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

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 \(\mathbb{Q}(χ)\) 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.

21 pages. Second version incorporating comments from Gabriel Navarro