The State of the Modular Atlas Project(Hiss).pdf

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1、WhatistheModularAtlasProject?TheStateoftheArtCondensationTheStateoftheModularAtlasProjectGerhardHissLehrstuhlDfürMathematikRWTHAachenLehrstuhlDfürMathematikRWTHAachenGroupsandComputationVColumbus,Ohio,March17–21,2008GerhardHissTheStateoftheModularAtlasProjectWhatistheModular

2、AtlasProject?TheModularAtlasTheStateoftheArtModularCharacterTablesCondensationWhatistheModularAtlasProject?Theprogramtocomputethemodular(=Brauer)charactertablesoftheATLASgroups.ATLAS:=AtlasofFiniteGroups,Conwayetal.,1986Start:PhD-thesisofGordonJamesonMathieugroups(1973)ForAt

3、lasgroupsuptoMcL(i.e.,order≤109):AnAtlasofBrauerCharacters,Jansenetal.,1995TheseandmoretablesarecollectedonthewebsiteoftheModularAtlasProject(http://www.math.rwth-aachen.de/˜MOC/)GerhardHissTheStateoftheModularAtlasProjectWhatistheModularAtlasProject?TheModularAtlasTheStateo

4、ftheArtModularCharacterTablesCondensationThePlayersWilsonWakiThackrayRybaParkerNoeskeNeunhöfferMüllerLuxLübeckJansenJamesH.andmanyothersGerhardHissTheStateoftheModularAtlasProjectWhatistheModularAtlasProject?TheModularAtlasTheStateoftheArtModularCharacterTablesCondensationMo

5、tivation:ClassifyingIrreducibleRepresentationsLetGbeafinitegroupandFafield.FactThereareonlyfinitelymanyirreducibleF-representationsofGuptoequivalence.AimsClassifyallirreduciblerepresentationsforagivengroupGandagivenfieldF.Describeallirreduciblerepresentationsofallfinitesimplegrou

6、ps.Useacomputerforsporadicsimplegroups.GerhardHissTheStateoftheModularAtlasProjectWhatistheModularAtlasProject?TheModularAtlasTheStateoftheArtModularCharacterTablesCondensationCharacters:ASimplificationThecharacteraffordedbytherepresentationX:G→GL(V)isthemap:χX:G→F,g7→Trace(X

7、(g)).Itisconstantonconjugacyclasses:aclassfunctiononG.Equivalentrepresentationshavethesamecharacter.FactIfFhascharacteristic0,thentwoF-representationsofGareequivalentifandonlyiftheircharactersareequal.GerhardHissTheStateoftheModularAtlasProjectWhatistheModularAtlasProject?Th

8、eModularAtlasTheStateoftheArtModularCharacterTablesCondensationBrauerCharac