The Quiver of the Semigroup Algebra of a Left Regu

arXiv:math/0608698v1[math.CO]28Aug2006THEQUIVEROFTHESEMIGROUPALGEBRAOFALEFTREGULARBANDFRANCOVSALIOLAAbstract.Recentlyithasbeennoticedthatmanyinterestingcombinatorialobjectsbelongtoaclassofsemigroupscalledleftregularbands,andthatrandomwalksonthesesemigroupsencodeseveralwell-knownrandomwalks.Forexample,thesetoffacesofahyperplanearrangementisendowedwithaleftregularbandstruc-ture.Thispaperstudiesthemodulestructureofthesemigroupalgebraofanarbitraryleftregularband,extendingresultsforthesemigroupalgebraofthefacesofahyperplanearrangement.Inparticular,adescriptionofthequiverofthesemigroupalgebraisgivenandtheCartaninvariantsarecomputed.Theseareusedtocomputethequiverofthefacesemigroupalgebraofahyperplanearrangementandtoshowthatthesemigroupalgebraofthefreeleftregularbandisisomorphictothepathalgebraofitsquiver.Contents1.Introduction22.LeftRegularBands33.RepresentationsoftheSemigroupAlgebra54.PrimitiveIdempotentsoftheSemigroupAlgebra65.ProjectiveIndecomposableModulesoftheSemigroupAlgebra96.TheQuiveroftheSemigroupAlgebra107.AnInductiveConstructionoftheQuiver118.Example:TheFreeLeftRegularBand139.Example:TheFaceSemigroupofaHyperplaneArrangement1410.Idempotentsinthesubalgebrask(yS)andkS≥X1511.CartanInvariantsoftheSemigroupAlgebra1612.Example:TheFaceSemigroupofaHyperplaneArrangement1813.Example:TheFreeLeftRegularBand1812FRANCOVSALIOLA14.FutureDirections2015.Appendix:ProofofLemma6.121References231.IntroductionAleftregularbandisasemigroupSsatisfyingx2=xandxyx=xyforallx,y∈S.RecentinterestinleftregularbandsandtheirsemigroupalgebrasaroseduetotheworkofK.S.Brown[Brown,2000],inwhichtherepresentationtheoryofthesemigroupalgebraisusedtostudyrandomwalksonthesemigroup.Thereareseveralinterestingexamplesofsuchrandomwalks,includingtherandomwalkonthechambersofahyperplanearrangement.Severaldetailedexamplesareincludedin[Brown,2000].Thestartingpointofthispaperisthefactthattheirreduciblerep-resentationsofthesemigroupalgebraofaleftregularbandareall1-dimensional.Thisimpliesthatthereisacanonicalquiver(adirectedgraph)associatedtotheleftregularband,andthatthesemigroupalgberaisaquotientofthepathalgebraofthequiver.ThispaperdeterminesacombinatorialdescriptionofthisquiverandtheCartaninvariantsofthesemigroupalgebrasandillustratesthetheorythroughdetailedexamples.Thepaperisstructuredasfollows.Section2recallsthedefinitionandcollectssomepropertiesofleftregularbands,andintroducestheexam-plesthatwillbeusedthroughoutthepaper.Section3describestheirre-duciblerepresentationsofthesemigroupalgebraofaleftregularband.InSection4acompletesystemofprimitiveorthogonalidempotentsforthesemigroupalgebraisexplicitlyconstructed.Section5describestheprojectiveindecomposablemodulesofthesemigroupalgebra.Sections6through9dealwithcomputingthequiverofthesemigroupalgebra.Sections10through13computetheCartaninvariantsofthesemigroupalgebras.Finally,Section14discussesfuturedirectionsforthisproject.THESEMIGROUPALGEBRAOFALEFTREGULARBAND32.LeftRegularBandsSee[Brown,2000,AppendixB]forfoundationsofleftregularbandsandforproofsofthestatementspresentedinthissection.AleftregularbandisasemigroupSsatisfyingthefollowingtwoproperties.(LRB1)x2=xforallx∈S.(LRB2)xyx=xyforallx,y∈S.DefinearelationontheelementsofSbyy≤xiffyx=x.Thisrelationisapartialorder(reflexive,transitiveandantisymmetric),soSisaposet.DefineanotherrelationontheelementsofSbyyxiffxy=x.Thisrelationisreflexiveandtransitive,butnotnecessarilyantisymmetric.ThereforewegetaposetLbyidentifyingxandyifxyandyx.Letsupp:S→Ldenotethequotientmap.LiscalledthesupportsemilatticeofSandsupp:S→Liscalledthesupportmap.Proposition2.1.IfSisaleftregularband,thenthereisasemilatticeLandasurjectionsupp:S→Lsatisfyingthefollowingpropertiesforallx,y∈S.(1)Ify≤x,thensupp(y)≤supp(x).(2)supp(xy)=supp(x)∨supp(y).(3)xy=xiffsupp(y)≤supp(x).(4)IfS′isasubsemigroupofS,thentheimageofS′inListhesupportsemilatticeofS′.Statement(1)saysthatsuppisanorder-preservingposetmap.(2)saysthatsuppisasemigroupmapwhereweviewLasasemigroupwithproduct∨.(3)followsfromtheconstructionofL,and(4)followsfromthefactthat(3)characterizesLuptoisomorphism.IfShasanidentityelementthenLhasaminimalelementˆ0.If,inaddition,Lisfinite,thenLhasamaximalelementˆ1,andisthereforealattice4FRANCOVSALIOLAabcacbbacbcacabcbaab5555ac ba555bc ca5555cb aSSSSSSSSSSSSSbckkkkkkkkkkkkk1abczzzzDDDDabDDDDDaczzzzzDDDDDbczzzzzaCCCCCCbc{{{{{{∅Figure1.TheposetofthefreeleftregularbandF({a,b,c})onthreegeneratorsanditssupportlattice.[Stanley,1997,Proposition3.3.1].InthiscaseListhesupportlatticeofS.Example2.2(TheFreeLeftRegularBand).ThefreeleftregularbandF(A)withidentityonafinitesetAisthesetofall(ordered)sequencesofdistinctelementsfromAwithmultiplicationdefinedby(a1,...,al)·(b1,...,bm)=(a1,...,al,b1,...,bm)✄where✄means“deleteanyelementthathasoccuredearlier”.Equiv-alently,F(A)isthesetofallwordsonthealphabetAthatdonotcontainanyrepeatedletters.TheemptysequenceisanelementofF(A),thereforeF(A)containsanidentityelement.ThesupportlatticeofF(A)isthelatticeLofsubsetsofAandthesupportmapsupp:F(A)→Asendsasequence(a1,...,al)tothesetofelementsinthesequence{a1,...,al