Heisenberg characters, unitriangular groups, and Fibonacci numbers

Let U-n(F-q) denote the group of unipotent n x n upper triangular matrices over a finite field with q elements. We show that the Heisenberg characters of Un+i (F-q) are indexed by lattice paths from the origin to the line x y = n using the steps (1, 0), (1, 1), (0, 1), (0.2), which are labeled in a certain way by nonzero elements of F-q. In particular, we prove for n >= 1 that the number of Heisenberg characters of Un+1 (F-q) is a polynomial in q - 1 with nonnegative integer coefficients and degree n, whose leading coefficient is the nth Fibonacci number. Similarly, we find that the number of Heisenberg supercharacters of U-n(F-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 U-n(F-q) consisting of matrices whose superdiagonal entries sum to zero are likewise all polynomials in q - 1 with nonnegative integer coefficients. (C) 2011 Elsevier Inc. All rights reserved.