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[德] 韦尔 著 / 世界图书出版公司 / 2011-01 / 平装
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Eversincetheyear1925,whenIsucceededindeterminingthecharactersofthesemi-simplecontinuousgroupsbyacombinationofE.Cartansinfini-tesimalmethodsandI.Schursintegralprocedure,Ihavelookedtowardthegoalofderivingthedecisiveresultsforthemostimportantofthesegroupsbydirectalgebraicconstruction,inparticularforthefullgroupofallnon-singu-larlineartransformationsandfortheorthogonalgroup.OwingmainlytoR.Brauersinterventionandcollaborationduringthelastfewyears,itnowappearsthatIhaveinmyhandsallthetoolsnecessaryforthispurpose.Thetaskmaybecharacterizedpreciselyasfollows:withrespecttotheassignedgroupoflineartransformationsintheunderlyingvectorspace,todecomposethespaceoftensorsofgivenrankintoitsirreducibleinvariantsubspaces.
TABLEOFCONTENTSPREFACETOTHEFIRSTEDITIONPREFACETOTHESECONDEDITIONCHAPTERIINTRODUCTION1.Fields,rings,ideals,polynomials2.Vectorspace3.Orthogonaltransformations,Euclideanvectorgeometry4.Groups,KleinsErlangerprogram..Quantities5.InvariantsandcovariantsCHAPTERIIVECTORINVARIANTS1.Remembranceofthingspast2.ThemainpropositionsofthetheoryofinvariantsA.FrostMAINTHEOREM3.Firstexample:thesymmetricgroup4.Capellisidentity5.ReductionofthefirstmainproblembymeansofCapellisidentities6.Secondexample:theunimodulargroup,.qL(n)7.Extensiontheorem.Thirdexample:thegroupofsteptransformations8.Ageneralmethodforincludingeontravariantarguments9.Fourthexample:theorthogonalgroupB.ACLOSE-UPOFTHEORTHOGONALGROUP10.Cayleysrationalparametrizationoftheorthogonalgroup11,Formalorthogonalinvariants12.Arbitrarymetricgroundform13.TheinfinitesimalstandpointC.THESECONDMAINTHEOREM14.Statementofthepropositionfortheunimodulargroup15.Capellisformalcongruence16.Proofofthesecondmaintheoremfortheunimodulargroup17.ThesecondmaintheoremfortheunimodulargroupCHAPTERIIIMATRICALGEBRASANDGROUPRINGSA.THEORYOFFULLYREDUCIBLEMATRICALGEBRAS1.Fundamentalnotionsconcerningmatricalgebras.TheSchurlemma2.Preliminaries3.Representationsofasimplealgebra4.Wedderburnstheorem5.ThefullyreduciblematricalgebraanditscommutatoralgebraB.THERINGOFAFINITEGROUPANDITSCOMMUTATORALGEBRA6.Statingtheproblem7.FullreducibilityofthegroupringTABLEOFCONTENTS8.Formallemmas.9.Reciprocitybetweengroupringandcommutatoralgebra10.AgeneralizationCHAPTERIVTHESYMMETRICGROUPANDTHEFULLLINEARGROUP1.Representationofafinitegroupinanalgebraicallyclosedfield2.TheYoungsymmetrizers.Acombinatoriallsmma3.Theirreduciblerepresentationsofthesymmetricgroup4.Decompositionoftensorspace5.Quantities.ExpansionCHAPTERVTHEORTHOGONALGROUPA.THEENVELOPINGALGEBRAANDTHEORTHOGONALIDEAL1.Vectorinvariantsoftheunimodulargroupagain2.Theenvelopingalgebraoftheorthogonalgroup3.Givingtheresultitsformalsetting4.Theorthogonalprimeideal5.AnabstractalgebrarelatedtotheorthogonalgroupB.THEIRREDUCIBLEREPRESENTATIONS6.Decompositionbythetraceoperation7.TheirreduciblerepresentationsofthefullorthogonalgroupC.THEPROPERORTHOGONALGROUP8.Cliffordstheorem9.RepresentationsoftheproperorthogonalgroupCHAPTERVITHESYMPLECTICGROUP1.Vectorinvariantsofthesymplecticgroup2.Parametrizationandunitaryrestriction3.EmbeddingalgebraandrepresentationsofthesymplecticgroupCHAPTERVIICHARACTERS1.Preliminariesaboutunitarytransformations2.Characterforsymmetrizationoralternationalone3.Averagingoveragroup4.Thevolumeelementoftheunitarygroup5.Computationofthecharacters6.ThecharactersofGL(n).Enumerationofcovariants7.Apurelyalgebraicapproach8.Charactersofthesymplecticgroup9.Charactersoftheorthogonalgroup10.DecompositionandX-multiplication11.ThePoinear~polynomial……
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