Combinatorics of Minuscule Representations

329 Pages·2013·1.69 MB·English

by  R. M. Green

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199 CombinatoricsofMinusculeRepresentations
CAMBRIDGE TRACTS IN MATHEMATICS
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Combinatorics of Minuscule
Representations
R.M.GREEN
UniversityofColoradoBoulder
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Contents
Introduction page1
1 ClassicalLiealgebrasandWeylgroups 5
1.1 Liealgebras 5
1.2 TheclassicalLiealgebras 8
1.3 ClassicalLiealgebrasandpartiallyorderedsets 11
1.4 ClassicalWeylgroupsandpartiallyorderedsets 16
1.5 Notesandreferences 18
2 Heapsovergraphs 19
2.1 Basicdefinitions 20
2.2 FullheapsoverDynkindiagrams 26
2.3 Localstructureoffullheaps 30
2.4 Quotientheaps 38
2.5 Notesandreferences 40
3 Weylgroupactions 42
3.1 Linearoperatorsandgroupactions 43
3.2 Properideals 55
3.3 Parabolicsubheaps 62
3.4 Notesandreferences 67
4 Lietheory 70
4.1 RepresentationsofLiealgebrasfromheaps 70
4.2 ReviewofLietheory 75
4.3 ReviewofWeylgroups 80
4.4 Stronglyorthogonalsets 86
4.5 Notesandreferences 88
5 Minusculerepresentations 89
5.1 Highestweightmodules 89
5.2 Weightsandheaps 92
5.3 Periodicityandtrivialization 96
5.4 Reflections 107
5.5 Minusculerepresentationsfromheaps 112
v
vi Contents
5.6 Invariantbilinearforms 115
5.7 Notesandreferences 116
6 FullheapsoveraffineDynkindiagrams 118
6.1 FullheapsintypeA(1) 118
l
6.2 ProperidealsintypeA(1) 125
l
6.3 SpinrepresentationsintypeD 129
l
6.4 TypesB(1)andD(2) 134
l l+1
6.5 FullheapsintypeE(1)andE(1) 139
6 7
6.6 TheclassificationoffullheapsoveraffineDynkindiagrams 144
6.7 Notesandreferences 147
7 Chevalleybases 148
7.1 Kac’sasymmetryfunction 148
7.2 RelationsinsimplylacedsimpleLiealgebras 153
7.3 Folding 160
7.4 Longandshortroots 165
7.5 Relationsinnon-simplylacedsimpleLiealgebras 175
7.6 Notesandreferences 181
8 CombinatoricsofWeylgroups 183
8.1 Minusculesystems 183
8.2 Weylgroupsaspermutationgroups 188
8.3 Idealsofroots 197
8.4 Weightpolytopes 202
8.5 Facesofweightpolytopes 207
8.6 Graphsfromminusculerepresentations 211
8.7 Notesandreferences 214
9 The28bitangents 216
9.1 TheGossetgraph 216
9.2 DelPezzosurfaces 219
9.3 Bitangents 226
9.4 Hesse–Cayleynotation 231
9.5 Steinercomplexes 237
9.6 Symplecticstructure 244
9.7 Notesandreferences 247
10 Exceptionalstructures 248
10.1 The27linesonacubicsurface 249
10.2 Combinatoricsofdoublesixes 253
10.3 2-graphs 258
10.4 Generalizedquadrangles 265
10.5 Higherinvariantforms 269
10.6 Notesandreferences 273
11 Furthertopics 275
11.1 MinusculeelementsofWeylgroups 275
Contents vii
11.2 Principalsubheapsasabstractposets 280
11.3 Gaussianposets 284
11.4 Jeudetaquin 289
11.5 Notesandreferences 296
AppendixA Posets,graphsandcategories 298
AppendixB Lietheoreticdata 304
References 307
Index 311

Introduction
Highestweightmodulesplayakeyroleintherepresentationtheoryofseveralclasses
of algebraic objects occurring in Lie theory, including Lie algebras, Lie groups,
algebraic groups, Chevalley groups and quantized enveloping algebras. In many of
the most important situations, the weights may be regarded as points in Euclidean
space, Rn, and there is a finite group (called a Weyl group) that acts on the set
of weights by linear transformations. The minuscule representations are those for
which the Weyl group acts transitively on the weights, and the highest weight of
sucharepresentationiscalledaminusculeweight.Theterm“minusculeweight”is
a translation of Bourbaki’s term poids minuscule [8, VIII, section 7.3]; the spelling
“miniscule” is also found in the literature, although less commonly, and Russian-
speaking authors often call minuscule weights microweights. The list of minuscule
representationsisasfollows:allfundamentalrepresentationsintypeA ,thenatural
n
representationsintypesC andD ,thespinrepresentationsintypesB andD ,the
n n n n
two27-dimensionalrepresentationsintypeE andthe56-dimensionalrepresentation
6
intypeE .
7
Minusculeweightsandminusculerepresentationsareimportantbecausetheyoccur
inawidevarietyofcontextsinmathematicsandphysics,especiallyinrepresentation
theory and algebraic geometry. Minuscule representations are the starting point of
StandardMonomialTheorydevelopedbyLakshmibai,Seshadriandothers[42],and
theyplayakeyroleinthegeometryofSchubertvarieties.
One of the advantages of minuscule representations is that they are often much
easier to understand than typical representations of the same objects. For example,
Seshadri [70] proves that standard monomials give a basis for the homogeneous
coordinateringofG/P intheminusculecase,whichissignificantlymoretractable
than the general case. Littelmann’s path model for representations of Lie algebras
[45] is also much simpler when minuscule representations are involved. Minuscule
representationshavemanyotheralgebraicapplications:theyareusefulinthetheory
ofChevalleygroups,wheretheyareusedtoconstructminusculeweightgeometries
[5, section 6], and they also appear in the context of Bruhat decompositions [86].
Certainquestionsintherepresentationtheoryofalgebraicgroupscanbereducedto
questionsaboutminusculerepresentations[80].InthetheoryofMacdonaldpolyno-
mials[85,section6],minusculeweightsareusedtodefinetheMacdonaldoperators.
1