Quivers, Geometric Invariant Theory, and Moduli of

arXiv:0712.0558v1[math.AG]4Dec2007QUIVERS,GEOMETRICINVARIANTTHEORY,ANDMODULIOFLINEARDYNAMICALSYSTEMSMARKUSBADERAbstract.Weusegeometricinvarianttheoryandthelanguageofquiverstostudycompactificationsofmodulispacesoflineardynamicalsystems.Ageneralapproachtothisproblemispresentedandappliedtotwowellknowncases:WeshowhowbothLomadze’sandHelmke’scompactificationarisesnaturallyasageometricinvarianttheoryquotient.Bothmodulispacesareproventobesmoothprojectivemanifolds.Furthermore,adescriptionofLomadze’scompactificationasaQuotschemeisgiven,whereasHelmke’scompactificationisshowntobeanalgebraicGrassmannbundleoveraQuotscheme.Thisgivesanalgebro-geometricdescriptionofbothcompactifications.Asanapplication,wedeterminethecohomologyringofHelmke’scompactificationandprovethatthetwocompactificationsarenotisomorphicwhenthenumberofoutputsispositive.0.IntroductionInthisarticle,westudyactionsofproductsofgenerallineargroupsonspacesofmatrices.Wepresentgeneraltechniquesfromalgebraicgeometrythatweapplytotwoconcreteexamples,namelytotwodifferentcompactificationsofthemodulispaceofcontrollablelineardynamicalsystems.Thefirstsectionintroducesgeometricinvarianttheoryandrepresentationtheoryofquiversinatutorialway.Weexplainhowthismachinerycanbeusedtosystematicallystudytheproblemofcompactifyingthemodulispaceoflineardynamicalsystems.Twoimportantresultsarepresentedandweexplainhowtheycanbeadaptedtocoverthecasesrelevanttocontroltheory.Inthesecondpartofthearticle,weshowhowboththeHelmkeandtheLomadzecompactificationcanbeconstructedasalgebraicvarietiesusingthismachinery.Weobtainanalgebro-geometricdescriptionofbothcompactificationsthatweusetostudyandcomparebothvarieties.ModulispacesoflineardynamicalsystemshavebeenintroducedtocontroltheorybyKalman[17]andHazewinkel[10].AsalgebraicvarietiestheyhavebeenconstructedandstudiedamongothersbyHazewinkelin[10],byByrnesandHurtin[3],byKalmanin[17]andbyTannenbaumin[33].Inalgebraicgeometrythemaintechniquetoconstructmodulispacesisasquotientsofalgebraicvarietiesunderalgebraicgroupactionsusinggeometricinvarianttheory.Let˜Σn,m,pdenotethespaceoflineardynamicalsystemsxt+1=Axt+Butyt=Cxt+Dut(1)withnstates,minputs,andpoutputs.Itisaspaceofmatrices˜Σn,m,p=kn×m×kn×p×km×n×km×p,wherekisafixed,algebraicallyclosedfieldofcharacteristic0.Letthegroupofinvertiblen×nmatricesGLnacton˜Σn,m,pbychangeofbasisinthestatespacekn:(2)g,(A,B,C,D)
7→gAg−1,gB,Cg−1,D
.ThecontrollablesystemsformaZariski-opensubsetwhichwedenotewith˜Σcn,m,p⊂˜Σn,m,p.Geometricinvarianttheoryprovidesthemeanstosystematicallyconstructsuchquotients.ItassociateswitheverycharacterofthegroupGLn,soinparticularwiththecharacterdet:GLn−→k∗,anopensubsetofstablepoints,andrealizesthealgebraicquotient{det−stablepoints}//GLn.ByrnesandHurt[3]werethefirsttonoticethatthedet-stablepointscoincidewiththecontrollablesystemsandthereforethatthemodulispaceofcontrollablelineardynamicalsystemscanberealizedasthequotientΣcn,m,p:=˜Σcn,m,p//GLnusingGIT.Thisquotientisnon-projective.Compactificationshavebeenintroducedbyseveralauthors,letusmentionHelmke[12],Lomadze[22],andRosenthal[30].2000MathematicsSubjectClassification.15A30,14L24,16G20.12MARKUSBADERByaquiverwemeananorientedgraph,thatisafinitesetofverticestogetherwithafinitesetoforientededgesbetweenthevertices.Toeveryvertexweassignadimension,andfurthermorewemarkasubsetofvertices.Thisdataisdescribedbydiagramslikethefollowing:(3)n•A::A::A::C◦mBooD








◦pThecorrespondingGITproblemisthefollowing:westudyrepresentationsofthequiverofthepre-scribeddimension.Inourconcreteexamplethismeansmatrices(A,B,C,D)∈˜Σn,m,p.ThegenerallineargroupGLnactsbychangeofbasisonthevectorspaceknassociatedwiththemarkedvertex.Thiscorrespondstothegroupactionintroducedin(2).IngeneralwewillbegivenaquiverQ,asubsetofmarkedverticesM,andadimensionvectorv(i.e.aprescribeddimensionateachvertex).WiththisdataweassociatearepresentationspaceRepvQwhichisalwaysaspaceofmatrices,agroupGLv,Mwhichisalwaysafiniteproductofgenerallineargroups,andanactionofthisgrouponthespaceofrepresentations.Inthatframework,theproblemofcompactifyingthemodulispaceoflineardynamicalsystemsbecomesthefollowing:Wearegivenaspaceofmatrices˜Σn,m,pwhichisthespaceofrepresentations˜Σn,m,p=RepvQforthequiverQandwithdimensionvectorvasintroducedindiagram(3).Wemarkonevertex-theonecorrespondingtothestatespacekn,andwearegivenacharacterχ=detofthegroupGLn=GLv,M.Thequotientspace{det−stablepointsinRepvQ}//GLv,M=Σcn,m,pisnotprojective.Thegoalistoreplacethedata(Q,v,M,χ)byanewquiver˜Q,anewdimensionvector˜v,anewsetofmarkedvertices˜M,andanewcharacter˜χofGL˜v,˜M,suchthatthequotient{˜χ−stablepointsinRep˜v˜Q}//GL˜v,˜MisprojectiveandcontainsthepreviousquotientΣcn,m,pasanopensubset.Tobemoreprecise,weneedtodothefollowing:(1)Findanewquiver˜Q,anewdimensionvector˜v,andanewsetofmarkedvertices˜M,togetherwithmorphismsϕ:GLQ,M−→GL˜Q,˜M,Φ:RepQ,v−→Rep˜Q,˜v,wherethelatterisaclosedembedding,equivariantwithrespecttoϕ.(2)GivenacharacterχofGLv,M(suchasdet:GLn−→k∗inoursituation),determinethechar-acters˜χofGL˜v,˜M,suchthatundertheembeddingΦtheχ-(semi)stablerepresentationswillbemappedtothesetof˜