On the character degrees of a Sylow $p$-subgroup of a finite Chevalley group $G(p^f)$ over a bad prime

TL;DR

This paper characterizes the irreducible characters of Sylow p-subgroups of finite Chevalley groups over bad primes, revealing the existence of characters with degrees involving division by p, and provides explicit constructions for these characters.

Contribution

It offers a uniform parametrization of irreducible characters for type G2 and demonstrates the existence of characters with degrees divisible by p in all Chevalley groups over bad primes.

Findings

Existence of characters with degree q^n/p for bad primes p.

Explicit construction of such characters via inflation and induction.

Uniform parametrization for type G2 when p ≥ 5.

Abstract

Let $q$ be a power of a prime $p$ and let $U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G(q)$ defined over the field with $q$ elements. We first give a parametrization of the set $\text{Irr}(U(q))$ of irreducible characters of $U(q)$ when $G(q)$ is of type $\mathrm{G}_2$. This is uniform for primes $p \ge 5$, while the bad primes $p=2$ and $p=3$ have to be considered separately. We then use this result and the contribution of several authors to show a general result, namely that if $G(q)$ is any finite Chevalley group with $p$ a bad prime, then there exists a character $\chi \in \text{Irr}(U(q))$ such that $\chi(1)=q^n/p$ for some $n \in \mathbb{Z}_{\ge_0}$. In particular, for each $G(q)$ and every bad prime $p$, we construct a family of characters of such degree as inflation followed by an induction of linear characters of an abelian subquotient $V(q)$ of $U(q)$.