000 03632nam a22004577a 4500
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008 121120s2013 xxk| s |||| 0|eng d
020 _a9781447148326
_9978-1-4471-4832-6
024 7 _a10.1007/978-1-4471-4832-6
_2doi
050 4 _aQA251.5
072 7 _aPBF
_2bicssc
072 7 _aMAT002010
_2bisacsh
082 0 4 _a512.46
_223
100 1 _aSchneider, Peter.
_eauthor.
245 1 0 _aModular Representation Theory of Finite Groups
_h[electronic resource] /
_cby Peter Schneider.
264 1 _aLondon :
_bSpringer London :
_bImprint: Springer,
_c2013.
300 _aVIII, 178 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
505 0 _aPrerequisites in module theory -- The Cartan{Brauer triangle -- The Brauer character -- Green's theory of indecomposable modules -- Blocks.
520 _aRepresentation theory studies maps from groups into the general linear group of a finite-dimensional vector space. For finite groups the theory comes in two distinct flavours. In the 'semisimple case' (for example over the field of complex numbers) one can use character theory to completely understand the representations. This by far is not sufficient when the characteristic of the field divides the order of the group. Modular representation theory of finite groups comprises this second situation. Many additional tools are needed for this case. To mention some, there is the systematic use of Grothendieck groups leading to the Cartan matrix and the decomposition matrix of the group as well as Green's direct analysis of indecomposable representations. There is also the strategy of writing the category of all representations as the direct product of certain subcategories, the so-called 'blocks' of the group. Brauer's work then establishes correspondences between the blocks of the original group and blocks of certain subgroups the philosophy being that one is thereby reduced to a simpler situation. In particular, one can measure how nonsemisimple a category a block is by the size and structure of its so-called 'defect group'. All these concepts are made explicit for the example of the special linear group of two-by-two matrices over a finite prime field. Although the presentation is strongly biased towards the module theoretic point of view an attempt is made to strike a certain balance by also showing the reader the group theoretic approach. In particular, in the case of defect groups a detailed proof of the equivalence of the two approaches is given. This book aims to familiarize students at the masters level with the basic results, tools, and techniques of a beautiful and important algebraic theory. Some basic algebra together with the semisimple case are assumed to be known, although all facts to be used are restated (without proofs) in the text. Otherwise the book is entirely self-contained.
650 0 _aMathematics.
650 0 _aAssociative rings.
650 0 _aRings (Algebra).
650 0 _aGroup theory.
650 1 4 _aMathematics.
650 2 4 _aAssociative Rings and Algebras.
650 2 4 _aGroup Theory and Generalizations.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9781447148319
856 4 0 _uhttp://dx.doi.org/10.1007/978-1-4471-4832-6
912 _aZDB-2-SMA
942 _2Dewey Decimal Classification
_ceBooks
999 _c43668
_d43668