On a conjecture of Navarro and Tiep on character fields

TL;DR

This paper proves Navarro and Tiep's conjecture on character fields for certain infinite families of finite quasi-simple groups using roots of unity theory and character classification, advancing understanding of character fields.

Contribution

It develops new theory on sums of roots of unity and applies existing character classification to verify the conjecture for specific group families.

Findings

Proved the conjecture for some infinite families of groups

Established partial results for general and special linear groups

Connected roots of unity sums with character field properties

Abstract

In 2021, Navarro and Tiep proposed a conjecture on character fields of finite quasi-simple groups. We develop some theory on sums of roots of unity and use this theory to prove the conjecture for some infinite families of finite quasi-simple groups with known character table. We then use the classification of the irreducible complex characters of the finite general linear groups developed by Green to obtain some partial results about the conjecture for the finite general and special linear groups in arbitrary dimension.