Character values and conductors of low-rank groups of Lie type

TL;DR

This paper investigates the conductors and field of values of irreducible characters of certain low-rank finite groups of Lie type, revealing that conductors are often realized at a single group element.

Contribution

It demonstrates that for specific rank 1 finite groups of Lie type, the conductor of a character is achieved at a single element, and the field of values can be generated by one value in some cases.

Findings

Conductor c(χ) is realized at a single group element for certain rank 1 groups.

In some cases, the field of values is generated by a single character value.

The results relate to a conjecture of W. Feit and extend understanding of character fields.

Abstract

Let $\chi$ be a complex irreducible character of a finite group $G$. The conductor of $\chi$, denoted $c(\chi)$, is the smallest positive integer $n$ such that $\chi(x)\in \mathbb{Q}(\exp({2\pi i/n}))$ for all $x\in G$. We show that for certain rank $1$ finite groups of Lie type, the conductor $c(\chi)$ is realized at a single group element; that is, there exists $g\in G$ such that $c(\chi)=c(\chi(g))$. In some quasisimple cases, we further prove that the field of values \(\mathbb{Q}(\chi)\) is generated by a single value. This phenomenon, which is related to a well-known conjecture of W.~Feit, was recently observed by Boltje \emph{et al.} in their reduction of the conjecture to finite simple groups. Our approach uses techniques from algebraic number theory together with the known character tables of these groups.