A solution to Brauer's Problem 14

TL;DR

This paper addresses Brauer's Problem 14 by providing a group-theoretic method to count irreducible characters with Frobenius-Schur indicator 1, linking character theory to counting solutions of a specific group equation.

Contribution

It introduces a novel group-theoretic approach to count irreducible characters with Frobenius-Schur indicator 1 using solutions to a particular product of squares equation.

Findings

Counts solutions to $g_1^2\ldots g_n^2=1$ in G

Relates character indicators to group element equations

Provides a new perspective on character theory

Abstract

It is well known that the number of real irreducible characters of a finite group G coincides with the number of real conjugacy classes of G. Richard Brauer has asked if the number of irreducible characters with Frobenius-Schur indicator 1 can also be expressed in group theoretical terms. We show that this can done by counting solutions of $g_1^2\ldots g_n^2=1$ with $g_1,\ldots,g_n\in G$.