Fields generated by characters of finite linear groups

TL;DR

This paper investigates the fields generated by character values of finite linear groups, extending previous results from alternating groups to general linear and special linear groups over finite fields.

Contribution

It generalizes the known results about fields generated by character values from alternating groups to finite general linear groups and special linear groups over finite fields.

Findings

Fields generated by characters of $ ext{GL}_m( extbf{F}_q)$ are characterized.

Extension of Thompson's speculation to linear groups is confirmed.

New techniques for analyzing character fields of finite linear groups are developed.

Abstract

In previous work, the authors confirmed the speculation of J. G. Thompson that certain multiquadratic fields are generated by specified character values of sufficiently large alternating groups $A_n$. Here we address the natural generalization of this speculation to the finite general linear groups $\mathrm{GL}_m\left(\mathbb{F}_q\right)$ and $\mathrm{SL}_2\left(\mathbb{F}_q\right)$.