Counting characters of small degree in upper unitriangular groups

TL;DR

This paper investigates the enumeration of irreducible characters of small degree in upper unitriangular groups over finite fields, providing recursive formulas for degrees 1, 2, and 3, and confirming Lehrer’s conjecture in these cases.

Contribution

It establishes recursive formulas for counting irreducible characters of degrees q, q^2, and q^3 in U_n, confirming Lehrer’s conjecture for these degrees.

Findings

Recursive formulas for degrees 1, 2, and 3

Confirmation of Lehrer’s conjecture for small degrees

Polynomial count of characters depending only on e and n

Abstract

Let $U_n$ denote the group of upper $n \times n$ unitriangular matrices over a fixed finite field $\mathbb{F}$ of order $q$. That is, $U_n$ consists of upper triangular $n \times n$ matrices having every diagonal entry equal to $1$. It is known that the degrees of all irreducible complex characters of $U_n$ are powers of $q$. It was conjectured by Lehrer that the number of irreducible characters of $U_n$ of degree $q^e$ is an integer polynomial in $q$ depending only on $e$ and $n$. We show that there exist recursive (for $n$) formulas that this number satisfies when $e$ is one of $1, 2$ and $3$, and thus show that the conjecture is true in those cases.