Combinatorial methods of character enumeration for the unitriangular group

TL;DR

This paper confirms a conjecture about the polynomial nature of the count of irreducible characters of the unitriangular group with degrees as powers of q, providing explicit formulas for degrees up to 8.

Contribution

It introduces an algorithm to compute explicit polynomial formulas for the number of irreducible characters of T_n(q) with degree q^e for e  8, confirming a conjecture in this range.

Findings

Number of irreducible characters with degree q^e is polynomial in q-1 for e  8.

Explicit bivariate polynomials in n and q are derived for character counts.

All irreducible characters with degree  q^8 are Kirillov functions.

Abstract

Let $\UT_n(q)$ denote the group of unipotent $n\times n$ upper triangular matrices over a field with $q$ elements. The degrees of the complex irreducible characters of $\UT_n(q)$ are precisely the integers $q^e$ with $0\leq e\leq \lfloor \frac{n}{2} \rfloor \lfloor \frac{n-1}{2} \rfloor$, and it has been conjectured that the number of irreducible characters of $\UT_n(q)$ with degree $q^e$ is a polynomial in $q-1$ with nonnegative integer coefficients (depending on $n$ and $e$). We confirm this conjecture when $e\leq 8$ and $n$ is arbitrary by a computer calculation. In particular, we describe an algorithm which allows us to derive explicit bivariate polynomials in $n$ and $q$ giving the number of irreducible characters of $\UT_n(q)$ with degree $q^e$ when $n>2e$ and $e\leq 8$. When divided by $q^{n-e-2}$ and written in terms of the variables $n-2e-1$ and $q-1$, these functions are actually bivariate polynomials with nonnegative integer coefficients, suggesting an even stronger conjecture concerning such character counts. As an application of these calculations, we are able to show that all irreducible characters of $\UT_n(q)$ with degree $\leq q^8$ are Kirillov functions. We also discuss some related results concerning the problem of counting the irreducible constituents of individual supercharacters of $\UT_n(q)$.