Combinatorial methods of character enumeration for the unitriangular group

Abstract

Let UT n (q) denote the group of unipotent n × 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 ≤ e ≤ ⌊ n 2 ⌋⌊ n−1 2 ⌋, 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 ≤ 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 ≤ 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 ≤ 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). IntroductionLet F q be a finite field with q elements and write UT n (q) to denote the unitriangular group of n × n upper triangular matrices over F q with all diagonal entries equal to 1. This is a Sylow p-subgroup of the general linear group GL(n, F q ), where p > 0 is the characteristic of F q . This work concerns the problem of counting the irreducible characters of UT n (q).By a result of Isaacs [20], the degrees of all (complex) irreducible characters of UT n (q) are powers of q. In fact, Huppert [19] has shown that the set of integers occurring as degrees of irreducible characters of UT n (q) is {q e : 0 ≤ e ≤ M n }, whereOne may therefore define N n (q) and N n,e (q) for each positive integer n, prime power q > 1, and integer e, as the numbers N n (q) = the number of irreducible characters (also, of conjugacy classes) of UT n (q), N n,e (q) = the number of irreducible characters of UT n (q) of degree q e .

Iterative character constructions for algebra groups

“…We carry out the details of this application in the supplementary work [16], where we explicitly construct irreducible characters of UT n (q) with values in arbitrarily large cyclotomic fields. A second application of the results herein appears in the paper [17], in which we consider the problem of counting the irreducible characters of UT n (q) with a fixed degree and derive polynomial expressions for the number of irreducible characters UT n (q) of degree ≤ q 8 .…”

On the Coadjoint Orbits of Maximal Unipotent Subgroups of Reductive Groups

“…More recently, Isaacs has refined Lehrer's conjecture to say that k(U n (q), q d ) viewed as a polynomial in q −1 has positive integer coefficients. Further recent progress has also been, for example, in [7], A. Evseev verified that Isaacs' conjecture holds for n ≤ 13, and, in [27], E. Marberg showed that Isaacs' conjecture holds for d ≤ 8.…”

“…It is shown by Marberg in [28] that the formula for χ O above does not always give an irreducible character of U n (q). We refer the reader to the introduction of [27] for a discussion about this question for U n (q). We note that there has been related recent work regarding the parametrization of the complex characters for Sylow p-subgroups of finite groups of Lie type; see for example [16], [17] and [24].…”

Supercharacters of Unipotent and Solvable Groups

“…In the paper [26], it is verified that the number N D,e (q) can be calculated in terms of irreducible representations of certain algebra C D (q). Denote by Cr(D) the set of all positive roots (i, j) such that there exist the roots (i, k), (j, l) from D, where i < j < k < l (i.e., the roots (i, k) and (j, l) produce a crossing).…”