Tensor products of unipotent characters of general linear groups over finite fields

TL;DR

This paper investigates the multiplicities of unipotent characters in tensor products of GL(n,q), proving they are polynomials with non-negative coefficients, and characterizing when these polynomials are non-zero.

Contribution

It establishes that these multiplicities are polynomials in q with non-negative coefficients and provides a symmetric group representation criterion for their non-vanishing.

Findings

Multiplicities are polynomials in q with non-negative integer coefficients.

A necessary and sufficient condition for non-zero polynomials is given in terms of symmetric group representations.

The degree of these polynomials is analyzed and characterized.

Abstract

We study multiplicities of unipotent characters in tensor products of unipotent characters of GL(n,q). We prove that these multiplicities are polynomials in q with non-negative integer coefficients. We study the degree of these polynomials and give a necessary and sufficient condition in terms of the representation theory of symmetric groups for these polynomials to be non-zero.