 (2021), 729–743; see also arXiv [math.CO], 28 Feb 2022.

[K-Y] W.-J. Kao, W.-H. Yu, Four-point semidefinite bound for equiangular

lines, preprint at arXiv:2203.05828 [math.CO], 11 Mar 2022.

[L-S] P.W.H. Lemmens and J.J. Seidel, Equiangular lines, J. Algebra 24

(1973), 494–512.

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

[Ma1] J. Martinet, Reduction Modulo 2 and 3 of Euclidean Lattices, J. Algebra 251 (2002), 864–887.

[Ma2] J. Martinet, Reduction Modulo 2 and 3 of Euclidean Lattices, II , Contemporary Math. 344 (2004), 265–273.

[Ma3] J. Martinet, Lattices, Graphs and Groups, in preparation.

[Ne] G.Nebe, Mail to the author, January 25th, 2024.

[PARI] PARI/GP, a computer algebra system designed for fast computations in number theory, url http://pari.math.u-bordeaux.fr/ .

[WebM] J. Martinet, A Catalogue of Perfect Lattices, https://jamartin.perso.math.cnrs.fr/Lattices/index.html.

[WebN-S] G. Nebe, N.J.A. Sloane, A Catalogue of Lattices,

https://www.math.rwth-aachen.de/ Gabriele.Nebe/LATTICES/.

  Email address: Jacques.Martinet@math.cnrs.fr