Table Of ContentGroupTheory December10,2002
Group Theory
Lie’s, Tracks, and Exceptional Groups
Predrag Cvitanovic´
GONE WITH THE WIND PRESS
ATLANTA AND COPENHAGEN
GroupTheory December10,2002
dedicatedtothememoryof
BorisWeisfeilerandWilliamE.Caswell
GroupTheory December10,2002
Contents
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 Invariants 19
3.3 Invariancegroups 23
3.4 Projectionoperators 24
3.5 Furtherinvariants 25
Chapter4. Diagrammaticnotation 27
4.1 Birdtracks 27
4.2 Clebsch-Gordancoefficients 29
4.3 Zero-andone-dimensionalsubspaces 31
4.4 Infinitesimaltransformations 32
4.5 Liealgebra 35
4.6 OtherformsofLiealgebracommutators 37
4.7 Irrelevancyofclebsches 38
4.8 Abriefhistoryofbirdtracks 39
Chapter5. Recouplings 41
5.1 Couplingsandrecouplings 41
5.2 Wigner3n-jcoefficients 44
5.3 Wigner-Eckarttheorem 45
Chapter6. Permutations 49
6.1 Symmetrization 49
6.2 Antisymmetrization 51
6.3 Levi-Civitatensor 53
6.4 Determinants 55
6.5 Characteristicequations 57
6.6 Fully(anti)symmetrictensors 57
Chapter7. Casimiroperators 59
GroupTheory December10,2002
ii CONTENTS
7.1 CasimirsandLiealgebra 60
7.2 Independentcasimirs 61
7.3 Casimiroperators 63
7.4 Dynkinindices 64
7.5 Quadratic,cubiccasimirs 69
7.6 Quarticcasimirs 70
7.7 Sundryrelationsbetweenquarticcasimirs 72
7.8 Identicallyvanishingtensors 75
7.9 Dynkinlabels 75
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 90
9.5 Reductionoftensorproducts 93
9.6 3-jsymbols 94
9.7 Characters 97
9.8 Mixedtwo-indextensors 97
9.9 Mixeddefining⊗adjointtensors 99
9.10 Two-indexadjointtensors 101
9.11 CasimirsforthefullysymmetricrepsofSU(n) 105
9.12 SU(n),U(n)equivalenceinadjointrep 107
Chapter10. Orthogonalgroups 115
10.1 Two-indextensors 116
10.2 Mixedadjoint⊗definingreptensors 117
10.3 Two-indexadjointtensors 118
10.4 Three-indextensors 121
10.5 Gravitytensors 124
10.6 SO(n)Dynkinlabels 127
Chapter11. Spinors 129
11.1 Spinography 130
11.2 Fierzingaround 133
11.3 Fierzcoefficients 137
11.4 6-jcoefficients 138
11.5 Exemplaryevaluations,continued 140
11.6 Invarianceofγ-matrices 140
11.7 Handedness 142
11.8 Kahanealgorithm 143
Chapter12. Symplecticgroups 145
12.1 Two-indextensors 146
GroupTheory December10,2002
CONTENTS iii
Chapter13. Negativedimensions 149
13.1 SU(n)=SU(−n) 150
13.2 SO(n)=Sp(−n) 151
Chapter14. Spinors’Sp(n)sisters 153
14.1 Spinsters 153
14.2 Racahcoefficients 158
14.3 Heisenbergalgebras 159
Chapter15. SU(n)familyofinvariancegroups 161
15.1 RepresentationsofSU(2) 161
15.2 SU(3)asinvariancegroupofacubicinvariant 163
15.3 Levi-CivitatensorsandSU(n) 166
15.4 SU(4)-SO(6)isomorphism 167
Chapter16. G2 familyofinvariancegroups 169
16.1 Jacobirelation 171
16.2 Alternativityandreductionoff-contractions 171
16.3 Primitivityimpliesalternativity 174
16.4 Sextonians 176
16.5 CasimirsforG2 178
16.6 Hurwitz’stheorem 179
16.7 RepresentationsofG2 180
Chapter17. E8 familyofinvariancegroups 181
17.1 Two-indextensors 182
17.2 DecompositionofSym3A 186
17.3 Decompositionof ⊗ 188
17.4 Diophantineconditions 190
17.5 Recentprogress 190
Chapter18. E6 familyofinvariancegroups 193
18.1 Reductionoftwo-indextensors 193
18.2 Mixedtwo-indextensors 194
18.3 DiophantineconditionsandtheE6family 196
18.4 Three-indextensors 197
18.5 Defining⊗adjointtensors 199
18.6 Two-indexadjointtensors 202
18.7 DynkinlabelsandYoungtableauxforE6 206
18.8 CasimirsforE6 207
18.9 SubgroupsofE6 210
18.10Springerrelation 211
Chapter19. F4 familyofinvariancegroups 213
19.1 Two-indextensors 213
19.2 Defining⊗adjointtensors 216
19.3 Two-indexadjointtensors 218
19.4 JordanalgebraandF4(26) 219
GroupTheory December10,2002
iv CONTENTS
Chapter20. E7 familyanditsnegativedimensionalcousins 221
20.1 Theantisymmetricquarticinvariant 222
20.2 FurtherDiophantineconditions 224
20.3 Liealgebraidentification 225
20.4 Symmetricquarticinvariant 228
20.5 Theextendedsupergravitiesandthemagictriangle 229
Chapter21. Exceptionalmagic 235
21.1 Magictriangle 235
21.2 Landsberg-Manivelconstruction 238
21.3 Epilogue 238
Chapter22. Magicnegativedimensions 243
22.1 E7andSO(4) 243
22.2 E6andSU(3) 243
AppendixA. Recursivedecomposition 245
AppendixB. PropertiesofYoungprojections 247
B.1 UniquenessofYoungprojectionoperators 247
B.2 Normalization 248
B.3 Orthogonality 249
B.4 Thedimensionformula 249
B.5 Literature 251
AppendixC. G2 calculations 253
C.1 EvaluationrulesforG2 253
C.2 G2,furthercalculations 255
Bibliography 259
GroupTheory December10,2002
CONTENTS v
ACKNOWLEDGMENTS
IwouldliketothankTonyKennedyforcoauthoringtheworkdiscussedinchapters
onspinors,spinstersandnegativedimensions;HenrietteElvangforcoauthoringthe
chapteronrepresentationsofU(n); DavidPritchardformuchhelpwiththeearly
versionsofthismanuscript;RogerPenroseforinventingbirdtracks(andthusmak-
ingthemrespectable)whileIwasstrugglingthroughgradeschool; PaulLauwers
forthebirdtracksrock-around-the-clock; Feza Gürseyand PierreRamond forthe
first lessons on exceptional groups; Susumu Okubo for inspiring correspondence;
BobPearsonforassortedbirdtrack,Youngtableauxandlatticecalculations;Bernard
JuliaforcommentsthatIhopetounderstandsomeday(aswellaswhydoeshenot
citemyworkonthemagictriangle?);P.HoweandL.Brinkforteachingmehowto
count(supergravitymultiplets),W.Siegelformanyhelpfulcriticismsofmysuper-
gravitymultipletcounting,E.CremmerforhospitalityatÉcoleNormaleSuperieure,
M.KontsevichforbringingtomyattentionthemorerecentworkofDeligne,Cohen
anddeMan;J.Landsbergforacknowledgingmysufferings;R.Abdelatif,G.M.Ci-
cuta, A. Duncan, E. Eichten, B. Durhuus, R. Edgar, M. Günaydin, K. Oblivia,
G. Seligman, A. Springer, L. Michel, R.L. Mkrtchyan, P.G.O. Freund, T. Gold-
man, R.J. Gonsalves, H. Harari, D. Milicˇic´, H. Pfeiffer, C. Sachrayda P. Sikivie,
S.A.Solla,G.Tiktopoulos,andB.W. Westburyfordiscussions(orcorrespondence).
Theappellation“birdtracks”isduetoBerniceDurandwhoenteredmyoffice,saw
theblackboardandaskedpuzzled: “Whatisthis? Footprintsleftbybirdsscurrying
alongasandybeach?”
I am grateful to Dorte Glass for typing most of the manuscript, to the Aksel
Tovborg Jensens Legat, Kongelige Danske Videnskabernes Selskab, for financial
supportwhichmadethetransformationfromhand-drawntodraftedbirdtrackspos-
sible,and,mostofall,tothegoodAndersJohansenwhodidmostofdrawingofthe
birdtracksandwithoutwhomthisbookwouldstillbebutacollectionofsquiggles
inmyfilingcabinet. CarolMonsrudandCecileGourgueshelpedwithtypingthe
earlyversionofthismanuscript.
The manuscript was written in stages, in Chewton-Mendip, Paris, Bures-sur-
Yvette,Rome,Copenhagen,Frebbenholm,Røros,Juelsminde,Göteborg-Copenhagen
train,CathayPacific(HongKong-Paris)andinnumerableotherairportsandplanes,
SjællandsOdde,Göteborg,Miramare,KurkelaandVirginiaHighlands. Iamgrate-
fultoT.Dorrian-Smith, R.delaTorre, BDC,N.-R.Nilsson, E.Høsøinen, family
Cvitanovic´, U.SelmerandfamilyHerlinfortheirkindhospitalityalongthislong
way.
IwouldlovetothankW.E.Caswellforaperspicaciousobservation,andB.Weis-
feilerfordelightfuldiscussionsatI.A.S,butitcannotbedone. In1985,whilehiking
inAndes,BoriswasallegedlykidnappedbytheChileanstatesecurity,andtortured
and executed by Nazis at Colonia Dignidad, Chile. Bill boarded flight AA77 on
September11,2001andwasflownintoPentagonbyanolesscharminggroupof
Islamicfanatics. Thisbookisdedicatedtotheirmemory.
GroupTheory December10,2002
GroupTheory December10,2002
Chapter One
Introduction
This monograph offers a derivation of all classical and exceptional semi-simple
Lie algebras through a classification of “primitive invariants”. Using somewhat
unconventionalnotationinspiredbytheFeynmandiagramsofquantumfieldtheory,
the invariant tensors are represented by diagrams; severe limits on what simple
groupscouldpossiblyexistarededucedbyrequiringthatirreduciblerepresentations
be of integer dimension. The method provides the full Killing-Cartan list of all
possiblesimpleLiealgebras,butfailstoprovetheexistenceofF ,E ,E andE .
4 6 7 8
Onesimplequantumfieldtheoryquestionstartedthisproject;whatisthegroup
theoreticfactorforthefollowingQuantumChromodynamicsgluonself-energydi-
agram
= ? (1.1)
I first computed the answer for SU(n). There was a hard way of doing it, using
Gell-Mann f and d coefficients. There was also an easy way, where one
ijk ijk
coulddoodleoneselftotheanswerinafewlines. Thisisthe“birdtracks”method
whichwillbedescribedhere. ItworksnicelyforSO(n)andSp(n)aswell. Out
of curiosity, I wanted the answer for the remaining five exceptional groups. This
engenderedfurtherthought,andthatwhichIlearnedcanbebetterunderstoodasthe
answertoadifferentquestion. Supposesomeonecameintoyourofficeandasked,
“OnplanetZ,mesonsconsistofquarksandantiquarks,butbaryonscontainthree
quarksinasymmetriccolorcombination. Whatisthecolorgroup?” Theanswer
isneithertrivial,norwithoutsomebeauty(planetZ quarkscancomein27colors,
andthecolorgroupcanbeE ).
6
Onceyouknowhowtoanswersuchgroup-theoreticalquestions,youcananswer
manyothers. Thismonographtellsyouhow. Likethebrain,itisdividedintotwo
halves;theploddinghalfandtheinterestinghalf.
Theploddinghalfdescribeshowgrouptheoreticcalculationsarecarriedoutfor
unitary,orthogonalandsymplecticgroups,chapters3–15. Exceptforthe“negative
dimensions”ofchapter13andthe“spinsters”ofchapter14,noneofthatisnew,but
themethodsarehelpfulincarryingoutdailychores, suchasevaluatingQuantum
Chromodynamics group theoretic weights, evaluating lattice gauge theory group
integrals,computing1/N corrections,evaluatingspinortraces,evaluatingcasimirs,
implementingevaluationalgorithmsoncomputers,andsoon.
The interesting half, chapters16–21, describes the “exceptional magic” (a new
constructionofexceptionalLiealgebras),the“negativedimensions”(relationsbe-
tweenbosonicandfermionicdimensions). Openproblemsandpersonalconfessions
arerelegatedtotheepilogue, sect.21.3. Themethodsusedareapplicabletofield
GroupTheory December10,2002
2 CHAPTER1
theoreticmodelbuilding. Regardlessoftheirpotentialapplications,theresultsare
sufficiently intriguing to justity this entire undertaking. In what follows we shall
forgetaboutquarksandquantumfieldtheory,andofferinsteadasomewhatunortho-
doxintroductiontothetheoryofLiealgebras. IfthestyleisnotBourbaki,itisnot
sobyaccident.
There are two complementary approaches to group theory. In the canonical
approach one chooses the basis, or the Clebsch-Gordan coefficients, as simply as
possible. ThisisthemethodwhichKilling[97]andCartan[20]usedtoobtainthe
completeclassificationofsemi-simpleLiealgebras,andwhichhasbeenbroughtto
perfectionbyDynkin[57]. Thereexistmanyexcellentreviewsofapplicationsof
Dynkindiagrammethodstophysics,suchasthereviewbySlansky[145].
Inthetensorialapproachpursuedhere,thebasesarearbitraryandeverystatement
isinvariantunderchangeofbasis. Tensorcalculusdealsdirectlywiththeinvariant
blocksofthetheoryandgivestheexplicitformsoftheinvariants,Clebsch-Gordan
series,evaluationalgorithmsforgrouptheoreticweights,etc.
The canonical approach is often impractical for computational purposes, as a
choiceofbasisrequiresaspecificcoordinatizationoftherepresentationspace. Usu-
ally,nothingthatwewanttocomputedependsonsuchacoordinatization;physical
predictions are pure scalar numbers (“color singlets”), with all tensorial indices
summedover. However,thecanonicalapproachcanbeveryusefulindetermining
chains of subgroup embeddings, we refer the reader to the Slansky review [145]
forsuchapplications. Hereweshallconcentrateontensorialmethods,borrowing
from Cartan and Dynkin only the nomenclature for identifying irreducible repre-
sentations. Extensive listings of these are given by McKay and Patera [109] and
Slansky[145].
Toappreciatethesenseinwhichcanonicalmethodsareimpractical,letusconsider
using them to evaluate the group-theoretic factor associated with diagram (1.1)
for the exceptional group E . This would involve summations over 8 structure
8
constants. TheCartan-Dynkinconstructionenablesustoconstructthemexplicitly;
anE structureconstanthasabout2483/6elements,andthedirectevaluationofthe
8
group-theoreticfactorfordiagram(1.1)istediousevenonacomputer. Anevaluation
in terms of a canonical basis would be equally tedious for SU(16); however, the
tensorialapproachillustratedbytheexampleofsect.2.2yieldstheanswerforall
SU(n)inafewsteps.
Simplicityofsuchcalculationsisonemotivationforformulatingatensorialap-
proachtoexceptionalgroups. Theotheristhedesiretounderstandtheirgeometrical
significance. TheKilling-CartanclassificationisbasedonamappingofLiealgebras
onto a Diophantine problem on the Cartan root lattice. This yields an exhaustive
classificationofsimpleLiealgebras,butgivesnoinsightintotheassociatedgeome-
tries. In the 19th century, the geometries or the invariant theory were the central
question and Cartan, inhis1894 thesis, made an attempttoidentifytheprimitive
invariants. MostoftheentriesinhisclassificationweretheclassicalgroupsSU(n),
SO(n)andSp(n). Ofthefiveexceptional algebras, Cartan[21]identifiedG as
2
thegroupofoctonionisomorphisms,andnotedalreadyinhisthesisthatE hasa
7
skew-symmetricquadraticandasymmetricquarticinvariant. Dickinson[51]char-
acterized E as a 27-dimensional group with a cubic invariant. The fact that the
6
