On the characters of the Sylow -subgroups of untwisted Chevalley groups

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$\text{Irr}(UY_{n}(q))$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$\unicode[STIX]{x1D6FC}$we establish a one-to-one correspondence between the minimal degree members of the family indexed by$\unicode[STIX]{x1D6FC}$and the linear characters of a certain subquotient$\overline{T}_{\unicode[STIX]{x1D6FC}}$of$UY_{n}(q)$. For$Y_{n}=A_{n}$our single root character construction recovers, among other things, the elementary supercharacters of these groups. Most importantly, though, this paper lays the groundwork for our classification of the elements of$\text{Irr}(UE_{i}(q))$,$6\leqslant i\leqslant 8$, and$\text{Irr}(UF_{4}(q))$.