000 04933nam a22004815i 4500
001 978-3-7643-9998-6
003 DE-He213
005 20181204133148.0
007 cr nn 008mamaa
008 100301s2009 sz | s |||| 0|eng d
020 _a9783764399986
_9978-3-7643-9998-6
024 7 _a10.1007/978-3-7643-9998-6
_2doi
040 _aISI Library, Kolkata
050 4 _aQA174-183
072 7 _aPBG
_2bicssc
072 7 _aMAT002010
_2bisacsh
072 7 _aPBG
_2thema
082 0 4 _a512.2
_223
100 1 _aPuig, Lluís.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
245 1 0 _aFrobenius Categories versus Brauer Blocks
_h[electronic resource] :
_bThe Grothendieck Group of the Frobenius Category of a Brauer Block /
_cby Lluís Puig.
264 1 _aBasel :
_bBirkhäuser Basel,
_c2009.
300 _aV, 498 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aProgress in Mathematics,
_x0743-1643 ;
_v274
505 0 _aGeneral notation and quoted results -- Frobenius P-categories: the first definition -- The Frobenius P-category of a block -- Nilcentralized, selfcentralizing and intersected objects in Frobenius P-categories -- Alperin fusions in Frobenius P-categories -- Exterior quotient of a Frobenius P-category over the selfcentralizing objects -- Nilcentralized and selfcentralizing Brauer pairs in blocks -- Decompositions for Dade P-algebras -- Polarizations for Dade P-algebras -- A gluing theorem for Dade P-algebras -- The nilcentralized chain k*-functor of a block -- Quotients and normal subcategories in Frobenius P-categories -- The hyperfocal subcategory of a Frobenius P-category -- The Grothendieck groups of a Frobenius P-category -- Reduction results for Grothendieck groups -- The local-global question: reduction to the simple groups -- Localities associated with a Frobenius P-category -- The localizers in a Frobenius P-category -- Solvability for Frobenius P-categories -- A perfect F-locality from a perfect Fsc -locality -- Frobenius P-categories: the second definition -- The basic F-locality -- Narrowing the basic Fsc-locality -- Looking for a perfect Fsc-locality.
520 _aThis book contributes to important questions in the representation theory of finite groups over fields of positive characteristic — an area of research initiated by Richard Brauer sixty years ago with the introduction of the blocks of characters. On the one hand, it introduces and develops the abstract setting of the Frobenius categories — also called the Saturated fusion systems in the literature — created by the author fifteen years ago for a better understanding of what was loosely called the local theory of a finite group around a prime number p or, later, around a Brauer block, and for the purpose of an eventual classification — a reasonable concept of simple Frobenius category arises. On the other hand, the book develops this abstract setting in parallel with its application to the Brauer blocks, giving the detailed translation of any abstract concept in the particular context of the blocks. One of the new features in this direction is a framework for a deeper understanding of one of the central open problems in modular representation theory, known as Alperin’s Weight Conjecture (AWC). Actually, this new framework suggests a more general form of AWC, and a significant result of the book is a reduction theorem of this form of AWC to quasi-simple groups. Although this book is a research monograph, all the arguments are widely developed to make it accessible to the interested graduate students and, at the same time, to put them on the verge of the research on this new subject: the third part of the book on the localities associated to a Frobenius category gives some insight on the open question about the existence and the uniquenes of a perfect locality — also called centric linking system in the literature. We have developed a long introduction to explain our purpose and to provide a guideline for the reader throughout the twenty four sections. A systematic appendix on the cohomology of categories completes the book.
650 0 _aGroup theory.
650 0 _aAlgebraic topology.
650 1 4 _aGroup Theory and Generalizations.
_0http://scigraph.springernature.com/things/product-market-codes/M11078
650 2 4 _aAlgebraic Topology.
_0http://scigraph.springernature.com/things/product-market-codes/M28019
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9783034600156
776 0 8 _iPrinted edition:
_z9783764399979
830 0 _aProgress in Mathematics,
_x0743-1643 ;
_v274
856 4 0 _uhttps://doi.org/10.1007/978-3-7643-9998-6
912 _aZDB-2-SMA
942 _cEB
950 _aMathematics and Statistics (Springer-11649)
999 _c426282
_d426282