The numbers game, geometric representations of Cox

arXiv:math/0610702v1[math.CO]23Oct2006Thenumbersgame,geometricrepresentationsofCoxetergroups,andDynkindiagramclassificationresultsRobertG.DonnellyDepartmentofMathematicsandStatistics,MurrayStateUniversity,Murray,KY42071AbstractThenumbersgameisaone-playergameplayedonafinitesimplegraphwithcertain“am-plitudes”assignedtoitsedgesandwithaninitialassignmentofrealnumberstoitsnodes.Themovesofthegamesuccessivelytransformthenumbersatthenodesusingtheamplitudesinacertainway.ThisgamehasbeenstudiedpreviouslybyProctor,Mozes,Bj¨orner,Eriks-son,andWildberger.WeshowthatthoseconnectedsuchgraphsforwhichthenumbersgamemeetsacertainfinitenessrequirementarepreciselytheDynkindiagramsassociatedwiththefinite-dimensionalcomplexsimpleLiealgebras.Asaconsequenceofourproofweobtaintheclassificationsofthefinite-dimensionalKac-MoodyalgebrasandofthefiniteWeylgroups.WeuseCoxetergrouptheorytoestablishamoregeneralresultthatappliestoEriksson’sE-games:anE-gamemeetsthefinitenessrequirementifandonlyifanaturallyassociatedCoxetergroupisfinite.ToprovethisandsomeotherfinitenessresultswefurtherdevelopEriksson’stheoryofageometricrepresentationofCoxetergroupsandobservesomecuriousdifferencesofthisrepresentationfromthestandardgeometricrepresentation.Keywords:numbersgame,generalizedCartanmatrix,Dynkindiagram,Coxetergraph,Coxeter/Weylgroup,geometricrepresentation,semisimpleLiealgebra,Kac-MoodyalgebraContents1.Introductionandstatementsofourmainresults2.Proofofourfirstmainresult3.Classificationsoffinite-dimensionalKac-MoodyalgebrasandfiniteWeylgroups4.Quasi-standardgeometricrepresentationsofCoxetergroupsandageneralizationtoE-games5.SomefurtherfinitenessaspectsofE-gameplay6.Comments1.IntroductionandmainresultsThenumbersgameisaone-playergameplayedonafinitesimplegraphwithweights(whichwecall“amplitudes”)onitsedgesandwithaninitialassignmentofrealnumbers(whichwecallinitial“populations”)toitsnodes.Attheoutset,eachofthetwoedgeamplitudes(oneforeachdirection)willbenegativeintegers;laterwewillrelaxthisintegralityrequirement.Themoveaplayercanmakeisto“fire”oneofthenodeswithapositivepopulation.Thismovetransformsthepopulationatthefirednodebychangingitssign,anditalsotransformsthepopulationateachadjacentnodeinacertainwayusinganamplitudealongtheincidentedge.Theplayerfiresthenodesinsomesequenceoftheplayer’schoosing,continuinguntilnonodehasapositivepopulation.ThisnumbersgameformulatedbyMozes[Moz]hasalsobeenstudiedbyProctor[Pr1],[Pr2],Bj¨orner[Bj¨or],[BB],Eriksson[Erik1],[Erik2],[Erik3],andWildberger[Wil1],[Wil2],[Wil3].Wildbergerstudiesadualversionwhichhecallsthe“mutationgame.”SeeAlonetal[AKP]forabriefandreadabletreatmentofthenumbersgameon“unweighted”cyclicgraphs.MuchofthenumbersgamediscussioninChapter4of[BB]canbefoundin[Erik2].Proctordevelopedthisprocessin[Pr1]1tocomputeWeylgrouporbitsofweightswithrespecttothefundamentalweightbasis.Forthisreasonwepreferhisperspectiveoffiringnodeswithpositive,asopposedtonegative,populations.Themotivatingquestionforthispaperis:forwhichsuchgraphsdoesthereexistanontrivialinitialassignmentofnonnegativepopulationssuchthatthenumbersgameterminatesinafinitenumberofsteps?Forgraphswithintegeramplitudes,ouranswertothisquestion(Theorem1.1)isthattheonlysuchconnectedgraphsaretheDynkindiagramsofFigure1.1.Moreover,fromEriksson’sStrongConvergenceTheorem(Theorem3.1of[Erik3])weareabletoconcludethatforanyinitialassignmentofpopulationstothenodesofaDynkindiagramandforanylegalsequenceofnodefirings,thenumbersgamewillterminateinthesamefinitenumberofstepsandultimatelyyieldateachnodethesamenonpositiveterminalpopulation.OurproofofTheorem1.1inSection2requiressomeCoxeter/Weylgrouptheory,butonlyimplicitly—inparticulartheproofofEriksson’sComparisonTheorem(Theorem4.5of[Erik2]).AsaconsequenceofourproofofTheorem1.1andwiththehelpofanotherresultofErikssonwere-deriveinSection3theclassificationsofthefinite-dimensionalKac-Moodyalgebras(thefinite-dimensionalcomplexsemisimpleLiealgebrascf.[Hum1],[Kac])andofthefiniteWeylgroups(thefinitecrystallographicCoxetergroupsof[Hum2]).Oursecondmainresult(Theorem1.3)answersourmotivatingquestionforaclassofgraphs(the“E-games”of[Erik2])whoseamplitudesareallowedtobecertainrealnumbers.TheclassificationobtainedinthistheoremusestheclassificationoffiniteirreducibleCoxetergroupsbyconnectedpositivedefiniteCoxetergraphs(cf.§2.3-2.7in[Hum2]).TheconnectiontoCoxetergroupsismadeviaaparticulargeometricrepresentationstudiedin[Erik2],referredtohereasa“quasi-standardgeometricrepresentation.”Thishasmanysimilaritiestothestandardgeometricrepresentation,butalsosomesurprisingdifferencesinregardtosomefinitenessproperties.Resultswedevelopaboutquasi-standardgeometricrepresentationsinSection4leadtoourproofofTheorem1.3andtosomefurtherE-gameresultsinSection5.Theseincludeamethodforcomputing(incertaincircumstances)thepositiverootsintherootsystemforaquasi-standardgeometricrepresentationofafiniteCoxetergroup(Theorem5.5)andaclassification(Theorem5.8)ofthoseE-gamesforwhichthechoicesofnodefiringsare“interchangeable”insomesense.ThelatterappliesaclassificationresultofStembridge[Stem]about“fullycommutative”elementsinfiniteCoxetergr