Representations of finite pattern groups

TL;DR

This paper establishes a natural correspondence between coadjoint orbits and irreducible representations of finite pattern groups over finite fields, providing explicit constructions for these representations.

Contribution

It introduces an explicit method to construct irreducible representations of finite pattern groups via coadjoint orbits and subgroup induction, clarifying their classification.

Findings

Bijection between coadjoint orbits and irreducible representations

Explicit construction of irreducible representations using subgroup induction

Classification of representations based on coadjoint orbit structure

Abstract

Let $G=1+A$ be a finite pattern group over the finite field ${\mathbb{F}}_q$. We give a natural bijection between coadjoint orbits of $G$ and its equivalent classes of irreducible representations. More precisely, given any $T\in A^t$, viewed as a representative of associated coadjoint orbit ${\mathfrak{O}}_T$ of $G$, we can explicitly construct a subgroup $H_T $ of $G$, such that ${\mathrm{Ind}}_{H_T}^G \psi_T$ is irreducible and ${\mathrm{Ind}}_{H_T}^G \psi_T \cong {\mathrm{Ind}}_{H_{T'}}^G \psi_{T'}$ if and only if $T$ and $ T'$ are in the same coadjoint orbit. Here $\psi_T(x)=\psi({\mathrm{tr}} Tx)\text{ for }x\in H_T,$ and $\psi$ is a fixed nontrivial additive character of ${\mathbb{F}}_q$.