Nonexistence of tight spherical designs

Akihiro Munemasa

joint work with Eiichi Bannai and Boris Venkov

1 Introduction and Preliminaries

A spherical design is a finite subset X of the unit sphere S^{n−1} ⊂ ℝ^{n} which approximates S^{n−1} (to a certain degree t ∈ ℕ, to be defined later). A spherical t-design is called tight if it has the smallest possible number of points among all t-designs in S^{n−1}. Before giving the exact definition, let us list some examples.

• A regular simplex in S^{n−1} is a tight spherical 2-design.

• X = \frac{1}{\sqrt{2}}E_{8}, where E_{8} ⊂ ℝ^{8} is the set of 240 roots (vectors of norm 2) of the root system of type E_{8}. X is a tight spherical 7-design.

• X = \frac{1}{2}Λ_{4} ⊂ S^{23}, where Λ ⊂ ℝ^{24} is the Leech lattice, Λ_{4} is the set of the 196560 shortest vectors (of norm 4) in Λ. X is a tight spherical 11-design.

Definition 1. A subset X ⊂ S^{n−1} is a spherical t-design if

\frac{1}{|X|} ∑_{x∈X} f(x) = \frac{1}{ω_{n−1}} ∫_{S^{n−1}} f(x) dS

for every polynomial f(x) = f(x_{1}, x_{2}, ⋯, x_{n}) of degree ≤ t, where

ω_{n−1} = ∫_{S^{n−1}} 1 dS

If X ⊂ S^{n−1} is a spherical t-design, then

|X| ≥ \binom{n − 1 + t/2}{t/2} + \binom{n + t/2 − 2}{t/2 − 1}

if t is even, and

|X| ≥ 2\binom{n − 1 + (t − 1)/2}{(t − 1)/2}

if t is odd. X is said to be tight if equality holds.

If X ⊂ S^{n−1} (n ≥ 3) is a tight spherical t-design, then

• (Bannai–Damerell, 1979–1980) t ≤ 5, t = 7 or t = 11.

• (Bannai–Sloane, 1981) t = 11 ⟹ Leech.

• t ≤ 3 ⟹ classified (trivial for t = 1, regular simplex for t = 2, generalized octahedron for t = 3).

• t = 5 ⟹ a derived configuration is a tight 4-design in S^{n−2}.

• t = 4 ⟹ an extended configuration is a tight 5-design in S^{n}.

So the existence of a tight spherical 4-design in S^{n−2} is equivalent to the existence of a tight spherical 5-design in S^{n−1}. We may restrict our attention to the existence problem of tight spherical 5- and 7-designss.

If X ⊂ S^{n−1} (n ≥ 3) is a tight spherical 5-design, then n = 3 and X is an icosahedron, or

n = d^{2} − 2 (d: odd)

= 7, 23, 47, …

\{(x, y) | x, y ∈ X\} = \{±1, ±\frac{1}{d}\}.

For n = 7, we have the derived E_{8}, and for n = 23, we have Co._{3} × 2 on 276 · 2.

If X ⊂ S^{n−1} (n ≥ 3) is a tight spherical 7-design, then

n = 3d^{2} − 4 = 8, 23, 44, 71, …

\{(x, y) | x, y ∈ X\} = \{0, ±1, ±\frac{1}{d}\}

For n = 8, we have E_{8}, and for n = 23, we have the derived Leech configuration on 4600 points.

2 Nonexistence of Tight Spherical 7-Designs

A tight spherical (2s + 1)-design X is antipodal, i.e., X = −X, and an antipodal subset X ⊂ S^{n−1} is a spherical (2s + 1)-design iff

\frac{1}{|X|} ∑_{x∈X} f(x) = \frac{1}{ω_{n−1}} ∫_{S^{n−1}} f(x) dS

for every homogeneous polynomial of degree 2, 4, …, 2s. For 7-designs, it suffices to take f to be homogeneous polynomials of degree 2, 4, 6. Take

f(x) = (α, x)^{2}, (α, x)^{4}, (α, x)^{6}

where α ∈ ℝ^{n}, x = (x_{1}, …, x_{n}), and (α, x) = ∑_{i=1}^{n} α_{i}x_{i}. The conditions derived by taking the above f are called the basic relations, and these are given below [11, 12].

\frac{1}{|X|} ∑_{x∈X}(α, x)^{2} = \frac{1}{n}(α, α),
\frac{1}{|X|} ∑_{x∈X}(α, x)^{4} = \frac{3}{n(n + 2)}(α, α)^{2},
\frac{1}{|X|} ∑_{x∈X}(α, x)^{6} = \frac{15}{n(n + 2)(n + 4)}(α, α)^{3}.

Suppose that X is a tight spherical 7-design. If one takes α ∈ X, then one obtains the consequences

n = 3d^{2} − 4,

x, y ∈ X ⟹ (x, y) = ±1, ±\frac{1}{d}, 0

but not more. To derive further restrictions, the key idea is to take α in the dual lattice of the lattice generated by X. Thus, for the convenience, we normalize the set X in such a way that it generates an integral lattice. If X ⊂ S^{n−1} is a tight spherical 7-design, then n = 3d^{2} − 4. Put D = \sqrt{d}X. Then

x, y ∈ D ⟹ (x, y) = ±d, ±1, 0.

Let Λ be the integral lattice in ℝ^{n} generated by D. The basic relations become:

\frac{1}{|D|} ∑_{x∈D}(α, x)^{2} = \frac{d}{n}(α, α)

\frac{1}{|D|} ∑_{x∈D}(α, x)^{4} = \frac{3d^{2}}{n(n + 2)}(α, α)^{2}

\frac{1}{|D|} ∑_{x∈D}(α, x)^{6} = \frac{15d^{3}}{n(n + 2)(n + 4)}(α, α)^{3}

for any α ∈ ℝ^{n}. Also,

|D| = |X| = \frac{n(n + 1)(n + 2)}{3}

After simplification, we get

∑_{x∈D}(α, x)^{2} = d(3d^{2} − 2)(d^{2} − 1)(α, α)
∑_{x∈D}(α, x)^{4} = 3d^{2}(d^{2} − 1)(α, α)^{2}
∑_{x∈D}(α, x)^{6} = 5d(d^{2} − 1)(α, α)^{3}

for all α ∈ ℝ^{n}.

If (α, x) is an integer for all x ∈ D, or equivalently, α belongs to the dual lattice of Λ = ⟨D⟩, then we can get some congruencial conditions. Assume

α ∈ Λ^{∗} = {α ∈ ℝ^{n} | (α, β) ∈ ℤ, ∀β ∈ Λ}
= {α ∈ ℝ^{n} | (α, x) ∈ ℤ, ∀x ∈ D}
⊃ Λ ⊃ D

For each k = 1, 2, …, define n_{k}(α) by

n_{k}(α) = \frac{1}{2}|\{x ∈ D | (α, x) = ±k\}| ∈ ℤ

Then the basic relations become:

∑_{k=1}^{∞} k^{2}n_{k}(α) = \frac{d}{2}(3d^{2} − 2)(d^{2} − 1)(α, α)
∑_{k=1}^{∞} k^{4}n_{k}(α) = \frac{3d^{2}}{2} (d^{2} − 1)(α, α)^{2}
∑_{k=1}^{∞} k^{6}n_{k}(α) = \frac{5d}{2}(d^{2} − 1)(α, α)^{3}

Example 1. d = 4, n = 44 (the smallest open case). Since

∑_{k=1}^{∞} k^{2}(k^{2} − 1)(k^{2} − 4)n_{k}(α) ≡ 0 (mod 24)

we have

\frac{5(α, α)(5(α, α)^{2} − 60(α, α) + 184)}{4} ∈ ℤ.

Similarly,

k^{2}(k^{2} − 1) ≡ 0 (mod 12)
k^{2}(k^{4} − 1) ≡ 0 (mod 60)

give more divisibility conditions, which imply (α, α) ∈ 2ℤ. We therefore obtain

(α, α) ∈ 2ℤ ∀α ∈ Λ^{∗} ⟹ Λ^{∗} is even ⟹ Λ^{∗} is integral ⟹ Λ = Λ^{∗} is an even unimodular lattice

of dimension n = 44

This is impossible, because an even unimodular lattice exists only when the dimension is a multiple of 8, and it is not the case for 44.

Example 2. d = 5. We claim min Λ = 5. Suppose

α ∈ Λ, (α, α) = min Λ ≤ 4.

Then for ∀x ∈ D, α + x ∈ Λ, so

(α, α) ≤ (α ± x, α ± x)
= (α, α) ± 2(α, x) + 5

Thus

|(α, x)| ≤ \frac{5}{2}

(α, x) ∈ {0, ±1, ±2}

This means

∑_{k=1}^{∞} k^{m}n_{k}(α) becomes n_{1}(α) + 2^{m}n_{2}(α)

This situation has been investigated by Martinet in a different context [9].

n_{1}(α) + 4n_{2}(α) = 3880(α, α)
n_{1}(α) + 16n_{2}(α) = 900(α, α)^{2}
n_{1}(α) + 64n_{2}(α) = 300(α, α)^{3}

Eliminating n_{1}(α), n_{2}(α) from these equations, we find a quadratic equation in (α, α) which has only imaginary solutions. This is a contradiction. Example 3. d = 6, n = 104 (still open).

• Λ is even unimodular.

• min Λ = 6, Λ_{6} = D.

• Λ_{8} = ∅.

• ϑ_{Λ} = 1 + 385840q^{3} + 153139896000q^{5} + ⋯ (uniquely determined modular form).

We now state our main result for the nonexistence of tight spherical 7designs. Let p a prime. For m ∈ ℤ, define

ν_{p}(m) = the largest integer ν such that p^{ν}|m Theorem 1. Let d > 1 be a positive integer, and suppose

ν_{2}(d) = 2, ν_{3}(d(d^{2} − 1)) < 4,
ν_{p}(d(d^{2} − 1)) < 3 ∀prime p ≥ 5.

Then a tight spherical 7-design in dimension n = 3d^{2} − 4 does not exist.

Proof. (Sketch). The lattice Λ becomes an even unimodular lattice of dimension n, but n ≡ 4 (mod 8), a contradiction.

Pétermann pointed out that the set of positive integers d satisfying the hypothesis of Theorem 1 has positive density in ℕ (in particular, is infinite).

3 Nonexistence of Tight Spherical 5-Designs

For spherical 5-design X, we have

\frac{1}{|X|} ∑_{x∈X}(α, x)^{2} = \frac{1}{n}(α, α)
\frac{1}{|X|} ∑_{x∈X}(α, x)^{4} = \frac{3}{n(n + 2)}(α, α)^{2}

for any α ∈ ℝ^{n}.

Suppose that X is a tight spherical 5-design. If one takes α ∈ X, then we obtain the consequences

n = d^{2} − 2, d = 2m + 1

x, y ∈ X ⟹ (x, y) = ±1, ±\frac{1}{d}

As we have done for tight spherical 7-designs, we normalize X to define an integral lattice Λ, and then take α in the dual lattice of Λ. If X ⊂ S^{n−1} (n > 3) is a tight spherical 5-design, then n = d^{2} − 2, d = 2m + 1. Put D = \sqrt{d}X. Then

x, y ∈ D ⟹ (x, y) = ±d, ±1

Let Λ be the integral lattice in ℝ^{n} generated by D. The basic relations read:

\frac{1}{|D|} ∑_{x∈D}(α, x)^{2} = \frac{d}{n}(α, α)

\frac{1}{|D|} ∑_{x∈D}(α, x)^{4} = \frac{3d^{2}}{n(n + 2)}(α, α)^{2}

for any α ∈ ℝ^{n}, and

|D| = |X| = n(n + 1).

After simplification, we get

∑_{x∈D}(α, x)^{2} = 4m(m + 1)(2m + 1)(α, α)
∑_{x∈D}(α, x)^{4} = 12m(m + 1)(α, α)^{2}

If α ∈ Λ^{∗}, then

∑_{k=1}^{∞} k^{2}n_{k}(α) = 2m(m + 1)(2m + 1)(α, α)

∑_{k=1}^{∞} k^{4}n_{k}(α) = 6m(m + 1)(α, α)^{2}

Since k^{4} − k^{2} ≡ 0 (mod 12), we have

\frac{m(m + 1)(α, α)(3(α, α) − (2m + 1))}{6} ∈ ℤ

This is much weaker than the divisibility condition for tight 7-designs, but we can get the following consequence:

m, m + 1 : square-free ⟹ (α, α) ∈ \frac{1}{3}ℤ

To get more restriction, we consider some other lattices related to Λ.

Λ ⊃ Λ_{+} : even sublattice

= {α ∈ Λ | (α, α) ∈ 2ℤ} is a sublattice of index 2 in Λ

Γ = \frac{1}{\sqrt{2}}Λ_{+}

Then Γ is an integral lattice and

det Γ = 2 · 3^{s} (s ∈ ℤ, s ≥ 0)

Theorem 2. Suppose

m = 2k, k ≡ 2 (mod 3), k, 2k + 1 are square-free.

Then there does not exist a tight spherical 5-design in ℝ^{n}, where n = (2m + 1)^{2} − 2.

The proof of this theorem is more complicated than that of Theorem 1, and requires the Milgram–Braun formula [10].

Example 4. The case m = 3 (n = 47) is not ruled out by Theorem 2. We give an outline of the proof of the nonexistence in this case. Suppose that there exists a tight spherical 5-design in ℝ^{47}. Then

Γ ⊂ ℝ^{47} has determinant 2

⟹ ∃Γ̃ ⊂ ℝ^{48} : even unimodular, by gluing ⟹ ϑΓ̃ is a modular form of weight 24

with respect to SL(2, ℤ)

But the analysis of the coefficients of ϑΓ̃ reveals that ϑΓ̃ can not be expressible as a linear combination of a known basis of the space of modular forms of weight 24. This is a contradiction.

We originally hoped to construct one in ℝ^{47}. Let X be a tight spherical 5-design in ℝ^{47}. Then a derived configuration (a hyperplane section of X) is a tight 4-design Y, |Y | = 1127. Angles among the elements of Y defines a strongly regular graph:

1  320
165  320
1  320

Then Venkov showed that the subgraph induced by “165” is a generalized quadrangle of order (4, 8).

Could our technique be applied to establish the uniqueness of a generalized quadrangle of order (4, 8)?

References

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