Frobenius categories versus Brauer blocks : the Grothendieck group of the Frobenius category of a Brauer block
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Bibliographic Information
Frobenius categories versus Brauer blocks : the Grothendieck group of the Frobenius category of a Brauer block
（Progress in mathematics, v. 274）
Birkhäuser, c2009
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
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Note
Includes bibliographical reference (p. [489]-492) and index
Description and Table of Contents
Description
I1 More than one hundred years ago, Georg Frobenius [26] proved his remarkable theorem a?rming that, for a primep and a ?nite groupG, if the quotient of the normalizer by the centralizer of anyp-subgroup ofG is a p-group then, up to a normal subgroup of order prime top,G is ap-group. Ofcourse,itwouldbeananachronismtopretendthatFrobenius,when doing this theorem, was thinking the category - notedF in the sequel - G where the objects are thep-subgroups ofG and the morphisms are the group homomorphisms between them which are induced by theG-conjugation. Yet Frobenius' hypothesis is truly meaningful in this category. I2 Fifty years ago, John Thompson [57] built his seminal proof of the nilpotencyoftheso-called Frobeniuskernelofa FrobeniusgroupGwithar- ments - at that time completely new - which might be rewritten in terms ofF; indeed, some time later, following these kind of arguments, George G Glauberman [27] proved that, under some - rather strong - hypothesis onG, the normalizerNofasuitablenontrivial p-subgroup ofG controls fusion inG, which amounts to saying that the inclusionN?G induces an ? equivalence of categoriesF =F .
Table of Contents
Introduction.- 1. General notation and quoted results.- 2. Frobenius P-categories: the first definition.- 3. The Frobenius P-category of a block.- 4. Nilcentralized and selfcentralizing objects in Frobenius P-categories.- 5. Alperin fusions in Frobenius P-categories.- 6. Exterior quotient of a Frobenius P-category over the selfcentralizing objects.- 7 Nilcentralized and selfcentralizing Brauer pairs in blocks.- 8. Decompositions for Dade P-algebras.- 9. Polarizations for Dade P-algebras.- 10. A gluing theorem for Dade P-algebras.- 11. The nilcentralized chain k*-functor of a block.- 12. Quotients and normal subcategories in Frobenius P-categories.- 13. The hyperfocal subcategory of a Frobenius P-category.- 14. The Grothendieck groups of a Frobenius P-category.- 15. Reduction results for the Grothendieck groups.- 16. The local-global question: reduction to the simple groups.- 17. Localities associated with Frobenius P-categories.- 18. The localizers in a Frobenius P-category.- 19 Solvability for Frobenius P-categories.- 20 A perfect F-locality from a perfect Fsc-locality.- 21. Frobenius P-categories: the second definition.- 22. The basic F-locality.- 23. Narrowing the basic Fsc-locality.- 24. Looking for a perfect Fsc-locality.- Appendix.- References.- Index.
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