Table Of ContentGroupTheory PUPcopyeditorversion8.7,September27,2007
Group Theory
GroupTheory PUPcopyeditorversion8.7,September27,2007
GroupTheory PUPcopyeditorversion8.7,September27,2007
Group Theory
Birdtracks, Lie’s, and Exceptional Groups
Predrag Cvitanovic´
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
GroupTheory PUPcopyeditorversion8.7,September27,2007
Copyright©2008byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress
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ISBN:978-0-691-11836-9(alk. paper)
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GroupTheory PUPcopyeditorversion8.7,September27,2007
dedicatedtothememoryof
BorisWeisfeilerandWilliam E.Caswell
GroupTheory PUPcopyeditorversion8.7,September27,2007
Contents
Acknowledgments xi
Chapter1. Introduction 1
Chapter2. Apreview 5
2.1 Basicconcepts 5
2.2 Firstexample: SU(n) 9
2.3 Secondexample: E6family 12
Chapter3. Invariantsandreducibility 15
3.1 Preliminaries 15
3.2 Definingspace,tensors,reps 19
3.3 Invariants 20
3.4 Invariancegroups 23
3.5 Projectionoperators 24
3.6 Spectraldecomposition 26
Chapter4. Diagrammaticnotation 28
4.1 Birdtracks 28
4.2 Clebsch-Gordancoefficients 30
4.3 Zero-andone-dimensionalsubspaces 33
4.4 Infinitesimaltransformations 33
4.5 Liealgebra 37
4.6 OtherformsofLiealgebracommutators 39
4.7 ClassificationofLiealgebrasbytheirprimitiveinvariants 39
4.8 Irrelevancyofclebsches 40
4.9 Abriefhistoryofbirdtracks 41
Chapter5. Recouplings 44
5.1 Couplingsandrecouplings 44
5.2 Wigner3n-jcoefficients 47
5.3 Wigner-Eckarttheorem 48
Chapter6. Permutations 51
6.1 Symmetrization 51
6.2 Antisymmetrization 53
6.3 Levi-Civitatensor 55
6.4 Determinants 57
6.5 Characteristicequations 59
GroupTheory PUPcopyeditorversion8.7,September27,2007
viii CONTENTS
6.6 Fully(anti)symmetrictensors 59
6.7 Identicallyvanishingtensors 60
Chapter7. Casimiroperators 62
7.1 CasimirsandLiealgebra 63
7.2 Independentcasimirs 64
7.3 Adjointrepcasimirs 66
7.4 Casimiroperators 67
7.5 Dynkinindices 68
7.6 Quadratic,cubiccasimirs 71
7.7 Quarticcasimirs 72
7.8 Sundryrelationsbetweenquarticcasimirs 73
7.9 Dynkinlabels 77
Chapter8. Groupintegrals 79
8.1 Groupintegralsforarbitraryreps 80
8.2 Characters 82
8.3 Examplesofgroupintegrals 83
Chapter9. Unitarygroups 85
9.1 Two-indextensors 85
9.2 Three-indextensors 86
9.3 Youngtableaux 87
9.4 Youngprojectionoperators 93
9.5 Reductionoftensorproducts 97
9.6 U(n)recouplingrelations 101
9.7 U(n)3n-jsymbols 102
9.8 SU(n)andtheadjointrep 106
9.9 Anapplicationofthenegativedimensionalitytheorem 107
9.10 SU(n)mixedtwo-indextensors 109
9.11 SU(n)mixeddefining⊗adjointtensors 110
9.12 SU(n)two-indexadjointtensors 113
9.13 CasimirsforthefullysymmetricrepsofSU(n) 117
9.14 SU(n),U(n)equivalenceinadjointrep 119
9.15 Sources 119
Chapter10. Orthogonalgroups 121
10.1 Two-indextensors 122
10.2 Mixedadjoint⊗definingreptensors 123
10.3 Two-indexadjointtensors 124
10.4 Three-indextensors 128
10.5 Gravitytensors 130
10.6 SO(n)Dynkinlabels 133
Chapter11. Spinors 135
11.1 Spinography 136
11.2 Fierzingaround 139
11.3 Fierzcoefficients 143
11.4 6-jcoefficients 144
11.5 Exemplaryevaluations,continued 146
GroupTheory PUPcopyeditorversion8.7,September27,2007
CONTENTS ix
11.6 Invarianceofγ-matrices 147
11.7 Handedness 148
11.8 Kahanealgorithm 149
Chapter12. Symplecticgroups 152
12.1 Two-indextensors 153
Chapter13. Negativedimensions 155
13.1 SU(n)=SU(−n) 156
13.2 SO(n)=Sp(−n) 158
Chapter14. Spinors’symplecticsisters 160
14.1 Spinsters 160
14.2 Racahcoefficients 165
14.3 Heisenbergalgebras 166
Chapter15. SU(n)familyofinvariancegroups 168
15.1 RepsofSU(2) 168
15.2 SU(3)asinvariancegroupofacubicinvariant 170
15.3 Levi-CivitatensorsandSU(n) 173
15.4 SU(4)–SO(6)isomorphism 174
Chapter16. G2 familyofinvariancegroups 176
16.1 Jacobirelation 178
16.2 Alternativityandreductionoff-contractions 178
16.3 Primitivityimpliesalternativity 181
16.4 CasimirsforG2 183
16.5 Hurwitz’stheorem 184
Chapter17. E8 familyofinvariancegroups 186
17.1 Two-indextensors 187
17.2 DecompositionofSym3A 190
17.3 Diophantineconditions 192
17.4 DynkinlabelsandYoungtableauxforE8 193
Chapter18. E6 familyofinvariancegroups 196
18.1 Reductionoftwo-indextensors 196
18.2 Mixedtwo-indextensors 197
18.3 DiophantineconditionsandtheE6family 199
18.4 Three-indextensors 201
18.5 Defining⊗adjointtensors 203
18.6 Two-indexadjointtensors 206
18.7 DynkinlabelsandYoungtableauxforE6 208
18.8 CasimirsforE6 211
18.9 SubgroupsofE6 214
18.10 Springerrelation 214
18.11 Springer’sconstructionofE6 215
Chapter19. F4 familyofinvariancegroups 217
19.1 Two-indextensors 217
GroupTheory PUPcopyeditorversion8.7,September27,2007
x CONTENTS
19.2 Defining⊗adjointtensors 220
19.3 JordanalgebraandF4(26) 223
19.4 DynkinlabelsandYoungtableauxforF4 224
Chapter20. E7 familyanditsnegative-dimensionalcousins 225
20.1 SO(4)family 226
20.2 Defining⊗adjointtensors 228
20.3 Liealgebraidentification 229
20.4 E7family 231
20.5 DynkinlabelsandYoungtableauxforE7 234
Chapter21. Exceptionalmagic 236
21.1 MagicTriangle 236
21.2 Abriefhistoryofexceptionalmagic 239
21.3 ExtendedsupergravitiesandtheMagicTriangle 242
Epilogue 243
AppendixA.Recursivedecomposition 244
AppendixB.PropertiesofYoungprojections 246
B.1 UniquenessofYoungprojectionoperators 246
B.2 Orthogonality 247
B.3 Normalizationandcompleteness 247
B.4 Dimensionformula 248
Bibliography 251
Index 271
GroupTheory PUPcopyeditorversion8.7,September27,2007
Acknowledgments
IwouldliketothankTonyKennedyforcoauthoringtheworkdiscussedinchapters
onspinors,spinstersandnegativedimensions;HenrietteElvangforcoauthoringthe
chapteronrepresentationsofU(n); DavidPritchardformuchhelpwiththeearly
versions of this manuscript; Roger Penrose for inventing birdtracks while I was
strugglingthroughgradeschool;PaulLauwersforthebirdtracks‘rockaroundthe
clock’;FezaGürseyandPierreRamondforthefirstlessonsonexceptionalgroups;
SusumuOkuboforinspiringcorrespondence;BobPearsonforassortedbirdtrack,
Young tableaux and lattice calculations; Bernard Julia for many stimulating in-
teractions; P. Howe and L. Brink for teaching me how to count (supergravity
multiplets); W. Siegel for helpfulcriticisms; E. Cremmer forhospitalityat École
NormaleSuperieure;M. Kontsevichforbringingtomyattentionthemorerecent
workofDeligne,Cohen,anddeMan;A.J.Macfarlane,H.Pfeifferandananony-
mousrefereeforcriticalreadingofthemanuscript;P.Cartierforwonderfulhistory
lessons;J.Landsbergforacknowledgingmysufferings;R.Abdelatif,G.M.Cicuta,
A. Duncan, B. Durhuus, R. Edgar, E. Eichten, P. G. O. Freund, S. Garoufalidis,
T. Goldman,R. J. Gonsalves, M. Günaydin,H.Harari, I.Khavkine, N. MacKay,
M. Marino, L. Michel, D. Milicˇic´, R. L. Mkrtchyan, K. Oblivia, M. Peskin, C.
Sachrajda, G. Seligman, P. Sikivie, A. Springer, A. Taylor, D. Thurston, G. Tik-
topoulos,andB.W.Westburyfordiscussionsand/orcorrespondence.
Theappellation“birdtracks”isdueto BerniceDurand,whoenteredmyoffice,
saw the blackboard and asked, puzzled: “What is this? Footprints left by birds
scurryingalongasandybeach?”
IamgratefultoDorteGlassfortypingmostofthemanuscript,totheAkselTovborg
Jensens Legat, Kongelige Danske Videnskabernes Selskab, for financial support
which made the transformation from hand-drawn to drafted birdtracks possible,
and,mostofall,tothegoodAndersJohansenfordrawingsome5,000birdtracks
– without him this book would still be but a collection of squiggles in my filing
cabinet. CarolMonsrudandCecileGourgueshelpedwithtypingtheearlyversion
ofthismanuscript.
The manuscript was written in stages, in Aspen, Chewton-Mendip, Princeton,
Paris,Bures-sur-Yvette,Rome,Copenhagen,Frebbenholm,Miramare,Røros,Juels-
minde,Göteborg–Copenhagentrain,CathayPacific(HongKong/Paris)andinnu-
merableotherairportsandplanes,SjællandsOdde,Göteborg,Miramare,Kurkela,
Cˇijovo, assorted Starbucks, Virginia Highlands, and Kostrena. I am grateful to
T. Dorrian-Smith, R. de la Torre, BDC, N.-R. Nilsson, U. Selmer, E. Høsøinen,
A.Wad,familiesCvitanovic´,andfamilyHerlinfortheirkindhospitalityalongthis
