Ranks for representations of GLn over finite fields, their agreement, and positivity of Fourier transform
| Autor: | Shamgar Gurevich, Roger Howe |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
0303 health sciences
Pure mathematics Finite group Rank (linear algebra) Group (mathematics) General Mathematics 010102 general mathematics General linear group 01 natural sciences 03 medical and health sciences Character (mathematics) Representation ring Order (group theory) 0101 mathematics Abelian group 030304 developmental biology Mathematics |
| Zdroj: | Indagationes Mathematicae. 32:1332-1371 |
| ISSN: | 0019-3577 |
| DOI: | 10.1016/j.indag.2021.07.002 |
| Popis: | In 1896 Frobenius showed that many important properties of a finite group could in principle 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 recent years, 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, 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. Following numerical and theoretical evidences, we conjectured that the two notions of rank 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 G L 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 m × n matrices over F q of low enough fixed rank. 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. |
| Databáze: | OpenAIRE |
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