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

2021/07/05 by Shamgar Gurevich, Roger Howe, Gurevich, Shamgar +2
Mathematics · Computer Science · #Finite Group Theory Research #Graph theory and applications #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.2107.02240

Abstract

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

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