Table Of ContentEncyclopaediaofMathematicalSciences
Volume135
InvariantTheoryandAlgebraicTransformationGroupsVI
SubseriesEditors:
R.V.Gamkrelidze V.L.Popov
Martin Lorenz
Multiplicative
Invariant Theory
123
Author
MartinLorenz
DepartmentofMathematics
TempleUniversity
Philadelphia,PA19122,USA
e-mail:lorenz@math.temple.edu
FoundingeditoroftheEncyclopaediaofMathematicalSciences:
R.V.Gamkrelidze
MathematicsSubjectClassification(2000):
Primary:13A50
Secondary:13H10,13D45,20C10,12F20
ISSN0938-0396
ISBN3-540-24323-2SpringerBerlinHeidelbergNewYork
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Tomy mother, Martha Lorenz,
and
tothe memory ofmyfather,
AdolfLorenz(1925 – 2001)
Preface
Multiplicative invariant theory, as a research area in its own right, is of relatively
recentvintage:thesystematicinvestigationofmultiplicativeinvariantswasinitiated
byDanielFarkasinthe1980s.Sincethenthesubjecthasbeenpursuedbyasmallbut
growingnumberofresearchers,andatthispointithasreachedastageinitsdevel-
opmentwhereacoherentaccountofthebasicresultsachievedthusfarisdesirable.
Suchisthegoalofthisbook.
The topic of multiplicative invariant theory is intimately tied to integral repre-
sentations of finite groups. Therefore, the field has a predominantly discrete, alge-
braicflavor.Geometry,specificallythetheoryofalgebraicgroups,entersthepicture
throughWeylgroupsandtheirrootlatticesaswellasviacharacterlatticesofalge-
braictori.
Ihavetriedtokeepthisbookreasonablyself-contained.Thecoreresultsonmul-
tiplicative invariants are presented with complete proofs often improving on those
found in the literature. The prerequisites from representation theory and the theory
ofrootsystemsareassembledearlyinthetext,forthemostpartwithreferencesto
CurtisandReiner[44],[45]andBourbaki[24].
Formultiplicativeinvariantalgebras,Chapters3–8giveanessentiallycomplete
accountofthestateofthesubjecttodate.Ontheotherhand,moreisknownabout
multiplicative invariant fields than what found its way into this book. The reader
maywishtoconsultthemonographsbySaltman[186]andVoskresenski˘ı[220]for
additionalinformation.Anovelfeatureofthepresenttextisthefullandstreamlined
derivation of the known rationality properties of the field of matrix invariants in
Chapter9.Thismaterialcouldheretoforeonlybefoundintheoriginalsourceswhich
arewidelyspreadintheliterature.
A more detailed overview of the contents of this book is offered in the Intro-
ductionbelow.Inaddition,eachofthesubsequentchaptershasitsownintroductory
section;thoseofChapters4–9arequiteextensive,givingcompletestatementsofthe
main results proved and delineating the pertinent algebraic background. The book
concludeswithachapteronresearchproblems.Ihopethatitwillstimulatefurther
interestinthefieldofmultiplicativeinvarianttheory.
VIII Preface
Acknowledgements
Dan Farkas’ early work on multiplicative invariants ([59], [60], [61]) provided the
initial impetus for my own research on the subject. I also owe an enormous debt
to Don Passman at whose invitation I spent a postdoctoral year at UW-Madison in
1978/9andwhohasbeenagoodfriendeversince.Inmanyways,Ihavetriedtoem-
ulatehisbook“InfiniteGroupRings”[147]inthesenotes.Morerecently,Ihaveben-
efittedfromtheinsightsofNicoleLemireandZinovyReichstein,boththroughtheir
own work on multiplicative invariant theory and through collaborations on related
topics.Forcommentsonearlierdraftsandhelpwithvariousaspectsofthisbook,I
wishtothankJacquesAlev,EstherBeneish,FrankDeMeyer,SteveDonkin,Victor
Guba,JensCarstenJantzen,GregorKemper,GabrieleNebe,WilhelmPlesken,and
DavidSaltman.
IamgratefultoVladimirPopovforinvitingmetowritethisaccountofmyfa-
voriteresearchareaandtotheanonymousrefereesfortheirconstructivesuggestions.
Whilewritingthisbook,IwassupportedinpartbytheNationalScienceFoun-
dationundergrantDMS-9988756andbytheLeverhulmeFoundation(ResearchIn-
terchangeGrantF/00158/X).
My greatest debt is to my family: to my daughter Esther for her unwavering
support over many years and long distances and to my wife Maria and our chil-
dren,Gabriel,DaliaandAidan,whoselovingandenergeticpresenceiskeepingme
grounded.
TempleUniversity,Philadelphia MartinLorenz
December2004
Contents
Introduction ....................................................... 1
NotationsandConventions .......................................... 9
1 GroupsActingonLattices ...................................... 13
1.1 Introduction ............................................... 13
1.2 G-Lattices ................................................ 13
1.3 Examples ................................................. 14
1.4 StandardLatticeConstructions ............................... 16
1.5 IndecomposableLattices .................................... 18
1.6 ConditioningtheLattice..................................... 20
1.7 ReflectionsandGeneralizedReflections........................ 21
1.8 LatticesAssociatedwithRootSystems ........................ 24
1.9 TheRootSystemAssociatedwithaFaithfulG-Lattice............ 27
1.10 FiniteSubgroupsofGLn((cid:0) ) ................................. 28
2 PermutationLatticesandFlasqueEquivalence .................... 33
2.1 Introduction ............................................... 33
2.2 PermutationLattices ........................................ 33
2.3 StablePermutationEquivalence .............................. 34
2.4 PermutationProjectiveLattices............................... 35
(cid:1)
2.5 Hi-trivial,FlasqueandCoflasqueLattices...................... 36
2.6 FlasqueandCoflasqueResolutions............................ 37
2.7 FlasqueEquivalence ........................................ 38
2.8 Quasi-permutationLatticesandMonomialLattices .............. 39
2.9 AnInvariantforFlasqueEquivalence.......................... 40
2.10 OverviewofLatticeTypes ................................... 42
2.11 RestrictiontotheSylowNormalizer........................... 43
2.12 SomeS -Lattices .......................................... 44
n
X Contents
3 MultiplicativeActions .......................................... 51
3.1 Introduction ............................................... 51
3.2 TheGroupAlgebraofaG-Lattice ............................ 51
3.3 ReductiontoFiniteGroups,(cid:0) -structure,andFiniteGeneration .... 52
3.4 UnitsandSemigroupAlgebras ............................... 53
3.5 Examples ................................................. 55
3.6 MultiplicativeInvariantsofWeightLattices..................... 61
3.7 PassagetoanEffectiveLattice................................ 63
3.8 TwistedMultiplicativeActions ............................... 64
3.9 HopfStructure............................................. 66
3.10 TorusInvariants............................................ 67
4 ClassGroup ................................................... 69
4.1 Introduction ............................................... 69
4.2 SomeExamples............................................ 70
4.3 KrullDomainsandClassGroups ............................. 70
4.4 Samuel’sExactSequence.................................... 72
4.5 GeneralizedReflectionsonRings ............................. 73
4.6 ProofofTheorem4.1.1...................................... 75
5 PicardGroup.................................................. 77
5.1 Introduction ............................................... 77
5.2 InvertibleModules ......................................... 78
5.3 TheSkewGroupRing ...................................... 79
5.4 TheTraceMap............................................. 80
5.5 TheKerneloftheMapPic(RG)→Pic(R) .................... 81
5.6 TheCaseofMultiplicativeActions............................ 83
6 MultiplicativeInvariantsofReflectionGroups..................... 85
6.1 Introduction ............................................... 85
6.2 ProofofTheorem6.1.1...................................... 86
6.3 ComputingtheRingofInvariants ............................. 87
6.4 SAGBIBases.............................................. 91
7 Regularity..................................................... 95
7.1 Introduction ............................................... 95
7.2 ProjectivityoverInvariants................................... 96
7.3 LinearizationbytheSliceMethod............................. 97
7.4 ProofofTheorem7.1.1...................................... 99
7.5 RegularityattheIdentity .................................... 100
8 TheCohen-MacaulayProperty .................................. 103
8.1 Introduction ............................................... 103
8.2 HeightandGrade .......................................... 104
8.3 LocalCohomology ......................................... 105
8.4 Cohen-MacaulayModulesandRings .......................... 106
Contents XI
8.5 TheCohen-MacaulayPropertyforInvariantRings............... 107
8.6 TheEllingsrud-SkjelbredSpectralSequences ................... 110
8.7 AnnihilatorsofCohomologyClasses .......................... 112
8.8 TheRestrictionMapforCohen-MacaulayInvariants ............. 114
8.9 TheCaseofMultiplicativeInvariants .......................... 116
8.10 ProofofTheorem8.1.1...................................... 118
8.11 Examples ................................................. 121
9 MultiplicativeInvariantFields................................... 125
9.1 Introduction ............................................... 125
9.2 StableIsomorphism ........................................ 128
9.3 RetractRationality ......................................... 128
9.4 The“No-nameLemma” ..................................... 131
9.5 FunctionFieldsofAlgebraicTori ............................. 133
9.6 SomeRationalityResultsforMultiplicativeInvariantFields....... 138
9.7 SomeConceptsfromAlgebraicGeometry...................... 141
9.8 TheFieldofMatrixInvariantsasaMultiplicativeInvariantField... 142
10 Problems...................................................... 149
10.1 TheCohen-MacaulayProblem ............................... 149
10.2 SemigroupAlgebras ........................................ 151
10.3 ComputationalIssues ....................................... 154
10.4 EssentialDimensionEstimates ............................... 155
10.5 RationalityProblems........................................ 159
References......................................................... 161
Index ............................................................. 173
Introduction
TheSetting
MultiplicativeactionsarisefromarepresentationG→GL(L)ofagroupGonalat-
ticeL.Thus,Lisafree(cid:0) -moduleoffiniterankonwhichGactsbyautomorphisms,
aG-latticeforshort.TheG-actiononLextendsuniquelytoanactionby (cid:0) -algebra
automorphismsonthegroupalgebra(cid:0) [L]overanychosencommutativebasering(cid:0) .
Multiplicativeinvarianttheoryisconcernedwiththestudyofthesubalgebra
(cid:0) [L]G ={f ∈ (cid:0) [L]|g(f)=f forallg ∈G}
ofallG-invariantelementsof(cid:0) [L],themultiplicativeinvariantalgebra(over(cid:0) )that
isassociatedwiththeG-latticeL.
The terminology “multiplicative”, introduced by Farkas [59], derives from the
fact that, inside (cid:0) [L], the lattice L becomes a multiplicative subgroup of the group
(cid:1) (cid:1)
U((cid:0) [L])ofunitsof (cid:0) [L].Indeed,identifyingLwith n bychoosinga -basis,the
(cid:1)
G-actiononLisgivenbymatricesinGLn( ),thegroupalgebra (cid:0) [L]becomesthe
Laurentpolynomialalgebra(cid:0) [x±1,...,x±1],andtheimageofLin(cid:0) [L]isthegroup
1 n
of monomials in the variables x and their inverses. For this reason, multiplicative
i
actions are sometimes called monomial actions or purely monomial actions (in or-
der to distinguish them from their twisted versions; see below). The terminologies
“exponentialactions”[24]or“latticeactions”canalsobefoundintheliterature.
Variousgeneralizationsofthisbasicset-upareofinterest,notablytheso-called
twistedmultiplicativeactions.Here,thethegroupGactsonthegroupring (cid:0) [L]of
alatticeLbyringautomorphismsthataremerelyrequiredtomap (cid:0) toitself.Thus,
GalsoactsonU((cid:0) [L])/U((cid:0) ).Foradomain (cid:0) ,thelattergroupisisomorphictoL,
therebymakingLaG-latticeinthetwistedsettingaswell.If (cid:0) isafield,thegroup
algebra(cid:0) [L]isoftenreplacedbyitsfieldoffractions,denotedby(cid:0) (L).AnyG-action
on(cid:0) [L]extendsuniquelyto(cid:0) (L).Inthecaseof(twisted)multiplicativeactions,the
resultingfields(cid:0) (L)withG-actionarecalled(twisted)multiplicativeG-fields.
(cid:1) (cid:1)
Example.LetS denotethesymmetricgroupon{1,2,3}andletL = a ⊕ a
3 1 2
bealatticeofrank2.Sendingthegeneratorss=(1,2)andt=(1,2,3)ofS tothe
3
