Article contents
On the characters of the Sylow
$p$-subgroups of untwisted Chevalley groups
$Y_{n}(p^{a})$
Published online by Cambridge University Press: 01 October 2016
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
$\text{Irr}(UF_{4}(q))$.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © The Author(s) 2016
References
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