Equiangular lines in Euclidean spaces Gary Greaves Tohoku - - PowerPoint PPT Presentation

Equiangular lines in Euclidean spaces

Gary Greaves

東北大学 Tohoku University

3rd June 2014

joint work with J. Koolen, A. Munemasa, and F. Szöllősi.

Gary Greaves — Equiangular lines in Euclidean spaces 1/13

Plan

◮ From lines to matrices; ◮ A contentious table; ◮ Seidel matrices with 3 eigenvalues; ◮ A strengthening of the relative bound.

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Equiangular line systems

◮ Let L be a system of n lines spanned by v1, . . . , vn ∈ Rd. ◮ L is equiangular if vi, vi = 1 and |vi, vj| = α

(α is called the common angle).

◮ Problem: given d, what is the largest possible number

N(d) of equiangular lines in Rd?

Example

◮ An orthonormal basis: n = d and α = 0. ◮ N(d) d.

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Seidel matrices

Let L be an equiangular line system of n lines in Rd with common angle α.

◮ Let M be the Gram matrix for the line system L. ◮ Then M is positive semidefinite with nullity n − d. ◮ Assume α > 0 and set S = (M − I)/α. ◮ S is a {0, ±1}-matrix with smallest eigenvalue −1/α

with multiplicity n − d.

◮ S = S(L) is called a Seidel matrix. ◮ Relation to graphs: S = J − I − 2A.

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Icosahedron

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i0 i1 i2 i3 i4 i5 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron

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i0 i1 i2 i3 i4 i5 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron

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i0 i1 i2 i3 i4 i5 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron

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i0 i1 i2 i3 i4 i5 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron

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i0 i1 i2 i3 i4 i5 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron

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i0 i1 i2 i3 i4 i5 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron

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i0 i1 i2 i3 i4 i5 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron

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i0 i1 i2 i3 i4 i5 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron

S =         1 1 1 1 1 1 1

−1 −1

1 1 1 1

−1 −1

1

−1

1 1

−1

1

−1 −1

1 1 1 1

−1 −1

1         ;

◮ Spectrum: {[−

√

5]3, [

√

5]3};

◮ n = 6, d = 3, and α = 1/

√

5.

◮ Question: for d = 3, can we do better than n = 6?

Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Upper bounds

Let L be an equiangular line system of n lines in Rd with smallest eigenvalue λ0.

◮ Gerzon ’73:

Absolute bound: n d(d + 1) 2 .

◮ van Lint and Seidel ’66: for λ0 < −

√

d + 2 Relative bound: n d(λ2

0 − 1)

λ2

0 − d .

◮ Neumann ’73:

If n > 2d then λ0 is an odd integer.

Gary Greaves — Equiangular lines in Euclidean spaces 6/13

Maximal sets of equiangular lines

Let L be an equiangular line system of n lines in Rd with common angle α.

d 2 3 4 5 6 7 – 13 14 15 16 17 18 19 20 n 3 6 6 10 16 28 28 36 40 48 48 72 90 30 42 51 61 76 96

Gary Greaves — Equiangular lines in Euclidean spaces 7/13

Maximal sets of equiangular lines

Let L be an equiangular line system of n lines in Rd with common angle α.

d 2 3 4 5 6 7 – 13 14 15 16 17 18 19 20 n 3 6 6 10 16 28 28 36 40 48 48 72 90 30 42 51 61 76 96

But according to wikipedia and the OEIS:

d 2 3 4 5 6 7 – 13 14 15 16 17 18 19 20 n 3 6 6 10 16 28 28 36 40 48 48 72 90 76 96

Gary Greaves — Equiangular lines in Euclidean spaces 7/13

Maximal sets of equiangular lines

Let L be an equiangular line system of n lines in Rd with common angle α.

d 2 3 4 5 6 7 – 13 14 15 16 17 18 19 20 n 3 6 6 10 16 28 28 36 40 48 48 72 90 29 41 51 61 76 96

But according to wikipedia and the OEIS:

d 2 3 4 5 6 7 – 13 14 15 16 17 18 19 20 n 3 6 6 10 16 28 28 36 40 48 48 72 90 76 96

Gary Greaves — Equiangular lines in Euclidean spaces 7/13

Properties of Seidel matrices with 3 eigenvalues

Let S be an n × n Seidel matrix with precisely 3 distinct eigenvalues λ < θ < η.

◮ tr S = 0, tr S2 = n(n − 1); ◮ det S ≡ (−1)n−1(n − 1) mod 4; ◮ (S − λI)(S − θI)(S − ηI) = 0.

Theorem

For primes p ≡ 3 mod 4, there do not exist any p × p Seidel matrices having precisely 3 distinct eigenvalues. Except for n = 4, they exist for all other n. n 3 4 5 6 7 8 9 10 11 12 # 1 2 2 3 4 10

Gary Greaves — Equiangular lines in Euclidean spaces 8/13

30 equiangular lines in R14?

Let S be an n × n Seidel matrix with spectrum λ(n−d)

< λ1 λ2 · · · λd.

Using the trace formulae, we have

d

∑

i=1

λi = −(n − d)λ0;

d

∑

i=1

λ2

i = n(n − 1) − (n − d)λ2 0.

Case: d = 14, n = 30, and λ0 = −5. Set µi = λi − 6. Then 1 =

d

∑

i=1

u2

i /d

d

• ∏ u2

i 1.

Hence ui ∈ {±1}.

Gary Greaves — Equiangular lines in Euclidean spaces 9/13

Strengthening the relative bound

Theorem

Let S be an n × n Seidel matrix with eigenvalues λ(n−d)

< λ1 λ2 · · · λd,

and suppose λ2

0 d + 2. If n =

• d(λ2

0−1)

λ2

0−d

• and some

integrality condition and nonzero condition are satisfied. Then S has at most 3 distinct eigenvalues.

◮ 30 lines in R14

—

{[−5]16, [5]9, [7]5};

◮ 42 lines in R16

—

{[−5]26, [7]7, [9]9}.

Gary Greaves — Equiangular lines in Euclidean spaces 10/13

Euler graphs

An Euler graph is a graph each of whose vertices have even valency.

Theorem (Mallows-Sloane ’75)

The number of switching classes of n × n Seidel matrices equals the number of Euler graphs on n vertices.

Theorem

Let S be a Seidel matrix with precisely 3 distinct eigenvalues. Then S is switching equivalent to a Seidel matrix S′ = J − I − 2A where A is the adjacency matrix of an Euler graph.

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30 and 42

Theorem

Let S be an n × n Seidel matrix with spec. {[λ]a, [µ]b, [ν]c}. Suppose n ≡ 2 mod 4, λ + µ ≡ 0 mod 4, and

|n − 1 + λµ| = 4.

Then |ν2 − (λ + µ)ν + λµ|/4 = n/c ∈ Z and

|ν| n/c − 1.

Gary Greaves — Equiangular lines in Euclidean spaces 12/13

30 and 42

Theorem

Let S be an n × n Seidel matrix with spec. {[λ]a, [µ]b, [ν]c}. Suppose n ≡ 2 mod 4, λ + µ ≡ 0 mod 4, and

|n − 1 + λµ| = 4.

Then |ν2 − (λ + µ)ν + λµ|/4 = n/c ∈ Z and

|ν| n/c − 1.

Corollary

The candidate Seidel matrices with spectra {[−5]16, [5]9, [7]5} and {[−5]26, [7]7, [9]9} do not exist.

Corollary

Regular graphs with spectra {[11]1, [2]16, [−3]9, [−4]4} and

{[12]1, [2]16, [−3]8, [−4]5} do not exist.

Gary Greaves — Equiangular lines in Euclidean spaces 12/13

Feasible Seidel matrices with 3 eigenvalues

n d λ µ ν Exist? 28 14 [−5]14 [3]7 [7]7 Y 30 14 [−5]16 [5]9 [7]5 N 40 16 [−5]24 [5]6 [9]10 ? 40 16 [−5]24 [7]15 [15]1 Y 42 16 [−5]26 [7]7 [9]9 N 48 17 [−5]31 [7]8 [11]9 Y 49 17 [−5]32 [9]16 [16]1 ? 48 18 [−5]30 [3]6 [11]12 ? 48 18 [−5]30 [7]16 [19]2 ? 54 18 [−5]36 [7]9 [13]9 ? 60 18 [−5]42 [11]15 [15]3 ? 72 19 [−5]53 [13]16 [19]3 Y 75 19 [−5]56 [10]1 [15]18 ? 90 20 [−5]70 [13]5 [19]15 ? 95 20 [−5]75 [14]1 [19]19 ?

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