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. Gary Greaves — Equiangular lines in Euclidean spaces 2/13

Equiangular line systems ◮ Let L be a system of n lines spanned by v 1 , . . . , v n ∈ R d . ◮ L is equiangular if � v i , v i � = 1 and |� v i , v j �| = α ( α is called the common angle ). ◮ Problem: given d , what is the largest possible number N ( d ) of equiangular lines in R d ? Example ◮ An orthonormal basis: n = d and α = 0 . ◮ N ( d ) � d . Gary Greaves — Equiangular lines in Euclidean spaces 3/13

Seidel matrices Let L be an equiangular line system of n lines in R d 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 − 2 A . Gary Greaves — Equiangular lines in Euclidean spaces 4/13

Icosahedron o0 o1 i0 o5 i1 i5 i2 i4 o2 i3 o4 o3 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron o0 o1 o5 i0 i1 i5 i2 i4 i3 o2 o4 o3 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron o0 o1 o5 i0 i1 i5 i2 i4 i3 o2 o4 o3 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron o0 o1 o5 i0 i1 i5 i2 i4 i3 o2 o4 o3 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron o0 o1 o5 i0 i1 i5 i2 i4 i3 o2 o4 o3 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron o0 o1 o5 i0 i1 i5 i2 i4 i3 o2 o4 o3 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron o0 o1 o5 i0 i1 i5 i2 i4 i3 o2 o4 o3 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron o0 o1 o5 i0 i1 i5 i2 i4 i3 o2 o4 o3 Gary Greaves — Equiangular lines in Euclidean spaces 5/13

Icosahedron   0 1 1 1 1 1 − 1 − 1 1 0 1 1     − 1 − 1 1 1 0 1   S = ;   1 − 1 1 0 1 − 1     − 1 − 1 1 1 0 1   1 1 − 1 − 1 1 0 √ √ 5 ] 3 , [ 5 ] 3 } ; ◮ Spectrum: { [ − √ ◮ 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 R d with smallest eigenvalue λ 0 . ◮ Gerzon ’73: n � d ( d + 1 ) Absolute bound : . 2 √ ◮ van Lint and Seidel ’66: for λ 0 < − d + 2 n � d ( λ 2 0 − 1 ) Relative bound : 0 − d . λ 2 ◮ Neumann ’73: If n > 2 d then λ 0 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 R d 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 R d 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 R d 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 S 2 = n ( n − 1 ) ; ◮ det S ≡ ( − 1 ) n − 1 ( n − 1 ) mod 4 ; ◮ ( S − λ I )( S − θ I )( S − η I ) = 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 # 0 0 1 2 0 2 3 4 0 10 Gary Greaves — Equiangular lines in Euclidean spaces 8/13

30 equiangular lines in R 14 ? Let S be an n × n Seidel matrix with spectrum λ ( n − d ) < λ 1 � λ 2 � · · · � λ d . 0 Using the trace formulae, we have d ∑ λ i = − ( n − d ) λ 0 ; i = 1 d λ 2 i = n ( n − 1 ) − ( n − d ) λ 2 ∑ 0 . i = 1 Case: d = 14 , n = 30 , and λ 0 = − 5 . Set µ i = λ i − 6 . Then d � u 2 ∑ d ∏ u 2 1 = i / d � i � 1. i = 1 Hence u i ∈ {± 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 ) < λ 1 � λ 2 � · · · � λ d , 0 � � d ( λ 2 0 − 1 ) and suppose λ 2 0 � d + 2 . If n = and some λ 2 0 − d integrality condition and nonzero condition are satisfied. Then S has at most 3 distinct eigenvalues. ◮ 30 lines in R 14 { [ − 5 ] 16 , [ 5 ] 9 , [ 7 ] 5 } ; — ◮ 42 lines in R 16 { [ − 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 − 2 A where A is the adjacency matrix of an Euler graph. Gary Greaves — Equiangular lines in Euclidean spaces 11/13

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 | ν 2 − ( 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 | ν 2 − ( 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 Exist? λ µ ν [ − 5 ] 14 [ 3 ] 7 [ 7 ] 7 28 14 Y [ − 5 ] 16 [ 5 ] 9 [ 7 ] 5 30 14 N [ − 5 ] 24 [ 5 ] 6 [ 9 ] 10 40 16 ? [ − 5 ] 24 [ 7 ] 15 [ 15 ] 1 40 16 Y [ − 5 ] 26 [ 7 ] 7 [ 9 ] 9 42 16 N [ − 5 ] 31 [ 7 ] 8 [ 11 ] 9 48 17 Y [ − 5 ] 32 [ 9 ] 16 [ 16 ] 1 49 17 ? [ − 5 ] 30 [ 3 ] 6 [ 11 ] 12 48 18 ? [ − 5 ] 30 [ 7 ] 16 [ 19 ] 2 48 18 ? [ − 5 ] 36 [ 7 ] 9 [ 13 ] 9 54 18 ? [ − 5 ] 42 [ 11 ] 15 [ 15 ] 3 60 18 ? [ − 5 ] 53 [ 13 ] 16 [ 19 ] 3 72 19 Y [ − 5 ] 56 [ 10 ] 1 [ 15 ] 18 75 19 ? [ − 5 ] 70 [ 13 ] 5 [ 19 ] 15 90 20 ? [ − 5 ] 75 [ 14 ] 1 [ 19 ] 19 95 20 ? Gary Greaves — Equiangular lines in Euclidean spaces 13/13