Weyl Tensors, Strongly Regular Graphs, Multiplicative Characters, and a Quadratic Matrix Equation
Abstract
We study solutions of a quadratic matrix equation arising in Riemannian geometry. Let $S$ be a real symmetric $n\times n$matrix with zeros on the diagonal and let $\theta$ be a real number. We construct nonzero solutions $(S,\theta)$ of the set of quadratic equations \[\sum_kS_{i,k}=0\quad\text{ and }\quad\sum_{k}S_{i,k}S_{k,j}+S_{i,j}^2=\theta S_{i,j}\text { for }i<j.\] Our solutions relate the equations to strongly regular graphs, to group rings, and to multiplicative characters of finite fields.
 Publication:

arXiv eprints
 Pub Date:
 April 2022
 DOI:
 10.48550/arXiv.2204.02706
 arXiv:
 arXiv:2204.02706
 Bibcode:
 2022arXiv220402706D
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Combinatorics;
 Mathematics  Differential Geometry;
 Mathematics  Number Theory;
 05C;
 53C;
 11T
 EPrint:
 v2: revised, expanded and corrected v3: revised, minor corrections, to appear in J. Algebra