Standard stratifications of EI categories and Alperin's weight conjecture

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Abstract

We characterize the finite EI categories whose representations are standardly stratified with respect to the natural preorder on the simple representations. The orbit category of a finite group with respect to any set of subgroups is always such a category. Taking the subgroups to be the p-subgroups of the group, we reformulate Alperin's weight conjecture in terms of the standard and proper costandard representations of the orbit category. We do this using the properties of the Ringel dual construction and a theorem of Dlab, which have elsewhere been described for standardly stratified algebras where there is a partial order on the simple modules. We indicate that these results hold in the generality of an algebra whose simple modules are preordered, rather than partially ordered.

Original languageEnglish (US)
Pages (from-to)4073-4091
Number of pages19
JournalJournal of Algebra
Volume320
Issue number12
DOIs
StatePublished - Dec 15 2008

Keywords

  • Alperin's weight conjecture
  • Category algebra
  • Orbit category
  • Representation
  • Ringel dual
  • Tilting module

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