On the characters of the Sylow p-subgroups of untwisted Chevalley groups Y_n(p^a)

TL;DR

This paper classifies irreducible characters of Sylow p-subgroups of untwisted Chevalley groups, focusing on those indexed by single roots, and establishes a correspondence with linear characters of certain subquotients, aiding future classifications.

Contribution

It introduces a novel parametrization of irreducible characters based on root antichains and constructs a correspondence for characters indexed by roots, extending previous supercharacter frameworks.

Findings

Established a one-to-one correspondence for minimal degree characters and linear characters of subquotients.

Recovered elementary supercharacters for type A groups.

Laid groundwork for classifying characters of other Sylow subgroups.

Abstract

Let $UY_n(q)$ be a Sylow p-subgroup of an untwisted Chevalley group $Y_n(q)$ of rank n defined over $\mathbb{F}_q$ where q is a power of a prime p. We partition the set $Irr(UY_n(q))$ of irreducible characters of $UY_n(q)$ into families indexed by antichains of positive roots of the root system of type $Y_n$. We focus our attention on the families of characters of $UY_n(q)$ which are indexed by antichains of length 1. Then for each positive root $\alpha$ we establish a one to one correspondence between the minimal degree members of the family indexed by $\alpha$ and the linear characters of a certain subquotient $\overline{T}_\alpha$ of $UY_n(q)$. For $Y_n = A_n$ our single root character construction recovers amongst other things the elementary supercharacters of these groups. Most importantly though this paper lays the groundwork for our classification of the elements of $Irr(UE_i(q))$, $6 \le i \le 8$ and $Irr(UF_4(q))$.