Heisenberg characters, unitriangular groups, and Fibonacci numbers

TL;DR

This paper explores the combinatorial structure of Heisenberg characters and supercharacters of upper triangular unipotent groups over finite fields, revealing connections to Fibonacci, Pell, and Delannoy numbers through polynomial enumeration.

Contribution

It introduces a new combinatorial indexing of Heisenberg characters using lattice paths and establishes polynomial formulas involving well-known number sequences.

Findings

Number of Heisenberg characters is a polynomial in q-1 with Fibonacci leading coefficient.

Number of supercharacters involves Delannoy numbers, providing a q-analogue of Pell numbers.

Counts of various characters are polynomials with nonnegative integer coefficients.

Abstract

Let $\UT_n(\FF_q)$ denote the group of unipotent $n\times n$ upper triangular matrices over a finite field with $q$ elements. We show that the Heisenberg characters of $\UT_{n+1}(\FF_q)$ are indexed by lattice paths from the origin to the line $x+y=n$ using the steps $(1,0), (1,1), (0,1), (1,1)$, which are labeled in a certain way by nonzero elements of $\FF_q$. In particular, we prove for $n\geq 1$ that the number of Heisenberg characters of $\UT_{n+1}(\FF_q)$ is a polynomial in $q-1$ with nonnegative integer coefficients and degree $n$, whose leading coefficient is the $n$th Fibonacci number. Similarly, we find that the number of Heisenberg supercharacters of $\UT_n(\FF_q)$ is a polynomial in $q-1$ whose coefficients are Delannoy numbers and whose values give a $q$-analogue for the Pell numbers. By counting the fixed points of the action of a certain group of linear characters, we prove that the numbers of supercharacters, irreducible supercharacters, Heisenberg supercharacters, and Heisenberg characters of the subgroup of $\UT_n(\FF_q)$ consisting of matrices whose superdiagonal entries sum to zero are likewise all polynomials in $q-1$ with nonnegative integer coefficients.