Associated Representations of finite pattern groups

TL;DR

This paper develops a method to construct and classify irreducible representations of finite pattern groups using an analogue of Kirillov's orbital method, providing explicit parameterizations and classifications.

Contribution

It introduces a construction based on Panov's associative polarization for finite pattern groups, enabling classification and parameterization of irreducible representations.

Findings

Classified irreducible representations of unipotent radicals of certain parabolic subgroups.

Parameterized irreducible characters of degree q via coadjoint orbits.

Extended the orbital method to finite pattern groups over finite fields.

Abstract

In this paper, we consider the construction of irreducible representations of finite pattern groups in terms of Panov's associative polarization, which is a finite-field analogue of Kirillov's orbital method. Using this construction, first, we are able to classify the irreducible representations of the unipotent radical of the standard parabolic subgroups of $\mathrm{GL}_n$ with 4 parts; second, we can parameterize irreducible characters of degree $q$ in terms of coadjoint orbits of cardinality $q^2$, for any finite pattern groups $G$ over $\mathbb{F}_q,$ where $\mathbb{F}_q$ is a finite field with $q$ elements.