Weyl Tensors, Strongly Regular Graphs, Multiplicative Characters, and a Quadratic Matrix Equation

TL;DR

This paper investigates solutions to a quadratic matrix equation linked to Riemannian geometry, revealing connections with strongly regular graphs, group rings, and multiplicative characters of finite fields.

Contribution

It constructs explicit solutions to a specific quadratic matrix equation and establishes novel links between these solutions and combinatorial and algebraic structures.

Findings

Constructed nonzero solutions of the quadratic matrix equations.

Linked solutions to strongly regular graphs and multiplicative characters.

Provided insights into the algebraic and geometric implications of the solutions.

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.