LEADER 04267nam 22006375 450 001 9910438154403321 005 20200630164953.0 010 $a1-4471-4832-0 024 7 $a10.1007/978-1-4471-4832-6 035 $a(CKB)2670000000308619 035 $a(EBL)1081784 035 $a(OCoLC)819652072 035 $a(SSID)ssj0000813523 035 $a(PQKBManifestationID)11510835 035 $a(PQKBTitleCode)TC0000813523 035 $a(PQKBWorkID)10785971 035 $a(PQKB)11055725 035 $a(DE-He213)978-1-4471-4832-6 035 $a(MiAaPQ)EBC1081784 035 $a(MiAaPQ)EBC6312087 035 $a(PPN)168294338 035 $a(EXLCZ)992670000000308619 100 $a20121120d2013 u| 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aModular Representation Theory of Finite Groups /$fby Peter Schneider 205 $a1st ed. 2013. 210 1$aLondon :$cSpringer London :$cImprint: Springer,$d2013. 215 $a1 online resource (182 p.) 300 $aDescription based upon print version of record. 311 $a1-4471-4831-2 320 $aIncludes bibliographical references (pages [175]-175) and index. 327 $aPrerequisites in module theory -- The Cartan{Brauer triangle -- The Brauer character -- Green's theory of indecomposable modules -- Blocks. 330 $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. 606 $aAssociative rings 606 $aRings (Algebra) 606 $aGroup theory 606 $aAssociative Rings and Algebras$3https://scigraph.springernature.com/ontologies/product-market-codes/M11027 606 $aGroup Theory and Generalizations$3https://scigraph.springernature.com/ontologies/product-market-codes/M11078 615 0$aAssociative rings. 615 0$aRings (Algebra). 615 0$aGroup theory. 615 14$aAssociative Rings and Algebras. 615 24$aGroup Theory and Generalizations. 676 $a512.46 700 $aSchneider$b Peter$4aut$4http://id.loc.gov/vocabulary/relators/aut$0350808 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910438154403321 996 $aModular Representation Theory of Finite Groups$92516813 997 $aUNINA