Counting characters of small degree in upper unitriangular groups

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.