Supersymmetric Schur functions and Lie superalgebra representations

221 Pages·2007·1.446 MB·English

by  Els Moens

FACULTEIT WETENSCHAPPEN
Supersymmetric Schur functions
and Lie superalgebra
representations
Els Moens
Promotor: Prof. dr. Joris Van der Jeugt
Proefschriftingediendtothetbehalenvandegraadvan
DoctorindeWetenschappen: Wiskunde
FaculteitWetenschappen
VakgroepToegepasteWiskundeenInformatica
Academiejaar2006–2007

Dankwoord
Bijdevoltooiingvanditdoctoraatsproefschriftrichtikgraageenwoordvandankaan
iedereendiemijdevoorbijejarenmetraadendaadheeftbijgestaan.
In de eerste plaats wil ik prof. Joris Van der Jeugt bedanken. Dankzij zijn exper-
tise in de wereld van de supersymmetrische functies en Lie superalgebra’s werd dit
proefschrift voor mij een ware ontdekkingstocht. Hij gaf mij zeer gerichte lees- en
onderzoekopdrachten zodat ik langzamerhand ook mijn weg vond in deze – voor mij
nogonbekende–wereld. Zonderzijnantwoordenopdievelevragen,debegeleiding,
detalrijkesuggestiesenhintenzoudezethesisnooittotstandzijngekomen. Bedankt,
Joris!
Verderwensikookde(ex-)collega’stebedankenvoordeaangenamewerkomge-
ving. Enkelen wens ik hier afzonderlijk te vermelden: mijn bureaugenoten door de
jaren heen: Karim, Tom, Annelies en Lizzy, zij moesten immers alles als eerste aan-
horen, van de vreugde bij het vinden van een bewijs, over het praten over koetjes en
kalfjes heen tot de stress bij het schrijven van een doctoraatsthesis; Stijn, voor het
luisterend oor, de vriendschap en de antwoorden op heel wat vragen; Glad, voor het
oplossenvanhetzoveelsteLATEX-probleem; entenslotteAnn, Wouter, Paul, Herbert,
KatiaenStefanievoordelogistiekesteunennogveelmeer.
Verderwilikookmijnouders, familieenvriendenbedanken. Mijnouderswilik
danken, omdat ze mij de kans hebben gegeven om te studeren, zij waren steeds daar
ommijtemotiverenenteblijvensteunen;familieenvriendenomdatjulliesteedsge-
interesseerdbleveninwatikdeed;enBartomdathijsteedsinmijisblijvengelovenen
omwillevandeonvoorwaardelijkesteunzodatikondanksallestochhebdoorgezet;en
tenslotte,onsdochtertjeJana,voordeontwapenendelachbijhethorenvanhetverhaal
vandedag.
ElsMoens,
Gent,december2006

Contents
Preface ix
1 SymmetricSchurFunctions 1
1.1 Partitionsandcompositepartitions . . . . . . . . . . . . . . . . . . . 1
1.1.1 Partitionsanddiagrams . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Skewdiagramsandtableaux . . . . . . . . . . . . . . . . . . 3
1.1.3 Compositepartitions . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Theringofsymmetricfunctions . . . . . . . . . . . . . . . . . . . . 5
1.3 SymmetricSchurfunctions . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 SymmetricSchurfunctionsindexedbyapartitionλ. . . . . . 7
1.3.2 SymmetricSchurfunctionsindexedbyacompositepartitionν¯;µ 11
1.4 Generatingfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 PropertiesofsymmetricSchurfunctions . . . . . . . . . . . . . . . . 15
1.5.1 Transitionmatrices . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.2 OperationsonsymmetricSchurfunctions . . . . . . . . . . . 16
1.5.3 Generalizationstothecaseofcompositepartitions . . . . . . 19
1.6 Appendix: Formulary . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.6.1 SymmetricSchurfunctionsindexedbyapartitionλ. . . . . . 24
1.6.2 SymmetricSchurfunctionsindexedbyacompositepartitionν¯;µ 26
2 SupersymmetricSchurFunctions 29
2.1 Supersymmetricfunctionsindexedbyλ . . . . . . . . . . . . . . . . 30
2.1.1 Elementaryandcompletesupersymmetricfunctions . . . . . 30
2.1.2 SupersymmetricSchurfunctions . . . . . . . . . . . . . . . . 31
2.1.3 Supertableaux. . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Supersymmetricbases . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.1 Supersymmetricpowerfunctions . . . . . . . . . . . . . . . 34
vi Contents
2.2.2 Themonomialsupersymmetricfunctionsandtheforgottensu-
persymmetricfunctions. . . . . . . . . . . . . . . . . . . . . 35
2.2.3 Generatingfunctions . . . . . . . . . . . . . . . . . . . . . . 38
2.3 SupersymmetricSchurfunctionsindexedbyν¯;µ . . . . . . . . . . . 40
2.3.1 Definitionandproperties . . . . . . . . . . . . . . . . . . . . 40
2.3.2 (m|n)-standardcompositepartitionsandsupertableaux . . . . 42
2.4 Appendix: Formulary . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4.1 Supersymmetricfunctionsindexedbya(skew)partition . . . 50
2.4.2 SupersymmetricSchurfunctionsindexedbyacompositepar-
titionν¯;µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Liesuperalgebrasandtheirrepresentations 55
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 DefinitionofLiesuperalgebras . . . . . . . . . . . . . . . . . . . . . 56
3.3 TheenvelopingalgebraofaLiesuperalgebra . . . . . . . . . . . . . 60
3.4 ACartansubalgebraandsimplerootsystems . . . . . . . . . . . . . 60
3.5 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5.1 Irreducibleandfaithfulrepresentations . . . . . . . . . . . . 63
3.5.2 Highestweightrepresentations . . . . . . . . . . . . . . . . . 64
3.5.3 Typicalandatypicalrepresentationsandcharacterformulas . 65
3.6 Highestweightandcompositepartitions . . . . . . . . . . . . . . . . 67
3.7 Covariant,contravariantandmixedtensormodules . . . . . . . . . . 70
3.8 AtypicalityandtheatypicalitymatrixofΛ . . . . . . . . . . . . . . . 71
4 AdeterminantalformulaforsupersymmetricSchurpolynomials 75
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Covariantmodulesforgl(m|n) . . . . . . . . . . . . . . . . . . . . . 78
4.3 Adeterminantalformulafors (x/y). . . . . . . . . . . . . . . . . . 87
λ
4.4 Fourcharacterizingproperties . . . . . . . . . . . . . . . . . . . . . 91
4.5 AproofofcoincidencewiththeSergeev-Pragaczformula . . . . . . . 95
5 DeterminantalformulaforcompositesupersymmetricS-functions 101
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2 Normally,criticallyandquasicriticallyrelatedroots . . . . . . . . . . 103
5.3 Someexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.4 Acharacterformulaforaspecialclassofcompositepartitions . . . . 111
5.5 Adeterminantalformulaforch(V ) . . . . . . . . . . . . . . . . . 115
ν¯;µ
5.6 Thecharacterformulaands (x/y) . . . . . . . . . . . . . . . . . . 118
ν¯;µ
Contents vii
6 Dimensionformulasforgl(m|n)representations 131
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 t-dimensionformulaforcovariantrepresentations . . . . . . . . . . . 132
6.2.1 Aformulaforthet-dimension . . . . . . . . . . . . . . . . . 135
6.2.2 Furthersimplifications,examplesandapplications . . . . . . 142
6.3 t-dimensionformulaformixedtensorrepresentations . . . . . . . . . 145
6.3.1 Aformulaforthet-dimension . . . . . . . . . . . . . . . . . 145
6.3.2 Thet-dimensioninthespecialcaseν¯;µ=(bc);(bc) . . . . . 151
A Nederlandstaligesamenvatting 175
A.1 SymmetrischeSchur-functies . . . . . . . . . . . . . . . . . . . . . . 175
A.1.1 Partitiesensamengesteldepartities . . . . . . . . . . . . . . 175
A.1.2 Deringvansymmetrischefuncties . . . . . . . . . . . . . . . 176
A.1.3 SymmetrischeSchur-functies . . . . . . . . . . . . . . . . . 177
A.2 SupersymmetrischeSchur-functies . . . . . . . . . . . . . . . . . . . 178
A.2.1 Supersymmetrischefunctiesge¨ındexeerddoorλ . . . . . . . 178
A.2.2 Supersymmetrischebasissen . . . . . . . . . . . . . . . . . . 180
A.2.3 SupersymmetrischeSchur-functiesge¨ındexeerddoorν¯;µ. . . 181
A.3 Liesuperalgebra’senrepresentaties . . . . . . . . . . . . . . . . . . 182
A.3.1 Definities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
A.3.2 Hoogstegewichtenversussamengesteldepartities . . . . . . 184
A.3.3 Covariante,contravarianteengemengdetensormodulen . . . 185
A.3.4 AtypicaliteitendeatypicaliteitsmatrixvanΛ . . . . . . . . . 185
A.4 EendeterminantformulevoorsupersymmetrischeS-functies . . . . . 186
A.4.1 Covariantemoduleningl(m|n)zijntam . . . . . . . . . . . . 186
A.4.2 Eendeterminantformulevoors (x/y) . . . . . . . . . . . . . 188
λ
A.4.3 Vierkarakteriserendeeigenschappen. . . . . . . . . . . . . . 189
A.4.4 Eendirectenonafhankelijkbewijs . . . . . . . . . . . . . . . 189
A.5 Determinantformulevoorsupersymmetrischefunctiess (x/y) . . . 190
ν¯;µ
A.5.1 Inleiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
A.5.2 Normaal,kritischenquasikritischgerelateerdewortels . . . . 190
A.5.3 Eenkarakterformulevoorkritischerepresentaties . . . . . . . 192
A.5.4 Dekarakterformuleens (x/y) . . . . . . . . . . . . . . . 194
ν¯;µ
A.6 Dimensieformulesvoorrepresentatiesingl(m|n) . . . . . . . . . . . 195
A.6.1 Inleiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
A.6.2 t-dimensievancovarianterepresentaties . . . . . . . . . . . . 195
A.6.3 t-dimensievangemengdetensorrepresentaties . . . . . . . . 197
B Erratum 199
viii Contents
Bibliography 201
Preface
“Fermionen und Bosonen sind fundamental verschiedene Typen von Quan-
tenteilchen. So wenig ein gewo¨hnlicher Spiegel einen Apfel wie eine Birne
aussehen la¨st, so wenig vermag eine der u¨blichen Symmetrien Fermionen in
Bosonenzuverwandeln. DasschaffterstderZauberspiegelderSupersymme-
trie.” (Prof.Dr.J.Jolie[30])
Lie superalgebras and their representations continue to play an important role in
theunderstandingandexploitationofsupersymmetryinphysicalsystems. TheLiesu-
peralgebrasunderconsiderationhere,namelygl(m|n)orsl(m|n)(sometimesdenoted
by U(m|n) or SU(m|n)), have applications in quantum mechanics [1, 42], nuclear
physics [6, 14, 27], string theory [19, 23], conformal field theory [21], supergrav-
ity [2, 35], M-theory [20], lattice QCD [7, 9, 15], solvable lattice models [62], spin
systems [24] and quantum systems [58]. Also their affine extensions [21, 24] or q-
deformations [1, 58] play an important role. In most of the applications, it are the
irreducible representations or “multiplets” of gl(m|n) (sometimes referred to as sim-
ple gl(m|n) modules) that play a role. But, despite the fact that Lie superalgebras
andtheirrepresentationshavebeenthesubjectofmuchattention,thereisnocomplete
description of the finite-dimensional complex irreducible representations of even the
simplestfamilyofbasicclassicalLiesuperalgebras,sl(m|n).
RepresentationtheoryofLiesuperalgebras,andinparticularofgl(m|n)oritssim-
ple counterpart sl(m|n), is not a straightforward copy of the corresponding theory
for simple Lie algebras. The development of gl(m|n) representation theory is quite
remarkable. Shortly after the classification of finite-dimensional simple Lie superal-
gebras [31, 59], Kac considered the problem of classifying all finite-dimensional ir-
reduciblerepresentationsofthebasicclassicalLiesuperalgebras[32]. Forasubclass
of these irreducible representations, known as “typical” representations, Kac derived
a character formula closely analogous to the Weyl character formula for irreducible
x Preface
representationsofsimpleLiealgebras[32]. Theproblemofobtainingacharacterfor-
mula for the remaining “atypical” irreducible representations has been the subject of
intensiveinvestigation.
From a computational and practical point of view, it is useful to identify charac-
ters with supersymmetric S-functions, since it is easy to work with S-functions, for
whichmanypropertiesareknown. Althoughthisidentificationholdsforcovariantand
contravariantirreduciblerepresentations[10,22],wherethecorrespondingS-function
islabelledbyasinglepartitionλ,itfailsformixedtensorirreduciblerepresentations,
wherethecorrespondingS-functionislabelledbyacompositepartitionν¯;µ.Theprob-
lemiswelldescribedandanalysedin[70],wherefurthermoreacharacterformulafor
atypical gl(m|n) irreducible representations is conjectured. Since then, some partial
solutions to this problem were given, e.g. for so-called generic representations [53],
forsinglyatypicalrepresentations[12,68,71],orfortamerepresentations[34]. More
recently,thecharacterproblemforgl(m|n)wasprincipallysolvedbySerganova[60],
whogaveanalgorithmtocomputecompositionfactormultiplicitiesofso-calledKac-
modules,andthusindirectlythecharacter. In[73],asubstantiallysimplermethodwas
conjectured to compute these composition factor multiplicities; this conjecture was
proved by Brundan [13]. Still, the method using composition factor multiplicities of
Kac-modules remains a rather indirect way of computing characters. Recently, there
wasafurtherbreakthroughforthisproblem. DevelopingontheworkofBrundan,Yu-
caiSuandZhang[66]managedtocomputethegeneralizedKazhdan-Lusztigpolyno-
mialsofgl(m|n)irreduciblerepresentations, leadingtoarelativelyexplicitcharacter
formula for all these irreducible representations, and thus proving that the character
formulaconjecturedin[70]holds.
ThemainideaofChapter1istofixsomenotationandterminology,andtodiscuss
brieflytheringofsymmetricfunctions. Asalreadymentioned, symmetricandsuper-
symmetricfunctionswillbeparametrizedbyeitherpartitionsorcompositepartitions.
Those objects play an important role in this thesis. So, we will give their definition,
their representation by means of Young diagrams and we will explain the notion of
(composite) tableaux. In the second section we consider the ring of symmetric func-
tions. Thebases,knownforthisring,arediscussedbriefly. Nexttothosebases,there
stillexistsanotherandevenmoreimportantbasis: thesymmetricSchurfunctions. As
severalbasesoftheringofsymmetricfunctionsaredefined,therelationbetweenthose
basesisgivenbymeansoftransitionmatrices. Duetotherelationbetweensymmetric
functionsandcharactersoftheirreduciblerepresentations,theremainderofthechapter
isdedicatedtothesymmetricSchurfunctions. Herewemakeadifferencebeweenthe
symmetric functions parametrized by either a partition or a composite partition. For
bothfamilies,wegivethedefinitionandalotofpropertiesandformulas. Someprop-