Orbit method for $p$-Sylow subgroups of finite classical groups

TL;DR

This paper develops a new orbit method-based approach to study the representation theory of p-Sylow subgroups of finite classical groups, introducing staircase orbits and decomposing supercharacters into orbit modules.

Contribution

It constructs a monomial module for these subgroups using a modified orbit method and classifies staircase orbits, providing a new framework for understanding their representations.

Findings

Constructed a monomial module isomorphic to the regular representation.

Classified staircase orbits of the U-action on the module.

Decomposed supercharacters into sums of orbit modules.

Abstract

For the $p$-Sylow subgroups $U$ of the finite classical groups of untwisted Lie type, $p$ an odd prime, we construct a monomial $\mathbb C U$-module $M$ which is isomorphic to the regular representation of $\mathbb C G$ by a modification of Kirillov's orbit method called monomial linearisation. We classify a certain subclass of orbits of the $U$-action on the monomial basis of $M$ consisting of so called staircase orbits and show, that every orbit module in $M$ is isomorphic to a staircase one. Finally we decompose the Andr\'e-Neto supercharacters of $U$ into a sum of $U$-characters afforded by staircase orbit modules contained in $M$.