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Algebras and Representation Theory

, Volume 20, Issue 4, pp 951–975 | Cite as

Decomposition of Tensor Products Involving a Steinberg Module

  • Tobias Kildetoft
Article

Abstract

We study the decomposition of tensor products between a Steinberg module and a costandard module, both as a module for the algebraic group G and when restricted to either a Frobenius kernel G r or a finite Chevalley group \(G(\mathbb {F}_q)\). In all three cases, we give formulas reducing this to standard character data for G. Along the way, we use a bilinear form on the characters of finite dimensional G-modules to give formulas for the dimension of homomorphism spaces between certain G-modules when restricted to either G r or \(G(\mathbb {F}_q)\). Further, this form allows us to give a new proof of the reciprocity between tilting modules and simple modules for G which has slightly weaker assumptions than earlier such proofs. Finally, we prove that in a suitable formulation, this reciprocity is equivalent to Donkin’s tilting conjecture.

Keywords

Algebraic groups Frobenius kernels Finite Chevalley groups Tilting modules 

Mathematics Subject Classification Primary (2010)

20G05 20C33 

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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Uppsala UniversityUppsalaSweden

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