Spectral Proofs of Maximality of Some Seidel Matrices

Abstract

A set of lines through the origin in a Euclidean space is equiangular if any pair from these lines forms the same angle. The problem to determine the maximum cardinality NðdÞ (d 2 Z 2) of a set of equiangular lines in R dates back to the result of Haantjes [8]. Also, the value NðdÞ is known for every d 17 (see [5, Table 1]). Some lower bounds of the values NðdÞ are given by constructing sets of equiangular lines for larger values of d. We are interested in whether the bounds can be improved, and in particular we will check whether some sets of equiangular lines can be extended. Lin and Yu [9, 10] defined a set X of equiangular lines of rank r to be saturated if there is no line l 62 X such that the union X [ flg is a set of equiangular lines of rank r. Here, the rank of a set of equiangular lines is the smallest dimension of Euclidean spaces into which these lines are isometrically embedded. By using a computer implementing their algorithm [10, p. 274], they verified in [10, Theorem 1 and the end of Sect. 3.2] that seven sets of equiangular lines are saturated. Their algorithm requires the computation of the clique numbers of graphs, which is known to be an NPcomplete problem. We will verify their results by investigating spectra, without a computer. We introduce Seidel matrices in connection with equiangular lines. A Seidel matrix is a symmetric matrix with zero diagonal and all off-diagonal entries 1. Note that if a Seidel matrix S has largest eigenvalue , then there exist vectors whose Gram matrix equals I S, which span a set of equiangular lines with common angle arccosð1= Þ. Cao, Koolen, Munemasa and Yoshino [2] defined a Seidel matrix S with largest eigenvalue to be maximal if there is no Seidel matrix S0 containing S as a proper principal submatrix with largest eigenvalue such that rankð I SÞ 1⁄4 rankð I S0Þ. In other words, the Seidel matrix obtained from a saturated set of equiangular lines is maximal. In this paper, we prove Theorem 1.1, which shows maximality of Seidel matrices with spectra in Table 1, with only the aid of spectra instead of a computer. Specifically, we use Cauchy’s interlacing theorem and the angles of matrices, which are used by Greaves and Yatsyna [6] in order to show some Seidel spectra do not exist. This method enables us to simultaneously verify maximality of some Seidel matrices having common spectra. For example, Szöll} osi and Östergård in [11, Theorem 5.2] showed that there exist, up to switching equivalence, at least 1045 Seidel matrices of order 28 with spectrum f1⁄25 ; 1⁄2 3 ; 1⁄2 7 g. Actually, Theorem 1.1 implies that these Seidel matrices are maximal.