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Decomposition of Tensor Products Involving a Steinberg Module

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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.

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Authors and Affiliations

  1. Uppsala University, Uppsala, Sweden

    Tobias Kildetoft

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  1. Tobias Kildetoft
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Correspondence to Tobias Kildetoft.

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Presented by Jon F. Carlson.

Supported in part by QGM (Centre for Quantum Geometry of Moduli Spaces) funded by the Danish National Research Foundation and in part by Knut and Alice Wallenbergs Foundation

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Kildetoft, T. Decomposition of Tensor Products Involving a Steinberg Module. Algebr Represent Theor 20, 951–975 (2017). https://doi.org/10.1007/s10468-017-9670-7

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  • DOI: https://doi.org/10.1007/s10468-017-9670-7

Keywords

Mathematics Subject Classification Primary (2010)

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