Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics PDF
Preview Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics
Fields Institute Communications 71 The Fields Institute for Research in Mathematical Sciences Mahir Can Zhenheng Li Benjamin Steinberg Qiang Wang Editors Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics Fields Institute Communications VOLUME 71 The Fields Institute for Research in Mathematical Sciences FieldsInstituteEditorialBoard: CarlR.Riehm,ManagingEditor EdwardBierstone,DirectoroftheInstitute MatheusGrasselli,DeputyDirectoroftheInstitute JamesG.Arthur,UniversityofToronto KennethR.Davidson,UniversityofWaterloo LisaJeffrey,UniversityofToronto BarbaraLeeKeyfitz,OhioStateUniversity ThomasS.Salisbury,YorkUniversity NorikoYui,Queen’sUniversity TheFieldsInstituteisacentreforresearchinthemathematicalsciences,locatedin Toronto,Canada.TheInstitutesmissionistoadvanceglobalmathematicalactivity intheareasofresearch,educationandinnovation.TheFieldsInstituteissupported bytheOntarioMinistryofTraining,CollegesandUniversities,theNaturalSciences and Engineering Research Council of Canada, and seven Principal Sponsoring Universities in Ontario (Carleton, McMaster, Ottawa, Toronto, Waterloo, Western andYork),aswellasbyagrowinglistofAffiliateUniversitiesinCanada,theU.S. andEurope,andseveralcommercialandindustrialpartners. Forfurthervolumes: http://www.springer.com/series/10503 Mahir Can • Zhenheng Li • Benjamin Steinberg Qiang Wang Editors Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics 123 TheFieldsInstituteforResearch intheMathematicalSciences Editors MahirCan ZhenhengLi DepartmentofMathematics DepartmentofMathematicalSciences TulaneUniversity UniversityofSouthCarolina NewOrleans,LA,USA Aiken,SC,USA BenjaminSteinberg QiangWang DepartmentofMathematics SchoolofMathematicsandStatistics CityCollegeofNewYork CarletonUniversity NewYork,NY,USA Ottawa,ON,Canada ISSN1069-5265 ISSN2194-1564(electronic) ISBN978-1-4939-0937-7 ISBN978-1-4939-0938-4(eBook) DOI10.1007/978-1-4939-0938-4 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2014941613 Mathematics Subject Classification (2010): 05A05, 05A16, 05A30, 05E05, 05E10, 06A06, 06A07, 11F85,14L10,14L30,14M17,14M27,14R20,14J60,16D80,16G99,16S99,16T10,20M14,20M30, 20M99,20M32,20G99,20G25,20G05,20M25,47D03,51F15,52B15,60J27 ©SpringerScience+BusinessMediaNewYork2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Coverillustration:DrawingofJ.C.FieldsbyKeithYeomans Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface The field of Algebraic Monoids owes a great deal to both Mohan Putcha and Lex Renner. In their hands, since the 1980s, this theory has turned into a full-fledged subjectwiththe2010MSCcode:20M32.Dedicatedtothe60thbirthdaysofPutcha andRenner,theInternationalWorkshoponAlgebraicMonoids,GroupEmbeddings, andAlgebraicCombinatoricstookplaceattheFieldsInstituteinToronto,Canada, fromJuly3toJuly6,2012. Thepurposeoftheworkshopwastostimulateresearchontheinterplaybetween algebraic monoids, group embeddings, and algebraic combinatorics by bringing together some of the principal investigators, junior researchers, and graduate students in these three areas, as well as to contribute to the increased synthesis of theseareas.Theresearchtalksgivenintheworkshopnotonlyreflectedthecurrent accomplishments of the invited speakers, but also outlined future directions of research.Asplanned,ithasledtoactivecollaborationsbetweentheparticipants.For example, shortly after the workshop Michel Brion and Lex Renner jointly proved that every algebraic monoid is strongly (cid:2)-regular, which settled a long standing openproblem. Theworkshophadtwomaincomponents.Thefirstcomponentconsistedofmini- courses on introductory topics for graduate students delivered by Michel Brion, EricJespers,andAnneSchilling.Thesetutorials,staggeredthroughoutthe4days, introduced the necessary background for the remaining 17 research talks, which formed the second component of the workshop. The invited talks were delivered byGeorgiaBenkart,NantelBergeron,TomDenton,StephenDoty,WenxueHuang, KiumarsKaveh,StuartMargolis,MohanPutcha,JanOknínski,LexRenner,Alvaro Rittatore, Yuval Roichman, Dewey Taylor, Ryan Therkelsen, Nicholas Thiéry, Sandeep Varma, and Monica Vazirani. The topics of these talks were diverse and varied. They included structure and representation theory of reductive algebraic monoids,monoidschemes,monoidsrelatedtoLietheory,equivariantembeddings of algebraic groups, constructions and properties of monoids from algebraic com- binatorics, endomorphism monoids induced from vector bundles, Hodge-Newton decompositionsofreductivemonoids,andapplicationsofmonoids. v vi Preface Putcha and Renner originated the systematic study of algebraic monoids inde- pendently around 1978. Putcha, at North Carolina State University, first obtained many foundational results from the semigroup point of view. He investigated Green’srelations,regularity,connectionsofregularmonoidstoreductivemonoids incharacteristiczero,semilattices,conjugacyclassesofidempotents,andsoon.In particular,heshowedtheexistenceofcross-sectionlatticesforirreduciblealgebraic monoids,connectingGreen’srelationstogroupactions.Soonafter,itbecameclear thatthisnotioniscloselyrelatedtoBorelsubgroups. AtthesametimeRennerbeganwritinghisthesisonthesubjectattheUniversity of British Columbia. This period witnessed the first wave of applications of alge- braicgeometryinalgebraicmonoids.Onepivotalresultwasthatreductivemonoids inanycharacteristicare(vonNeumann)regular,aresultthatdramaticallyinfluenced thelaterdevelopments.Inlate1980s,usingtheconnectionofcross-sectionlattices to Borel subgroups as a starting point, Renner classified all normal, semisimple monoids numerically in the spirit of classical Lie theory. As a consequence, J- irreduciblealgebraicmonoids(monoidswithauniqueJ-class)appeared.Hethen foundananalogueoftheBruhatdecompositionforreductivemonoidswithequally strikingconsequencesbyintroducingtheconceptofaRennermonoid,whichplays thesameroleformonoidsthattheWeylgroupdoesforgroups. Meanwhile, Putcha developed the monoid analogue of Tits’s theory of groups withBNpair,monoidsofLietype.Healsodiscoveredthateveryreductivemonoid hasatypemap,whichisthemonoidanalogueoftheDynkindiagramandbecomes the most important combinatorial invariant in the structure theory of reductive monoids.Aroundthistimeperiod,PutchaandRennertogetherdeterminedexplicitly thetypemapofJ-irreduciblealgebraicmonoids. In1990s,PutchagaveaclassificationofmonoidsofLietype,investigatedhighest weightcategoriesofrepresentations,andstudiedmonoidHeckealgebras.Oknínski andPutchashowedthateverycomplexrepresentationofafinitemonoidofLietype iscompletelyreducible,inparticularprovingthatthecomplexalgebraofthemonoid ofn(cid:2)nmatricesoverafinitefieldissemisimple.PutchaandRennersystematically studiedthecanonicalcompactificationofafinitegroupofLietypeandfoundthat the restriction of any irreducible modular representation of a finite monoid of Lie type to its unit group is still irreducible. Furthermore, they computed the number ofsuchirreduciblemodularrepresentations.Rennerintroducedtheconceptoffinite reductive monoids and studied modular representations of such monoids, showing thateachfinitereductivemonoidisamonoidofLietype.Heobtainedananalogue of the Tits system for reductive monoids by introducing a length function on the monoids. In the first decade of the twenty-first century, Putcha explored shellability, Bruhat-Chevalley order, root semigroups in reductive monoids, and parabolic monoids. Renner investigated Betti numbers of rationally smooth group embed- dings, blocks and representations of algebraic monoids, cellular decompositions (analogous to Schubert cells) of compactifications of a reductive group, descent systemsforBruhatposets,andH-polynomials. Preface vii Indeed,thetheoryoflinearalgebraicmonoidshasbeendevelopedsignificantly over the past three decades, due in large part to the efforts of Putcha and Renner. Meanwhile, it has also attracted researchers from different areas of mathematics becauseofitsconnectionstoalgebraicgroupemdeddings,algebraiccombinatorics, convex geometry, groups with BN-pairs, Lie theory, Kazhdan-Lusztig theory, semigrouptheory,andtoricvarietiesamongothers. Algebraic group embedding theory studies compactifications of algebraic groups. It incorporates torus embeddings and reductive monoids, and it provides us with a large and important class of spherical varieties. Some aspects of representation theory are related to the geometry of group embeddings, especially throughtheexamplesoflinearalgebraicmonoids. Algebraic combinatorics, which is concerned with discrete objects such as posets, permutations, and polytopes, is an ever-growing field of mathematics with increasingimportanceinotherdisciplinesincludingquantumchemistry,statistical biology, statistical physics, theoretical computer science, and so forth. Many questions in the combinatorial representation theory of algebraic monoids remain open. This volume contains the refereed proceedings of the workshop; all the papers were strictly refereed and are previously unpublished. We thank all the 35 partic- ipants including students, research experts, and speakers from Belgium, Canada, China, France, Israel, Poland, Turkey, Uruguay, and USA, and especially the authors whose papers are included here. We also thank all the referees who spent their valuable time reviewing these papers and providing useful suggestions for their improvement. We are grateful to the Fields Institute and the National Science Foundation of USA for the funding and support of this workshop. We thank the editorial staff of the Fields Institute Communications Series, as well as that of Springer, especially Ms. Debbie Iscoe and Dr. Carl Riehm for their kind cooperation,help,andguidanceinthepreparationofthisvolume. NewOrleans,USA MahirCan Aiken,USA ZhenhengLi NewYork,USA BenjaminSteinberg Ottawa,Canada QiangWang Contents OnAlgebraicSemigroupsandMonoids....................................... 1 MichelBrion AlgebraicSemigroupsAreStrongly(cid:2)-Regular .............................. 55 MichelBrionandLexE.Renner ReesTheoremandQuotientsinLinearAlgebraicSemigroups............. 61 MohanS.Putcha RepresentationsofReductiveNormalAlgebraicMonoids.................. 87 StephenDoty OnLinearHodgeNewtonDecompositionforReductiveMonoids......... 97 SandeepVarma TheStructureofAffineAlgebraicMonoidsinTermsofKernelData ..... 119 WenxueHuang AlgebraicMonoidsandRennerMonoids ..................................... 141 ZhenhengLi,ZhuoLi,andYou’anCao ConjugacyDecompositionofCanonicalandDualCanonicalMonoids.... 189 RyanK.Therkelsen TheEndomorphismsMonoidofaHomogeneousVectorBundle........... 209 L.Brambila-PazandAlvaroRittatore OnCertainSemigroupsDerivedfromAssociativeAlgebras................ 233 JanOknin´ski TheBettiNumbersofSimpleEmbeddings.................................... 247 LexE.Renner SL -RegularSubvarietiesofCompleteQuadrics............................. 271 2 MahirBilenCanandMichaelJoyce ix
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