Product Expansions of q-Character Polynomials

TL;DR

This paper studies the algebraic structure of q-character polynomials, revealing polynomial behaviors in q, explicit formulas for structure constants, and stability phenomena related to matrix statistics and representation theory of $Gl_n(F_q)$.

Contribution

It introduces universal polynomials governing the product structure of q-character polynomials and proves stability phenomena in matrix statistics and representation multiplicities.

Findings

Statistics are periodically polynomial in q.

Explicit formulas for structure constants $P_{\lambda,\mu}(q)$ and $R_{\lambda,\mu}^{ u}(q)$.

Stability phenomena in joint moments and tensor product multiplicities.

Abstract

The ring of q-character polynomials is a q-analog of the classical ring of character polynomials for the symmetric groups. This ring consists of certain class functions defined simultaneously on the groups $Gl_n(F_q)$ for all n, which we also interpret as statistics on matrices. Here we evaluate these statistics on all matrices and work towards computing the structure constants of the product in this ring. We show that the statistics are periodically polynomial in q, and governed by universal polynomials $P_{\lambda,\mu}(q)$ which we compute explicitly, indexed by pairs of integer partitions. The product structure is similarly polynomial in q in many cases, governed by polynomials $R_{\lambda,\mu}^{\nu}(q)$ indexed by triples of partitions, which we compute in some cases. Our calculations seem to exhibit several unexpected patterns. Mainly, we conjecture that certain indecomposable statistics generate the whole ring, and indeed prove this for statistics associated with matrices consisting of up to 2 Jordan blocks. Furthermore, the coefficients we compute exhibit surprising stability phenomena, which in turn reflect stabilizations of joint moments as well as multiplicities in the irreducible decomposition of tensor products of representations of $Gl_n(F_q)$ for $n\gg 1$. We use this stabilization to compute the correlation of the number of unipotent Jordan blocks of two sizes.