  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.
