Equiangular lines in Euclidean spaces
Gary Greaves
東北大学　Tohoku University
14th August 2015
joint work with J. Koolen, A. Munemasa, and F. Szöllősi.
Gary Greaves — Equiangular lines in Euclidean spaces
1/23
Plan
I
From lines to matrices;
I
Upper bounds for N (d);
I
Lower bounds for N (d);
I
Seidel matrices with 3 eigenvalues;
I
A strengthening of the relative bound;
I
New upper bounds.
Gary Greaves — Equiangular lines in Euclidean spaces
2/23
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 ?
Questions
I
Martin (2015): Find, as many as you can, equiangular
lines in Rd .
I
Yu (2015): Determine N (14) and N (16).
Gary Greaves — Equiangular lines in Euclidean spaces
3/23
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.
Gary Greaves — Equiangular lines in Euclidean spaces
4/23
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. Smallest eigenvalue λ0 = −1/α.
I
S = S(L) is called a Seidel matrix.
I
Relation to graphs: S = J − I − 2A.
Gary Greaves — Equiangular lines in Euclidean spaces
4/23
Basic properties of Seidel matrices
Let S be an n × n Seidel matrix.
I
tr S = 0, tr S2 = n(n − 1);
I
GG, AM, JK, FS (2015+): det S ≡4 (−1)n−1 (n − 1);
I
Neumann (1973): If S has an even integer eigenvalue λ
then λ has multiplicity at most 1.
I
2A = J − I − S.
Gary Greaves — Equiangular lines in Euclidean spaces
5/23
Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
6/23
Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
6/23
Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
6/23
Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
6/23
Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
6/23
Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
6/23
Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
6/23
Icosahedron
o0
o1
o5
i0
i1
i5
i2
i4
o2
i3
o4
o3
Gary Greaves — Equiangular lines in Euclidean spaces
6/23
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
6/23
Upper bounds for N (d)
Let L be an equiangular line system of n lines in Rd whose
Seidel matrix has smallest eigenvalue λ0 .
I Gerzon (1973):
Absolute bound:
I
d(d + 1)
.
2
Van Lint and Seidel (1966): for λ20 > d + 2
Relative bound:
I
n6
n6
d(λ20 − 1)
.
λ20 − d
Neumann (1973):
If n > 2d then λ0 is an odd integer.
Gary Greaves — Equiangular lines in Euclidean spaces
7/23
Upper bounds for N (d)
Let L be an equiangular line system of n lines in Rd whose
Seidel matrix has smallest eigenvalue λ0 .
I Van Lint and Seidel (1966): for λ2 > d + 2
0
Relative bound:
n6
d(λ20 − 1)
.
λ20 − d
I
Neumann (1973):
If n > 2d then λ0 is an odd integer.
I
Barg and Yu (2013): SDP upper bounds for d 6 136.
I
Okuda and Yu (2015+): ‘New relative bound’.
Gary Greaves — Equiangular lines in Euclidean spaces
7/23
Lower Bounds
I
√
Lemmens and Seidel (1973): N (d) > d d.
I
de Caen (2000):
N (d) > m2 /2 for d = 3m/2 − 1 (m = 4t ).
I
Szöllősi (2011): =⇒ N (d) > (d + 2)2 /72.
Gary Greaves — Equiangular lines in Euclidean spaces
8/23
Lower Bounds
I
√
Lemmens and Seidel (1973): N (d) > d d.
I
de Caen (2000):
N (d) > m2 /2 for d = 3m/2 − 1 (m = 4t ).
I
Szöllősi (2011): =⇒ N (d) > (d + 2)2 /72.
I
GG, AM, JK, FS (2015+): N (d) >
Gary Greaves — Equiangular lines in Euclidean spaces
32d2 +328d+296
.
1089
8/23
Lower Bounds
I
de Caen (2000):
N (d) > m2 /2 for d = 3m/2 − 1 (m = 4t ).
I
Szöllősi (2011): =⇒ N (d) > (d + 2)2 /72.
I
GG, AM, JK, FS (2015+): N (d) >
32d2 +328d+296
.
1089
Proposition (GG, AM, JK, FS 2015+)
For
(a)
(b)
(c)
each t > 1 and m = 4t there exists an equiangular set of
m(m/2 + 1) lines in dimension R3m/2+1 ; and
m(m/2 + 1) − 1 lines in dimension R3m/2 ; and
mj lines in dimension Rm+j−1 for every 1 6 j 6 m/2.
Gary Greaves — Equiangular lines in Euclidean spaces
8/23
Illustration of the proof
Let m = 4.

1 0 0
0 1 0

0 0 1
0 0 0
There is a set of m/2 + 1 MUBs of R4 .
 
 

0
1 1 1 1
1 1 1 1
 


0
 , 1 1 1 −1 −1 , 1 −1 −1 1 1
0 2 1 −1 1 −1 2 −1 1 −1 1
1
1 −1 −1 1
−1 1 1 −1
Gary Greaves — Equiangular lines in Euclidean spaces
9/23
Illustration of the proof
Let m = 4.

1 0 0
0 1 0

0 0 1
0 0 0










2
0
0
0
0
2
0
0
There is a set of m/2 + 1 MUBs of R4 .
 
 

0
1 1 1 1
1 1 1 1
 


0
 , 1 1 1 −1 −1 , 1 −1 −1 1 1
0 2 1 −1 1 −1 2 −1 1 −1 1
1
1 −1 −1 1
−1 1 1 −1
0
0
2
0
0
0
0
2
1
1
1
1
1
1
−1
−1
Gary Greaves — Equiangular lines in Euclidean spaces
1
−1
1
−1
1
−1
−1
1
1
−1
−1
−1
1
−1
1
1
1
1
−1
1

1
1

1

−1 




9/23
Illustration of the proof
Let m = 4.

1 0 0
0 1 0

0 0 1
0 0 0
There is a set of m/2 + 1 MUBs of R4 .
 
 

0
1 1 1 1
1 1 1 1
 


0
 , 1 1 1 −1 −1 , 1 −1 −1 1 1
0 2 1 −1 1 −1 2 −1 1 −1 1
1
1 −1 −1 1
−1 1 1 −1

2
0
0
0 1
1
1
1 1
1
1
1
 0
2
0
0 1
1
1
1 −1 −1 −1 −1


 0
1 −1
−1

0
2
0
1
1
1
1
1
−
−


√1  0
1 −1
1
1 −1 

√0 √0 √2 1 −1 −1
6 √
 2
0
0
0 
2
2
2 √0 √0 √0 √0 0


 0
0
0
0 2
2
2
2 √0 √0 √0 √0 
0
0
0
0 0
0
0
0 2
2
2
2

Gary Greaves — Equiangular lines in Euclidean spaces
9/23
Illustration of the proof
I
We have 12 lines in R7 ;

2
0
0
0 1
1
1
1 1
1
1
1
 0
2
0
0 1
1
1
1 −1 −1 −1 −1


 0
1 −1
−1

0
2
0
1
1
1
1
1
−
−


√1  0
1 −1
1
1 −1 

√0 √0 √2 1 −1 −1
6 √
 2
0
0
0 
2
2
2 √0 √0 √0 √0 0


 0
0
0
0 2
2
2
2 √0 √0 √0 √0 
0
0
0
0 0
0
0
0 2
2
2
2

Gary Greaves — Equiangular lines in Euclidean spaces
9/23
Illustration of the proof
I
We have 12 lines in R7 ;
I
We have 11 lines in R7 ;

2
0
0
0 1
1
1
1 1
1
1
1
 0
2
0
0 1
1
1
1 −1 −1 −1 −1


 0
1 −1
−1

0
2
0
1
1
1
1
1
−
−


√1  0
1 −1
1
1 −1 

√0 √0 √2 1 −1 −1
6 √
 2
0
0
0 
2
2
2 √0 √0 √0 √0 0


 0
0
0
0 2
2
2
2 √0 √0 √0 √0 
0
0
0
0 0
0
0
0 2
2
2
2

Gary Greaves — Equiangular lines in Euclidean spaces
9/23
Illustration of the proof
I
We have 12 lines in R7 ;
I
We have 11 lines in R6 ;
Left nullspace:
√ 
2
 0 

*
 0 +
 
 0 
 
 0 
 
 −1 
−1

2
0
0
0 1
1
1
1 1
1
1
1
 0
2
0
0 1
1
1
1 −1 −1 −1 −1


 0
1 −1
−1

−
−
0
2
0
1
1
1
1
1


√1  0
1 −1
1
1 −1 

√0 √0 √2 1 −1 −1
6 √
 2
0
0
0 
2
2
2 √0 √0 √0 √0 0


 0
0
0
0 2
2
2
2 √0 √0 √0 √0 
0
0
0
0 0
0
0
0 2
2
2
2

Gary Greaves — Equiangular lines in Euclidean spaces
9/23
Illustration of the proof
I
We have 12 lines in R7 ;
I
We have 11 lines in R6 ;
I
We have 8 lines in R6 ;
Left nullspace:
√ 
2
 0 

*
 0 +
 
 0 
 
 0 
 
 −1 
−1

2
0
0
0 1
1
1
1 1
1
1
1
 0
2
0
0 1
1
1
1 −1 −1 −1 −1


 0
1 −1
−1

−
−
0
2
0
1
1
1
1
1


√1  0
1 −1
1
1 −1 

√0 √0 √2 1 −1 −1
6 √
 2
0
0
0 
2
2
2 √0 √0 √0 √0 0


 0
0
0
0 2
2
2
2 √0 √0 √0 √0 
0
0
0
0 0
0
0
0 2
2
2
2

Gary Greaves — Equiangular lines in Euclidean spaces
9/23
Illustration of the proof
I
We have 12 lines in R7 ;
I
We have 11 lines in R6 ;
I
We have 8 lines in R5 ;
Left nullspace:
  √ 
0
2
0  0 



* 0  
+
   0 
0 ,  0 
   
1  0 
   
0  −1 
0
−1

2
0
0
0 1
1
1
1 1
1
1
1
 0
2
0
0 1
1
1
1 −1 −1 −1 −1


 0
1 −1
−1

−
−
0
2
0
1
1
1
1
1


√1  0
1 −1
1
1 −1 

√0 √0 √2 1 −1 −1
6 √
 2
0
0
0 
2
2
2 √0 √0 √0 √0 0


 0
0
0
0 2
2
2
2 √0 √0 √0 √0 
0
0
0
0 0
0
0
0 2
2
2
2

Gary Greaves — Equiangular lines in Euclidean spaces
9/23
Illustration of the proof
  √ 
0
2
0  0 



* 0  
+
   0 
0 ,  0 
   
1  0 
   
0  −1 
0
−1
I
We have 12 lines in R7 ;
I
We have 11 lines in R6 ;
Left nullspace:
I
We have 8 lines in R5 ;
<– Same as de Caen

2
0
0
0 1
1
1
1 1
1
1
1
 0
2
0
0 1
1
1
1 −1 −1 −1 −1


 0
1 −1
−1

0
2
0
1
1
1
1
1
−
−


√1  0
1 −1
1
1 −1 

√0 √0 √2 1 −1 −1
6 √
 2
0
0
0 
2
2
2 √0 √0 √0 √0 0


 0
0
0
0 2
2
2
2 √0 √0 √0 √0 
0
0
0
0 0
0
0
0 2
2
2
2

Gary Greaves — Equiangular lines in Euclidean spaces
9/23
Seidel matrices with few eigenvalues
Gary Greaves — Equiangular lines in Euclidean spaces
10/23
Seidel matrices with two distinct eigenvalues
Let L be an equiangular line system of n lines in Rd with
common angle α.
I If n meets either the absolute bound or the relative bound
then S(L) has precisely two distinct eigenvalues.
I
Related to strongly regular graphs (regular graphs with 3
eigenvalues).
Gary Greaves — Equiangular lines in Euclidean spaces
11/23
Seidel matrices with two distinct eigenvalues
Let L be an equiangular line system of n lines in Rd with
common angle α.
I If n meets either the absolute bound or the relative bound
then S(L) has precisely two distinct eigenvalues.
I
Related to strongly regular graphs (regular graphs with 3
eigenvalues).
Seidel (1995):
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
11/23
Seidel matrices with two distinct eigenvalues
Let L be an equiangular line system of n lines in Rd with
common angle α.
I If n meets either the absolute bound or the relative bound
then S(L) has precisely two distinct eigenvalues.
I
Related to strongly regular graphs (regular graphs with 3
eigenvalues).
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 50 61 76 96
Gary Greaves — Equiangular lines in Euclidean spaces
11/23
Seidel matrices with two distinct eigenvalues
Let L be an equiangular line system of n lines in Rd with
common angle α.
I If n meets either the absolute bound or the relative bound
then S(L) has precisely two distinct eigenvalues.
I
Related to strongly regular graphs (regular graphs with 3
eigenvalues).
d 2
n 3
3
6
4
6
5 6
10 16
2
2
2
2
2
7 – 13 14 15 16 17 18 19 20
28
28 36 40 48 48 72 90
30
42 50 61 76 96
2
2 2
Gary Greaves — Equiangular lines in Euclidean spaces
11/23
Seidel matrices with two distinct eigenvalues
Let L be an equiangular line system of n lines in Rd with
common angle α.
I If n meets either the absolute bound or the relative bound
then S(L) has precisely two distinct eigenvalues.
I
Related to strongly regular graphs (regular graphs with 3
eigenvalues).
d 2
n 3
3
6
4
6
5 6
10 16
2
2
2
3
2
2
7 – 13 14 15 16 17 18 19 20
28
28 36 40 48 48 72 90
30
42 50 61 76 96
2
2 2
3
3 3 3 3
Gary Greaves — Equiangular lines in Euclidean spaces
11/23
Seidel matrices with two distinct eigenvalues
Let L be an equiangular line system of n lines in Rd with
common angle α.
I If n meets either the absolute bound or the relative bound
then S(L) has precisely two distinct eigenvalues.
I
Related to strongly regular graphs (regular graphs with 3
eigenvalues).
d 2
n 3
3
6
4
6
5 6
10 16
2
2
2
3
2
2
7 – 13 14 15 16 17 18 19 20
28
28 36 40 48 48 72 90
30
42 50 61 76 96
2
2 2
3
3 3 3 3
4
Gary Greaves — Equiangular lines in Euclidean spaces
11/23
Seidel matrices with two distinct eigenvalues
Let L be an equiangular line system of n lines in Rd with
common angle α.
I If n meets either the absolute bound or the relative bound
then S(L) has precisely two distinct eigenvalues.
I
Related to strongly regular graphs (regular graphs with 3
eigenvalues).
d 2
n 3
3
6
4
6
5 6
10 16
2
2
2
3
2
2
7 – 13 14 15 16 17 18 19 20
28
28 36 40 48 48 72 90
29
41 50 61 76 96
2
2 2
3
3 3 3 3
4
Gary Greaves — Equiangular lines in Euclidean spaces
11/23
Seidel matrices with 3 eigenvalues: constructions
SRG construction
Let A be the adjacency matrix of an n-vertex strongly regular
graph with spectrum {[k]1 , [θ ]a , [τ ]b }.
I S = J − I − 2A has at most 3 distinct eigenvalues:
I
{[n − 1 − 2k]1 , [−1 − 2θ ]a , [−1 − 2τ ]b };
Gary Greaves — Equiangular lines in Euclidean spaces
12/23
Seidel matrices with 3 eigenvalues: constructions
SRG construction
Let A be the adjacency matrix of an n-vertex strongly regular
graph with spectrum {[k]1 , [θ ]a , [τ ]b }.
I S = J − I − 2A has at most 3 distinct eigenvalues:
I
{[n − 1 − 2k]1 , [−1 − 2θ ]a , [−1 − 2τ ]b };
Tensor construction
Let S be an n × n Seidel matrix with spectrum
{[λ0 ]n−d , [λ1 ]d }.
I Jm ⊗ (S − In ) + Inm has at most 3 distinct eigenvalues:
I
{[m(λ0 − 1) + 1]n−d , [1]n(m−1) , [m(λ1 − 1) + 1]d }.
Gary Greaves — Equiangular lines in Euclidean spaces
12/23
From two eigenvalues to three: coclique removal
Let S be an n × n Seidel matrix with spectrum
{[λ0 ]n−d , [λ1 ]d }.
Take a graph G in the switching class of S. For any vertex v
we have the following lemma.
Lemma
The Seidel matrix corresponding to G\{v} has spectrum
{[λ0 ]n−d−1 , [λ1 ]d−1 , [λ0 + λ1 ]1 }.
Gary Greaves — Equiangular lines in Euclidean spaces
13/23
From two eigenvalues to three: coclique removal
Let S be an n × n Seidel matrix with spectrum
{[λ0 ]n−d , [λ1 ]d }.
Take a graph G in the switching class of S. For any vertex v
we have the following lemma.
Lemma
The Seidel matrix corresponding to G\{v} has spectrum
{[λ0 ]n−d−1 , [λ1 ]d−1 , [λ0 + λ1
Gary Greaves — Equiangular lines in Euclidean spaces
]1
}.
13/23
From two eigenvalues to three: coclique removal
Let S be an n × n Seidel matrix with spectrum
{[λ0 ]n−d , [λ1 ]d }.
Suppose that a graph G in the switching class of S that
contains a coclique C on c 6 min(n − d, d) vertices.
Theorem (GG, AM, JK, FS 2015+)
The Seidel matrix corresponding to G\C has spectrum
{[λ0 ]n−d−c , [λ1 ]d−c , [λ0 + λ1 + 1 − c]1 , [λ0 + λ1 + 1]c−1 }.
Gary Greaves — Equiangular lines in Euclidean spaces
13/23
From two eigenvalues to three: coclique removal
Theorem (GG, AM, JK, FS 2015+)
The Seidel matrix corresponding to G\C has spectrum
{[λ0 ]n−d−c , [λ1 ]d−c , [λ0 + λ1 + 1 − c]1 , [λ0 + λ1 + 1]c−1 }.
Example (cf. Tremain 2008)
Start with Seidel matrix S with spectrum {[−5]21 , [7]15 }.
spec(S\K1 ) = {[−5]20 , [2]1 , [7]14 }
spec(S\K2 ) = {[−5]19 , [1]1 , [3]1 , [7]13 }
spec(S\K3 ) = {[−5]18 , [0]1 , [3]2 , [7]12 }
spec(S\K4 ) = {[−5]17 , [−1]1 , [3]3 , [7]11 }
spec(S\K5 ) = {[−5]16 , [−2]1 , [3]4 , [7]10 }
spec(S\K6 ) = {[−5]15 , [−3]1 , [3]5 , [7]9 }
spec(S\K7 ) = {[−5]14 , [−4]1 , [3]6 , [7]8 }
spec(S\K8 ) = {[−5]13 , [−5]1 , [3]7 , [7]7 }
Gary Greaves — Equiangular lines in Euclidean spaces
13/23
From two eigenvalues to three: coclique removal
Theorem (GG, AM, JK, FS 2015+)
The Seidel matrix corresponding to G\C has spectrum
{[λ0 ]n−d−c , [λ1 ]d−c , [λ0 + λ1 + 1 − c]1 , [λ0 + λ1 + 1]c−1 }.
Example (cf. Tremain 2008)
Start with Seidel matrix S with spectrum {[−5]21 , [7]15 }.
spec(S\K1 ) = {[−5]20 , [2]1 , [7]14 }
spec(S\K2 ) = {[−5]19 , [1]1 , [3]1 , [7]13 }
spec(S\K3 ) = {[−5]18 , [0]1 , [3]2 , [7]12 }
spec(S\K4 ) = {[−5]17 , [−1]1 , [3]3 , [7]11 }
spec(S\K5 ) = {[−5]16 , [−2]1 , [3]4 , [7]10 }
spec(S\K6 ) = {[−5]15 , [−3]1 , [3]5 , [7]9 }
spec(S\K7 ) = {[−5]14 , [−4]1 , [3]6 , [7]8 }
spec(S\K8 ) = {[−5]13 , [−5]1 , [3]7 , [7]7 } = {[−5]14 , [3]7 , [7]7 }
Gary Greaves — Equiangular lines in Euclidean spaces
13/23
From two eigenvalues to three: coclique removal
Theorem (GG, AM, JK, FS 2015+)
The Seidel matrix corresponding to G\C has spectrum
{[λ0 ]n−d−c , [λ1 ]d−c , [λ0 + λ1 + 1 − c]1 , [λ0 + λ1 + 1]c−1 }.
I
Resulting Seidel matrix has at most 3 eigenvalues when
c = λ1 + 1 or c = d.
I
Only two eigenvalues when c = d = λ1 + 1.
I
This technique is useful if we know something about the
size of a coclique in G.
Gary Greaves — Equiangular lines in Euclidean spaces
13/23
Properties of Seidel matrices with 3 eigenvalues
Theorem (GG, AM, JK, FS 2015+)
Let S be an n × n Seidel matrix with precisely 3 distinct
eigenvalues. Then S has a rational eigenvalue.
Furthermore, if n ≡ 3 mod 4 then every eigenvalue of S is
rational.
Gary Greaves — Equiangular lines in Euclidean spaces
14/23
Properties of Seidel matrices with 3 eigenvalues
Theorem (GG, AM, JK, FS 2015+)
Let S be an n × n Seidel matrix with precisely 3 distinct
eigenvalues. Then S has a rational eigenvalue.
Furthermore, if n ≡ 3 mod 4 then every eigenvalue of S is
rational.
Theorem (GG, AM, JK, FS 2015+)
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
Gary Greaves — Equiangular lines in Euclidean spaces
8
2
9
3
10
4
11
0
12
10
14/23
Strengthening the relative bound
Gary Greaves — Equiangular lines in Euclidean spaces
15/23
Strengthening the relative bound
Theorem (Relative bound)
Let L be an equiangular line system of n lines in Rd whose
Seidel matrix has smallest eigenvalue λ0 and suppose
λ20 > d + 2.
d(λ20 − 1)
.
n6
λ20 − d
Equality implies that S has 2 distinct eigenvalues.
Gary Greaves — Equiangular lines in Euclidean spaces
16/23
Strengthening the relative bound
Theorem (Relative bound)
Let L be an equiangular line system of n lines in Rd whose
Seidel matrix has smallest eigenvalue λ0 and suppose
λ20 > d + 2.
d(λ20 − 1)
.
n6
λ20 − d
Equality implies that S has 2 distinct eigenvalues.
GG, AM, JK, FS (2015+):
d(λ20 −1)
and there exists µ ∈ Z satisfying
Suppose n =
λ2 −d
0
certain conditions.
Then S has at most 4 distinct eigenvalues.
{[λ0 ]n−d , [µ − 1]a , [µ]b , [µ + 1]d−a−b }.
Gary Greaves — Equiangular lines in Euclidean spaces
16/23
Case study: equiangular lines in R14
I
Suppose there is n > 2 · 14 equiangular lines in R14 .
I
Neumann (1973): λ0 is an odd integer.
I
Lemmens and Seidel (1973): N3 (14) = 28.
I
Relative bound =⇒ λ0 = −5, N5 (14) 6 30.54 . . . .
I
Suppose we have n = 30 (d = 14), with corresponding
Seidel matrix S having eigenvalues
(n−d)
λ0
< λ1 6 λ2 6 · · · 6 λd .
Gary Greaves — Equiangular lines in Euclidean spaces
17/23
Case study: equiangular lines in R14
I
Lemmens and Seidel (1973): N3 (14) = 28.
I
Relative bound =⇒ λ0 = −5, N5 (14) 6 30.54 . . . .
I
Suppose we have n = 30 (d = 14), with corresponding
Seidel matrix S having eigenvalues
(n−d)
λ0
I
< λ1 6 λ2 6 · · · 6 λd .
Using the trace formulae, we have
d
∑ λi = −(n − d)λ0 = 80;
i=1
d
∑ λ2i = n(n − 1) − (n − d)λ20 = 470.
i=1
Gary Greaves — Equiangular lines in Euclidean spaces
17/23
Case study: equiangular lines in R14
I
Suppose we have n = 30 (d = 14), with corresponding
Seidel matrix S having eigenvalues
(n−d)
λ0
I
< λ1 6 λ2 6 · · · 6 λd .
Using the trace formulae, we have
d
∑ λi = −(n − d)λ0 = 80;
i=1
d
∑ λ2i = n(n − 1) − (n − d)λ20 = 470.
i=1
Then
d
1=
∑ (λi − 6)2 /d
i=1
Gary Greaves — Equiangular lines in Euclidean spaces
17/23
Case study: equiangular lines in R14
I
Suppose we have n = 30 (d = 14), with corresponding
Seidel matrix S having eigenvalues
(n−d)
λ0
I
< λ1 6 λ2 6 · · · 6 λd .
Using the trace formulae, we have
d
∑ λi = −(n − d)λ0 = 80;
i=1
d
∑ λ2i = n(n − 1) − (n − d)λ20 = 470.
i=1
Then
d
1=
∑ (λi − 6)2 /d >
i=1
Gary Greaves — Equiangular lines in Euclidean spaces
q
d
∏(λi − 6)2 > 1.
17/23
Case study: equiangular lines in R14
I
Suppose we have n = 30 (d = 14), with corresponding
Seidel matrix S having eigenvalues
(n−d)
λ0
I
< λ1 6 λ2 6 · · · 6 λd .
Using the trace formulae, we have
d
∑ λi = −(n − d)λ0 = 80;
i=1
d
∑ λ2i = n(n − 1) − (n − d)λ20 = 470.
i=1
Then
d
1=
∑ (λi − 6)2 /d >
i=1
q
d
∏(λi − 6)2 > 1.
Hence (λi − 6) ∈ {±1}.
Gary Greaves — Equiangular lines in Euclidean spaces
17/23
Relative bound in low dimensions
d
λ0
d(λ20 −1)
λ20 −d
14
15
16
17
18
19
20
−5
−5
−5
−5
−5
−5
−5
30
36
42
51
61
76
96
Spectrum
{[−5]16 , [5]9 , [7]5 }
{[−5]21 , [7]15 }
{[−5]26 , [7]7 , [9]9 }
{[−5]34 , [10]17 }
{[−5]43 , [11]9 , [12]1 , [13]8 }
{[−5]57 , [15]19 }
{[−5]76 , [19]20 }
Gary Greaves — Equiangular lines in Euclidean spaces
18/23
Relative bound in low dimensions
d
λ0
d(λ20 −1)
λ20 −d
14
15
16
17
18
19
20
−5
−5
−5
−5
−5
−5
−5
30
36
42
51
61
76
96
Spectrum
{[−5]16 , [5]9 , [7]5 }
{[−5]21 , [7]15 }
{[−5]26 , [7]7 , [9]9 }
{[−5]34 , [10]17 }
{[−5]43 , [11]9 , [12]1 , [13]8 }
{[−5]57 , [15]19 }
{[−5]76 , [19]20 }
Gary Greaves — Equiangular lines in Euclidean spaces
18/23
Relative bound in low dimensions
I
d
λ0
d(λ20 −1)
λ20 −d
14
15
16
17
18
19
20
−5
−5
−5
−5
−5
−5
−5
30
36
42
51
61
76
96
Spectrum
{[−5]16 , [5]9 , [7]5 }
{[−5]21 , [7]15 }
{[−5]26 , [7]7 , [9]9 }
{[−5]34 , [10]17 }
{[−5]43 , [11]9 , [12]1 , [13]8 }
{[−5]57 , [15]19 }
{[−5]76 , [19]20 }
Seidel matrices cannot have even eigenvalues with
multiplicity greater than 1.
2A = J − I − S.
Gary Greaves — Equiangular lines in Euclidean spaces
18/23
Relative bound in low dimensions
I
d
λ0
d(λ20 −1)
λ20 −d
14
15
16
17
18
19
20
−5
−5
−5
−5
−5
−5
−5
30
36
42
51
61
76
96
Spectrum
{[−5]16 , [5]9 , [7]5 }
{[−5]21 , [7]15 }
{[−5]26 , [7]7 , [9]9 }
{[−5]34 , [10]17 }
{[−5]43 , [11]9 , [12]1 , [13]8 }
{[−5]57 , [15]19 }
{[−5]76 , [19]20 }
Seidel matrices cannot have even eigenvalues with
multiplicity greater than 1.
2A = J − I − S.
Gary Greaves — Equiangular lines in Euclidean spaces
18/23
Euler graphs
An Euler graph is a graph each of whose vertices have even
valency.
Theorem (Mallows-Sloane 1975)
The number of switching classes of n × n Seidel matrices
equals the number of Euler graphs on n vertices.
Theorem (GG, AM, JK, FS 2015+)
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.
Gary Greaves — Equiangular lines in Euclidean spaces
19/23
Killing Seidel matrices with three eigenvalues
Theorem (GG, AM, JK, FS 2015+)
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.
Gary Greaves — Equiangular lines in Euclidean spaces
20/23
Killing Seidel matrices with three eigenvalues
Theorem (GG, AM, JK, FS 2015+)
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.
Gary Greaves — Equiangular lines in Euclidean spaces
20/23
Case study: equiangular lines in R14
I
Let S be the Seidel matrix with spectrum
{[−5]16 , [5]9 , [7]5 }.
I
The matrix M := S2 − 25I has spectrum {[0]25 , [24]5 }.
I
M = 28J − 24I − 2(AJ + JA) + 4(A2 + A) ≡ 0 mod 4.
I
M is positive semidefinite and
Mii = 4 =⇒ Mij ∈ {0, ±4}.
I
Deduce M is switching equivalent to 4J6 ⊗ I5 .
I
Evaluating [S3 ]11 in two different ways gives the bound.
Gary Greaves — Equiangular lines in Euclidean spaces
21/23
Conclusion
I
N (14) ∈ {28, 29} and N (16) ∈ {40, 41}.
I
I
Show that 30 (42) equiangular lines in R14 (R16 ) implies
that the associated Seidel matrix has spectrum
{[−5]16 , [5]9 , [7]5 } ({[−5]26 , [7]7 , [9]9 }).
Kill the corresponding Seidel matrices with three
eigenvalues.
I
Can we construct or kill other Seidel matrices with three
eigenvalues?
I
What about {[−5]43 , [11]9 , [12]1 , [13]8 }? (N (18) = 61?)
Gary Greaves — Equiangular lines in Euclidean spaces
22/23
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
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
Gary Greaves — Equiangular lines in Euclidean spaces
Existence
Y
N
?
Y
N
Y
?
?
?
?
?
Y
?
?
?
Remark
coclique removal?
from {[−5]57 , [15]19 }
from {[−5]76 , [19]20 }
23/23