Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics

360 Pages·2014·4.524 MB·English

by  Mahir Can, Zhenheng Li, Benjamin Steinberg, Qiang Wang (eds.)

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
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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)
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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