An Explicit Presentation of the Grothendieck Ring of Finitely Generated F_q[SL(2,F_q)]-Modules

2011/10/31 by Davide A. Reduzzi, Reduzzi, Davide A.
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT) #Rings, Modules, and Algebras #math.RT

paper · pdf · doi:10.48550/arxiv.1110.6881

11 pages. Comments are welcome

arxiv created 2011/10/31 · openalex publication_date 2011/10/31 · arxiv updated 2011/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be a prime and q=p^g. We show that the Grothendieck ring of finitely generated F_q[SL(2,F_q)]-modules is naturally isomorphic to the quotient of the polynomial algebra Z[x] by the ideal generated by f^[g](x)-x, where f(x)=sum_j=0^floor(p/2)(-1)^j(p/(p-j))((p-j); j)x^p-2j, and the superscript [g] denotes g-fold composition of polynomials. We conjecture that a similar result holds for simply connected semisimple algebraic groups defined and split over a finite field.

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