New results on equiangular lines or How I caught a gold fish? - PowerPoint PPT Presentation

New results on equiangular lines or ‘How I caught a gold fish?’ Ferenc Szöll˝ osi szoferi@gmail.com Department of Communications and Networking, Aalto University Talk at CSD8 2017 Mons Ferenc Szöll˝ osi (ComNet, Aalto University) Equiangular lines CSD8, 2017/08/23 1 / 32

Overview This talk covers some of my old and new results on equiangular lines. An overview, based on Equiangular lines in Euclidean spaces , arXiv:1402.6429 (jointly with Greaves, Koolen, and Munemasa) There are no 21 real equiangular lines in R 12 (with α = 1 / 5), based on Enumeration of Seidel matrices , arXiv:1703.02943 (jointly with Östergård) There are at least 54 real equiangular lines in R 18 , based on A remark on a construction of D.S. Asche , arXiv:1703.04505 � There are no 8 complex equiangular lines with angle 5 / 21, based on All complex equiangular tight frames in dimension 3, arXiv:1402.6429 Ferenc Szöll˝ osi (ComNet, Aalto University) Equiangular lines CSD8, 2017/08/23 2 / 32

Real equiangular lines Definition A set of n lines, spanned by the unit vectors v 1 , v 2 , . . . , v n ∈ R d is called equiangular, if there is a common angle α such that � � | = α , i � = j ∈ { 1 , . . . , n } . | v i , v j Examples: The n vectors of an orthonormal basis in R n (with α = 0); n = 3 vectors in R 2 formed by rotation of 120 degrees (with α = 1 / 2), aka ‘the complex 3rd roots of unity’; n = 6 lines passing through the antipodal vertices of the √ icosahedron in R 3 (with α = 1 / 5). Problem For d ≥ 2 fixed, what is the maximum number of real equiangular lines, N ( d ) ? Ferenc Szöll˝ osi (ComNet, Aalto University) Equiangular lines CSD8, 2017/08/23 3 / 32

Historical remarks Regarding the real case Research initiated by Haantjes 1948 Seidel and coauthors 1966–1995 Conway and Taylor 1972–1992 Makhnev 2002; and Bannai, Munemasa, Venkov 2005 Renewed interest in the complex case (quantum tomography) Godsil, Roy 2005 Appleby, Zauner 2000s Scott, Grassl 2010s This used to be a “sleepy field” (with very little activity) because it was believed that [at least in the real case] “things have already been completed in the ’70s (and what is left open is hopeless anyways), see reference [X], and/or ask Prof. Y about it”. The sleeper, however, has awakened! (Recent explosion in terms of the number of new papers). Ferenc Szöll˝ osi (ComNet, Aalto University) Equiangular lines CSD8, 2017/08/23 4 / 32

Representing the lines We assume α > 0 (we exclude the case of the o.n.b) The unit vectors v i ∈ R d can be represented by column vectors (w.r.t. a fixed basis). F = [ v 1 , . . . , v n ] is the frame operator. � � The Gram matrix G := F T F ( [ G ] i , j := ). v i , v j The Seidel matrix S := ( G − I ) /α ( | S i , j | = 1 for i � = j and S i , i = 0). The adjacency matrix A = ( J − S − I ) / 2 of the ambient graph. Note: If v i is replaced by − v i we obtain the same configuration. Permutation of the vectors within themselves yields the same configuration. Actually, we consider the equivalence classes of these objects. Definition Any two configurations { v 1 , . . . , v n } and {± v σ ( 1 ) , . . . , ± v σ ( n ) } are called equivalent ( σ ∈ S n ). Ferenc Szöll˝ osi (ComNet, Aalto University) Equiangular lines CSD8, 2017/08/23 5 / 32

n = 6 vectors in R 4 , α = 1 / 3 Assume that v 1 , v 2 , v 3 , and v 4 are linearly independent.   1 1 1 1 1 1   3 3 3 3 3       √ √ √ √ √   2 2 2 2 2 2 0 − − − −   3 3 3 3 3   F = [ v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ] :=     √ √ 6 6  0 0 0 0  − 3 3       √ √ 6 6 0 0 0 0 − 3 3 F is the frame operator, a short-fat matrix. In most cases it is inconvenient to work with F . Basis dependent representation. F is d × n . Column permutation and column negation does not change the line system what F represents. In this representation n , d , and α are visible. Ferenc Szöll˝ osi (ComNet, Aalto University) Equiangular lines CSD8, 2017/08/23 6 / 32

Gram matrices   1 1 1 1 1 1 3 3 3 3 3     1 − 1 − 1 − 1 − 1 1   3 3 3 3 3     1 − 1 1 − 1 1 1   3 3 3 3 3 G = F T F =     1 − 1 1 1 − 1 1   3 3 3 3 3     1 − 1 − 1 1 1   1  3 3 3 3 3  1 − 1 1 − 1 1 1 3 3 3 3 3 G is n × n , positive semidefinite. | G i , j | = α , G i , i = 1. rank ( G ) = d (the dimension is no longer visible). F is essentially the Cholesky-decomposition of G . Since the vectors of F live in R 4 , we have rank ( G ) = 4. In particular, there is some 4 × 4 nonvanishing principal minor. det ( M 4 ) = 32 / 81 > 0, where M 4 is the leading 4 × 4 principal minor. Ferenc Szöll˝ osi (ComNet, Aalto University) Equiangular lines CSD8, 2017/08/23 7 / 32

Seidel matrices   0 1 1 1 1 1  1 0 − 1 − 1 − 1 − 1      1 − 1 0 1 − 1 1   S = ( G − I ) /α =   1 − 1 1 0 1 − 1     1 − 1 − 1 1 0 1 1 − 1 1 − 1 1 0 S is of n × n . S = S T . S ii = 0, S ij = ± 1 for i � = j . Neither the dimension d nor the common angle α is visible. Lemma The smallest eigenvalue of S , λ min ( S ) = − 1 /α of multiplicity n − d . Proof: Recall, that rank ( G ) = d , so G has exactly n − d eigenvalues 0 (and all other eigenvalues are positive). Then G − I has exactly n − d eigenvalues − 1, and finally S = ( G − I ) /α has exactly n − d eigenvalues − 1 /α . This is its smallest eigenvalue. Example: Λ( S ) = {− 3 , − 3 , 1 , 1 , 1 , 3 } = { [ − 3 ] 2 , [ 1 ] 3 , [ 3 ] 1 } . Ferenc Szöll˝ osi (ComNet, Aalto University) Equiangular lines CSD8, 2017/08/23 8 / 32

Graphs Let J be the matrix with J ij = 1.   0 0 0 0 0 0  0 0 1 1 1 1      0 1 0 0 1 0   A = ( J − S − I ) / 2 =   0 1 0 0 0 1     0 1 1 0 0 0 0 1 0 1 0 0 A is of n × n A = A T A ii = 0, A ij ∈ { 0 , 1 } . The matrix A is called the adjacency matrix of the ambient graph (or underlying graph) Γ( S ) of the Seidel matrix S . Remark: Γ depends on the concrete representation of S . Remark: there is no (clear) correspondence between the eigenvalues of A , and the eigenvalues of S . √ Example: Λ( A ) = { [ − 1 ] 2 , [ 0 ] 1 , [ 1 ] 1 , [( 1 ± 17 ) / 2 ] 1 } . Ferenc Szöll˝ osi (ComNet, Aalto University) Equiangular lines CSD8, 2017/08/23 9 / 32

Upper bounds Lemma[Gerzon’s bound, 1971] Let d ≥ 2. Then N ( d ) ≤ d ( d + 1 ) / 2. Proof: counting argument. The right hand side is the dimension of the real vectorspace of real symmetric matrices. If there is equality, then d = 2 , 3 or ( d + 2 ) is the square of an odd integer. For d = 2 , 3 , 7 , 23 we do have equality. For d = 47 we don’t have! (Makhnev, 2002) Example[ n = 28 lines in R 7 , α = 1 / 3] Let v = [ − 3 , − 3 , 1 , 1 , 1 , 1 , 1 , 1 ] ∈ R 8 , and let V := { σ ( v ): σ ∈ S 8 } . √ � 8 � = 28, and � v , σ ( v ) � = ± 8. Normalize by 24 to get unit | V | = 2 vectors. Observe that � σ ( v ) , [ 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ] � = 0, so we are in R 7 . Ferenc Szöll˝ osi (ComNet, Aalto University) Equiangular lines CSD8, 2017/08/23 10 / 32

Upper bounds (ctd.) Lemma[Relative bound] √ d + 2 then n ≤ d ( 1 − α 2 ) / ( 1 − d α 2 ) . If α ≤ 1 / When n > 2 d then α is 1 / ( 2 k + 1 ) for some k ∈ N . √ The condition α ≤ 1 / d + 2 is essential. Equality iff the Seidel matrix has exactly two distinct eigenvalues. √ The interesting cases are usually those where α > 1 / d + 2. Message: when the angle is “too” small, then there are not so many lines. For maximizing n in R d , the choice of α is crucial. Example[ n = 16 lines in R 6 with α = 1 / 3] Let d = 6, and assume that n > 2 d = 12. Then 1 /α ∈ { 3 , 5 , . . . } and in √ 6 + 2. Therefore n ≤ 6 ( 1 − α 2 )( 1 − 6 α 2 ) = 16. Take particular α ≤ 1 / a symmetric Hadamard matrix H with constant diagonal − 1 of order 16. It has spectrum Λ( H ) = { [ − 4 ] 10 , [ 4 ] 6 } . Then consider S := H + I . Ferenc Szöll˝ osi (ComNet, Aalto University) Equiangular lines CSD8, 2017/08/23 11 / 32

Upper bounds (ctd.) There are other ad-hoc bounds for (i) small d ; and for (ii) specific values of α (Barg, King, Okuda with coauthors). Lemma[Lemmens–Seidel, 1973] Let α = 1 / 3, and d ≥ 15. Then n ≤ 2 ( d − 1 ) . Proof: Lengthy, yet elementary argument. Proposition[Neumaier, 1989] Let α = 1 / 5, and d large. Then n ≤ ⌊ 3 ( d − 1 ) / 2 ⌋ . Proof: Graph theory (correspondence with Dynkin graphs). Theorem[Bukh, 2015] If α is fixed, then n ≤ c ( α ) · d . Proof: Clever probablistic argument. Message: when the angle is “too” large, then there are not so many lines either. Ferenc Szöll˝ osi (ComNet, Aalto University) Equiangular lines CSD8, 2017/08/23 12 / 32