An Explicit Presentation of the Grothendieck Ring of Finitely Generated F_{q}[SL(2,F_{q})]-Modules

Details An Explicit Presentation of the Grothendieck Ring of Finitely Generated F_{q}[SL(2,F_{q})]-Modules An Explicit Presentation of the Grothendieck Ring of Finitely Generated F_{q}[SL(2,F_{q})]-Modules

2011-10-31

arxiv.org/abs/1110.6881v1

Mathematics

Representation Theory

11 pages. Comments are welcome

Scientific paper

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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