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Some remarks on the ring of twisted tilting modules for algebraic groups

Authors
Journal
Journal of Algebra
0021-8693
Publisher
Elsevier
Publication Date
Volume
321
Issue
11
Identifiers
DOI: 10.1016/j.jalgebra.2008.08.023
Keywords
  • Rational Modules
  • Twisted Tilting Modules
  • Character Formulas
  • Special Linear Group
Disciplines
  • Mathematics

Abstract

Abstract In a recent paper [S. Doty, A. Henke, Decomposition of tensor products of modular irreducibles for SL 2, Q. J. Math. 56 (2005) 189–207], Doty and Henke give a decomposition of the tensor product of two rational simple modules for the special linear group of degree 2 over an algebraically closed field of characteristic p > 0 . In performing this calculation it proved useful to know that the simple modules are twisted tensor products of tilting modules. It seems natural therefore to consider the ring of twisted tilting modules for a semisimple group G (a subring of the representation ring of G). However, we quickly specialize to the case in which G is the special linear group of degree 2. We show that (in this case) the ring is reduced and describe associated varieties. We give formulas from which one may determine the multiplicities of the indecomposable module summands of the tensor product of twisted tilting modules.

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