Ranks for Representations of GL(n) Over Finite Fields, their Agreement, and Positivity of Fourier Transform

TL;DR

This paper proves that two notions of rank for irreducible representations of GL(n) over finite fields agree for large q, using Fourier analysis and positivity properties of matrix rank sets.

Contribution

It establishes the agreement of Fourier theoretic and algebraic rank notions for low-rank representations of GL(n) over finite fields, confirming a conjecture for large q.

Findings

Proof of the rank agreement conjecture for large q

Identification of positivity properties of Fourier transforms of low-rank matrices

Application of rank theory to counting problems in finite fields

Abstract

In [Frobenius1896] it was shown that many important properties of a finite group could be examined using formulas involving the character ratios of group elements, i.e., the trace of the element acting in a given irreducible representation, divided by the dimension of the representation. In [Gurevich-Howe15] and [Gurevich-Howe17], the current authors introduced the notion of rank of an irreducible representation of a finite classical group. One of the motivations for studying rank was to clarify the nature of character ratios for certain elements in these groups. In fact in the above cited papers, two notions of rank were given. The first is the Fourier theoretic based notion of U-rank of a representation, which comes up when one looks at its restrictions to certain abelian unipotent subgroups. The second is the more algebraic based notion of tensor rank which comes up naturally when one attempts to equip the representation ring of the group with a grading that reflects the central role played by the few "smallest" possible representations of the group. In [Gurevich-Howe17] we conjectured that the two notions of rank mentioned just above agree on a suitable collection called "low rank" representations. In this note we review the development of the theory of rank for the case of the general linear group GL_n over a finite field F_q, and give a proof of the "agreement conjecture" that holds true for sufficiently large q. Our proof is Fourier theoretic in nature, and uses a certain curious positivity property of the Fourier transform of the set of matrices of low enough fixed rank in the vector space of matrices of size m x n over F_q. In order to make the story we are trying to tell clear, we choose in this note to follow a particular example that shows how one might apply the theory of rank to certain counting problems.