Associated Representations of finite pattern groups

TL;DR

The paper develops a finite-field analogue of Kirillov's orbit method for finite pattern groups using Panov's associative polarization, enabling explicit construction and classification of irreducible representations. It classifies irreps of the unipotent radical of standard parabolic subgroups with four blocks and shows that, for any finite pattern group over $\mathbb{F}_q$, irreducibles of degree $q$ correspond bijectively to coadjoint orbits of size $q^2$ via polarizations. A key result demonstrates that the unipotent radical $U_{n_1,n_2,n_3,n_4}$ is of good type, allowing irreps to be realized as inductions from linear characters associated with associative polarizations, and the number of degree-$q$ irreps matches the count of $q^2$-sized coadjoint orbits. Together, these findings extend the orbit method to a broad class of finite pattern groups and provide a concrete framework for counting and parameterizing degree-$q$ irreducibles in terms coadjoint orbits.

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.

Theorems & Definitions (22)

• Lemma 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 2.4
• Proposition 2.5
• Proposition 2.6
• Theorem 2.7
• Definition 2.8
• Theorem 2.9
• Definition 3.1