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
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Plan
I
From lines to matrices;
I
A contentious table;
I
Seidel matrices with 3 eigenvalues;
I
A strengthening of the relative bound.
Gary Greaves — Equiangular lines in Euclidean spaces
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Equiangular line systems
I
Let L be a system of n lines spanned by v1 , . . . , vn ∈ Rd .
I
L is equiangular if hvi , vi i = 1 and |hvi , vj i| = α
(α is called the common angle).
I
Problem: given d, what is the largest possible number
N (d) of equiangular lines in Rd ?
Example
I
An orthonormal basis: n = d and α = 0.
I
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.
I Let M be the Gram matrix for the line system L.
I
Then M is positive semidefinite with nullity n − d.
I
Assume α > 0 and set S = (M − I )/α.
I
S is a {0, ±1}-matrix with smallest eigenvalue −1/α
with multiplicity n − d.
I
S = S(L) is called a Seidel matrix.
I
Relation to graphs: S = J − I − 2A.
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Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
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Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
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Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
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Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
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Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
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Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
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Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
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Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
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Icosahedron

0 1
1
1
1
1
1 0
1 −1 −1 1 


1 1
0
1 −1 −1
;

S=

1
−
1
1
0
1
−
1


1 −1 −1 1
0
1 
1 1 −1 −1 1
0

I
√
√
Spectrum: {[− 5]3 , [ 5]3 };
I
√
n = 6, d = 3, and α = 1/ 5.
I
Question: for d = 3, can we do better than n = 6?
Gary Greaves — Equiangular lines in Euclidean spaces
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Upper bounds
Let L be an equiangular line system of n lines in Rd with
smallest eigenvalue λ0 .
I Gerzon ’73:
I
I
d(d + 1)
.
2
√
van Lint and Seidel ’66: for λ0 < − d + 2
Absolute bound:
n6
Relative bound:
n6
d(λ20 − 1)
.
λ20 − d
Neumann ’73:
If n > 2d then λ0 is an odd integer.
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Maximal sets of equiangular lines
Let L be an equiangular line system of n lines in Rd with
common angle α.
d 2
n 3
3
6
4
6
5 6
10 16
7 – 13 14 15 16 17 18 19 20
28
28 36 40 48 48 72 90
30
42 51 61 76 96
Gary Greaves — Equiangular lines in Euclidean spaces
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Maximal sets of equiangular lines
Let L be an equiangular line system of n lines in Rd with
common angle α.
d 2
n 3
3
6
4
6
5 6
10 16
7 – 13 14 15 16 17 18 19 20
28
28 36 40 48 48 72 90
30
42 51 61 76 96
But according to wikipedia and the OEIS:
d 2
n 3
3
6
4
6
5 6
10 16
7 – 13 14 15 16 17 18 19 20
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
n 3
3
6
4
6
5 6
10 16
7 – 13 14 15 16 17 18 19 20
28
28 36 40 48 48 72 90
29
41 51 61 76 96
But according to wikipedia and the OEIS:
d 2
n 3
3
6
4
6
5 6
10 16
7 – 13 14 15 16 17 18 19 20
28
28 36 40 48 48 72 90
76 96
Gary Greaves — Equiangular lines in Euclidean spaces
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Properties of Seidel matrices with 3 eigenvalues
Let S be an n × n Seidel matrix with precisely 3 distinct
eigenvalues λ < θ < η.
I tr S = 0, tr S2 = n(n − 1);
I
det S ≡ (−1)n−1 (n − 1) mod 4;
I
(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
# 0
4
0
5
1
6
2
7
0
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2
9
3
10
4
11
0
12
10
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30 equiangular lines in R14?
Let S be an n × n Seidel matrix with spectrum
(n−d)
λ0
< λ1 6 λ2 6 · · · 6 λd .
Using the trace formulae, we have
d
∑ λi = −(n − d)λ0;
i=1
d
∑ λ2i = n(n − 1) − (n − d)λ20.
i=1
Case: d = 14, n = 30, and λ0 = −5. Set µi = λi − 6. Then
q
d
1 = ∑ u2i /d > d ∏ u2i > 1.
i=1
Hence ui ∈ {±1}.
Gary Greaves — Equiangular lines in Euclidean spaces
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Strengthening the relative bound
Theorem
Let S be an n × n Seidel matrix with eigenvalues
(n−d)
< λ1 6 λ2 6 · · · 6 λd ,
d(λ20 −1)
2
and some
and suppose λ0 > d + 2. If n =
λ2 −d
λ0
0
integrality condition and nonzero condition are satisfied.
Then S has at most 3 distinct eigenvalues.
I
30 lines in R14
—
{[−5]16 , [5]9 , [7]5 };
I
42 lines in R16
—
{[−5]26 , [7]7 , [9]9 }.
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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
S0 = 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| 6 n/c − 1.
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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| 6 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.
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Feasible Seidel matrices with 3 eigenvalues
n
28
30
40
40
42
48
49
48
48
54
60
72
75
90
95
d
14
14
16
16
16
17
17
18
18
18
18
19
19
20
20
λ
[−5]14
[−5]16
[−5]24
[−5]24
[−5]26
[−5]31
[−5]32
[−5]30
[−5]30
[−5]36
[−5]42
[−5]53
[−5]56
[−5]70
[−5]75
µ
[3]7
[5]9
[5]6
[7]15
[7]7
[7]8
[9]16
[3]6
[7]16
[7]9
[11]15
[13]16
[10]1
[13]5
[14]1
Gary Greaves — Equiangular lines in Euclidean spaces
ν
[7]7
[7]5
[9]10
[15]1
[9]9
[11]9
[16]1
[11]12
[19]2
[13]9
[15]3
[19]3
[15]18
[19]15
[19]19
Exist?
Y
N
?
Y
N
Y
?
?
?
?
?
Y
?
?
?
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