Classification of Sylow classes of parabolic and reflection subgroups in unitary reflection groups

TL;DR

This paper classifies minimal parabolic and reflection subgroups containing an -Sylow subgroup in finite unitary reflection groups, aiding the understanding of their -Sylow structures and connections to modular representation theory.

Contribution

It provides a classification of minimal parabolic and reflection subgroups containing -Sylow subgroups, enhancing understanding of subgroup structures in unitary reflection groups.

Findings

Classification of minimal -Sylow containing subgroups

Most such subgroups are the entire parabolic subgroup

Connection established with modular representation theory

Abstract

Let $\ell$ be a prime divisor of the order of a finite unitary reflection group. We classify up to conjugacy the parabolic and reflection subgroups that are minimal with respect to inclusion, subject to containing an $\ell$-Sylow subgroup. The classification assists in describing the $\ell$-Sylow subgroups of unitary reflection groups up to group isomorphism. This classification also relates to the modular representation theory of finite groups of Lie type. We observe that unless a parabolic subgroup minimally containing an $\ell$-Sylow subgroup is $G$ itself, the reflection subgroup within the parabolic minimally containing an $\ell$-Sylow subgroup is the whole parabolic subgroup.