On the number of irreducible representations for finite unipotent Heisenberg-type groups

TL;DR

This paper develops a method to count irreducible representations of certain unipotent groups, showing the count is a polynomial in q-1 with nonnegative coefficients, confirming Isaacs' conjecture.

Contribution

It introduces a way to compute the number of irreducible representations for generalized Heisenberg groups and related subgroups, extending the orbit method approach.

Findings

Number of irreducible representations is a polynomial in q-1.

The polynomial has nonnegative integer coefficients.

Results confirm Isaacs' conjecture for these groups.

Abstract

Let $U$ be an algebraic subgroup of the group of $n\times n$ upper-triangular matrices with units on the diagonal over a finite field of large enough characteristic, and $\mathfrak{n}$ be the Lie algebra of $U$. The main tool in representation theory of $U$ is the orbit method, which classifies irreducible representations of the group $U$ in terms of coadjoint orbits on the dual space $\mathfrak{n}^*$. We consider two types of generalizations of the Heisenberg group, namely, generalized Heisenberg groups defined with an arbitrary bilinear form, and certain subgroups in maximal unipotent subgroups of classical orthogonal algebraic groups. We provide a way to calculate the number of irreducible representations of such groups. It turned out that this number is a polynomial in $q-1$ with nonnegative integer coefficients, which agrees with Isaacs' conjecture.