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In [Frobenius1896] it was shown that many important properties of a finitegroup could be examined using formulas involving the character ratios of groupelements, i.e., the trace of the element acting in a given irreduciblerepresentation, divided by the dimension of the representation. In[Gurevich-Howe15] and [Gurevich-Howe17], the current authors introduced thenotion of rank of an irreducible representation of a finite classical group.One of the motivations for studying rank was to clarify the nature of characterratios 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 notionof U-rank of a representation, which comes up when one looks at itsrestrictions to certain abelian unipotent subgroups. The second is the morealgebraic based notion of tensor rank which comes up naturally when oneattempts to equip the representation ring of the group with a grading thatreflects the central role played by the few "smallest" possible representationsof the group. In [Gurevich-Howe17] we conjectured that the two notions of rankmentioned just above agree on a suitable collection called "low rank"representations. In this note we review the development of the theory of rankfor the case of the general linear group GL_n over a finite field F_q, and givea proof of the "agreement conjecture" that holds true for sufficiently large q.Our proof is Fourier theoretic in nature, and uses a certain curious positivityproperty of the Fourier transform of the set of matrices of low enough fixedrank in the vector space of matrices of size m x n over F_q. In order to makethe story we are trying to tell clear, we choose in this note to follow aparticular example that shows how one might apply the theory of rank to certaincounting problems.

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