NumberDB$^\beta$ Advanced search Tags Tables Help Login
Quantum dimensions of SU(N) WZW representations
edit · history · discussion · files · short url · algebraic groups physics
Numbers
$N$
$k$
$(a_1,\ldots,a_{N-1})$ 
$d_{N,k}(\lambda)$
2
1
$(0)$:
1
comment: Vacuum representation.
2
1
$(1)$:
1
comment: Defining representation.
2
2
$(0)$:
1
comment: Vacuum representation.
2
2
$(1)$:
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
comment: Defining representation.
2
2
$(2)$:
1
2
3
$(0)$:
1
comment: Vacuum representation.
2
3
$(1)$:
1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
comment: Defining representation.
2
3
$(2)$:
1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
2
3
$(3)$:
1
2
4
$(0)$:
1
comment: Vacuum representation.
2
4
$(1)$:
1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
comment: Defining representation.
2
4
$(2)$:
2
2
4
$(3)$:
1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
2
4
$(4)$:
1
2
5
$(0)$:
1
comment: Vacuum representation.
2
5
$(1)$:
1.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
comment: Defining representation.
2
5
$(2)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
2
5
$(3)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
2
5
$(4)$:
1.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
2
5
$(5)$:
1
2
6
$(0)$:
1
comment: Vacuum representation.
2
6
$(1)$:
1.847759065022573512256366378793576573644833251727284972230195462561070015002204717429679869700689192
comment: Defining representation.
2
6
$(2)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
2
6
$(3)$:
2.613125929752753055713286346854374307167522376698539055097796733816162082922384101903707440385256286
2
6
$(4)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
2
6
$(5)$:
1.847759065022573512256366378793576573644833251727284972230195462561070015002204717429679869700689192
2
6
$(6)$:
1
2
7
$(0)$:
1
comment: Vacuum representation.
2
7
$(1)$:
1.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
comment: Defining representation.
2
7
$(2)$:
2.532088886237956070404785301110833347871664914160790491708090569284310777713749447056458553361096987
2
7
$(3)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
2
7
$(4)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
2
7
$(5)$:
2.532088886237956070404785301110833347871664914160790491708090569284310777713749447056458553361096987
2
7
$(6)$:
1.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
2
7
$(7)$:
1
2
8
$(0)$:
1
comment: Vacuum representation.
2
8
$(1)$:
1.902113032590307144232878666758764286811397268251500444894611288860306340170387003437585621941622763
comment: Defining representation.
2
8
$(2)$:
2.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
2
8
$(3)$:
3.077683537175253402570290576036909824006702143537792427039156250374863288495090918454593721667105435
2
8
$(4)$:
3.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
2
8
$(5)$:
3.077683537175253402570290576036909824006702143537792427039156250374863288495090918454593721667105435
2
8
$(6)$:
2.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
2
8
$(7)$:
1.902113032590307144232878666758764286811397268251500444894611288860306340170387003437585621941622763
2
8
$(8)$:
1
2
9
$(0)$:
1
comment: Vacuum representation.
2
9
$(1)$:
1.918985947228994779780736114132655398124909696844323910088990117990380616592573333941008561347214472
comment: Defining representation.
2
9
$(2)$:
2.682507065662362337723623297838735435026584996841075797285300235363302213803284546486173045339570923
2
9
$(3)$:
3.228707415119564907894586259065242504492492095518180765301581327002170796491283118125350354924154132
2
9
$(4)$:
3.513337091666135188782171596297981842074594817770149422137746900120181775693030525928915329171499370
2
9
$(5)$:
3.513337091666135188782171596297981842074594817770149422137746900120181775693030525928915329171499370
2
9
$(6)$:
3.228707415119564907894586259065242504492492095518180765301581327002170796491283118125350354924154132
2
9
$(7)$:
2.682507065662362337723623297838735435026584996841075797285300235363302213803284546486173045339570923
2
9
$(8)$:
1.918985947228994779780736114132655398124909696844323910088990117990380616592573333941008561347214472
2
9
$(9)$:
1
2
10
$(0)$:
1
comment: Vacuum representation.
2
10
$(1)$:
1.931851652578136573499486399457794735267809678016809100804686152620846427959711032695123483716140904
comment: Defining representation.
2
10
$(2)$:
2.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
2
10
$(3)$:
3.346065214951231622301175123667492813837481553393757173981365890611578906421818071545511018043782477
2
10
$(4)$:
3.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
2
10
$(5)$:
3.863703305156273146998972798915589470535619356033618201609372305241692855919422065390246967432281808
2
10
$(6)$:
3.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
2
10
$(7)$:
3.346065214951231622301175123667492813837481553393757173981365890611578906421818071545511018043782477
2
10
$(8)$:
2.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
2
10
$(9)$:
1.931851652578136573499486399457794735267809678016809100804686152620846427959711032695123483716140904
2
10
$(10)$:
1
2
11
$(0)$:
1
comment: Vacuum representation.
2
11
$(1)$:
1.941883634852104054313964552587578454499730211478007177175289052954083417520013353865149212816768523
comment: Defining representation.
2
11
$(2)$:
2.770912051306419791800751044030197757210996832695069803604807617453515114067455725067764595826494947
2
11
$(3)$:
3.438905131194306251583225751990281222192633391829659445314551531248746832228887006053038040811733689
2
11
$(4)$:
3.907041544768731396824366162285231006277981937602106142564132620588678472149837508518493095628983277
2
11
$(5)$:
4.148114905279377503522501537190318170825344256057248952158030268622267632183957902749997791919063505
2
11
$(6)$:
4.148114905279377503522501537190318170825344256057248952158030268622267632183957902749997791919063505
2
11
$(7)$:
3.907041544768731396824366162285231006277981937602106142564132620588678472149837508518493095628983277
2
11
$(8)$:
3.438905131194306251583225751990281222192633391829659445314551531248746832228887006053038040811733689
2
11
$(9)$:
2.770912051306419791800751044030197757210996832695069803604807617453515114067455725067764595826494947
2
11
$(10)$:
1.941883634852104054313964552587578454499730211478007177175289052954083417520013353865149212816768523
2
11
$(11)$:
1
2
12
$(0)$:
1
comment: Vacuum representation.
2
12
$(1)$:
1.949855824363647214036263365987862434465571601239994875296159150175291863268807587400224916241473850
comment: Defining representation.
2
12
$(2)$:
2.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
2
12
$(3)$:
3.513518789299706831453152419335977934930240638657369933257429066265475326536680470801960930023455815
2
12
$(4)$:
4.048917339522305313522214407023369723596387786056518510838223724592572145768854500154607924090195574
2
12
$(5)$:
4.381286267534823072404689085032695444150222094232289686146523673329881829843077313843528973572368019
2
12
$(6)$:
4.493959207434934122100019536016959242529098923585608421462197756387414609825949138303770029306341487
2
12
$(7)$:
4.381286267534823072404689085032695444150222094232289686146523673329881829843077313843528973572368019
2
12
$(8)$:
4.048917339522305313522214407023369723596387786056518510838223724592572145768854500154607924090195574
2
12
$(9)$:
3.513518789299706831453152419335977934930240638657369933257429066265475326536680470801960930023455815
2
12
$(10)$:
2.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
2
12
$(11)$:
1.949855824363647214036263365987862434465571601239994875296159150175291863268807587400224916241473850
2
12
$(12)$:
1
2
13
$(0)$:
1
comment: Vacuum representation.
2
13
$(1)$:
1.956295201467611275857133495739199064919475617725354215770355327281193666240190243999517050909571273
comment: Defining representation.
2
13
$(2)$:
2.827090915285201791004255143970634355881620918754949090121999576130229764205262543363568001786599990
2
13
$(3)$:
3.574329190217506124061720330104837182639784797531117077905803949986454129059092693706724255098962411
2
13
$(4)$:
4.165352128002918218656801805344195303080787356674540443756906280706162967626550337655877848506780126
2
13
$(5)$:
4.574329190217506124061720330104837182639784797531117077905803949986454129059092693706724255098962411
2
13
$(6)$:
4.783386116752813066861388639709833420801096536480303305892354903411423430445452787363085052696171263
2
13
$(7)$:
4.783386116752813066861388639709833420801096536480303305892354903411423430445452787363085052696171263
2
13
$(8)$:
4.574329190217506124061720330104837182639784797531117077905803949986454129059092693706724255098962411
2
13
$(9)$:
4.165352128002918218656801805344195303080787356674540443756906280706162967626550337655877848506780126
2
13
$(10)$:
3.574329190217506124061720330104837182639784797531117077905803949986454129059092693706724255098962411
2
13
$(11)$:
2.827090915285201791004255143970634355881620918754949090121999576130229764205262543363568001786599990
2
13
$(12)$:
1.956295201467611275857133495739199064919475617725354215770355327281193666240190243999517050909571273
2
13
$(13)$:
1
2
14
$(0)$:
1
comment: Vacuum representation.
2
14
$(1)$:
1.961570560806460898252364472268478073947867461786672190005832177090613027099210127830129971706601527
comment: Defining representation.
2
14
$(2)$:
2.847759065022573512256366378793576573644833251727284972230195462561070015002204717429679869700689192
2
14
$(3)$:
3.624509785411551372409941227504289587424989085761172116898081357545888867388494815793679809722562827
2
14
$(4)$:
4.261972627395668561058055103003274652214505127104233045406875200551802493464311756280067404028330765
2
14
$(5)$:
4.735650251450755821895602855401355336174863467270780208746388413951787362543454952559929570425090696
2
14
$(6)$:
5.027339492125848104514975071064072385737194252075487128274476471806894561384491140754094974712897859
2
14
$(7)$:
5.125830895483012357592172592355399818030246702774689824255392592941314000115304113010580188893271459
2
14
$(8)$:
5.027339492125848104514975071064072385737194252075487128274476471806894561384491140754094974712897859
2
14
$(9)$:
4.735650251450755821895602855401355336174863467270780208746388413951787362543454952559929570425090696
2
14
$(10)$:
4.261972627395668561058055103003274652214505127104233045406875200551802493464311756280067404028330765
2
14
$(11)$:
3.624509785411551372409941227504289587424989085761172116898081357545888867388494815793679809722562827
2
14
$(12)$:
2.847759065022573512256366378793576573644833251727284972230195462561070015002204717429679869700689192
2
14
$(13)$:
1.961570560806460898252364472268478073947867461786672190005832177090613027099210127830129971706601527
2
14
$(14)$:
1
2
15
$(0)$:
1
comment: Vacuum representation.
2
15
$(1)$:
1.965946199367803556563897689710397432197445750131265751994760918407815705104488434859368972258708455
comment: Defining representation.
2
15
$(2)$:
2.864944458808711609146231783643126772525175555890233856496700023721072093139288996256149425700859702
2
15
$(3)$:
3.666380470827031860832185535609101549138766407647091446734908231243231836311338504040547634489579939
2
15
$(4)$:
4.342962293250029840995300403388422984045010795306429302810992645113017730230805320390104504660992535
2
15
$(5)$:
4.871649743585544639189361845582782792376695571143920027439895219835746889056135071996631307884544920
2
15
$(6)$:
5.234439004803106375788215502147396694900366342485284288811668390816183230430263647071020737457449895
2
15
$(7)$:
5.418975723729710366267517716456409655626400909996930955194753450658458378472757558334217030460507964
2
15
$(8)$:
5.418975723729710366267517716456409655626400909996930955194753450658458378472757558334217030460507964
2
15
$(9)$:
5.234439004803106375788215502147396694900366342485284288811668390816183230430263647071020737457449895
2
15
$(10)$:
4.871649743585544639189361845582782792376695571143920027439895219835746889056135071996631307884544920
2
15
$(11)$:
4.342962293250029840995300403388422984045010795306429302810992645113017730230805320390104504660992535
2
15
$(12)$:
3.666380470827031860832185535609101549138766407647091446734908231243231836311338504040547634489579939
2
15
$(13)$:
2.864944458808711609146231783643126772525175555890233856496700023721072093139288996256149425700859702
2
15
$(14)$:
1.965946199367803556563897689710397432197445750131265751994760918407815705104488434859368972258708455
2
15
$(15)$:
1
2
16
$(0)$:
1
comment: Vacuum representation.
2
16
$(1)$:
1.969615506024416118733486049179046027341286503439684837580051504711655199886072478549356820112239798
comment: Defining representation.
2
16
$(2)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
2
16
$(3)$:
3.701666313593293412260932390684918394284091757250065465635858484163588216794872515630503006869488374
2
16
$(4)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
2
16
$(5)$:
4.987241532966372064906219210499445260099211525661429046285812993505813889276789795051479462304677077
2
16
$(6)$:
5.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
2
16
$(7)$:
5.671281819617709530994418439863964421625378260689750303215909988875243416680944994179859826981728172
2
16
$(8)$:
5.758770483143633536216437109298925879744832537057858532361146651096884843983557835798983559351792270
2
16
$(9)$:
5.671281819617709530994418439863964421625378260689750303215909988875243416680944994179859826981728172
2
16
$(10)$:
5.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
2
16
$(11)$:
4.987241532966372064906219210499445260099211525661429046285812993505813889276789795051479462304677077
2
16
$(12)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
2
16
$(13)$:
3.701666313593293412260932390684918394284091757250065465635858484163588216794872515630503006869488374
2
16
$(14)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
2
16
$(15)$:
1.969615506024416118733486049179046027341286503439684837580051504711655199886072478549356820112239798
2
16
$(16)$:
1
2
17
$(0)$:
1
comment: Vacuum representation.
2
17
$(1)$:
1.972722606805444747205018389638134221456963006405752633487330270126908132988639203256955994148291016
comment: Defining representation.
2
17
$(2)$:
2.891634483401269358039331428569883055647623776881406958719626556788741715865585083406568561583409722
2
17
$(3)$:
3.731670109218422889986727899401816455872742785294640612492731784689741555302342610634390185089665987
2
17
$(4)$:
4.469915502194056556477293727368064540512996468623490515477523342138609871584386442056950778273409234
2
17
$(5)$:
5.086233252469905039511029868393067892703782277694912325807871935406635091695721207535723725376811071
2
17
$(6)$:
5.563811818438910305900819220760441540070767465690784995647264564041806655131561853904642969333974667
2
17
$(7)$:
5.889624101775843954544713187878302197217253315688352338974360666456995643311183869107314805499639369
2
17
$(8)$:
6.054782792720508603745401056353182652614420584367360700775416495357083617756140668029385804630795242
2
17
$(9)$:
6.054782792720508603745401056353182652614420584367360700775416495357083617756140668029385804630795242
2
17
$(10)$:
5.889624101775843954544713187878302197217253315688352338974360666456995643311183869107314805499639369
2
17
$(11)$:
5.563811818438910305900819220760441540070767465690784995647264564041806655131561853904642969333974667
2
17
$(12)$:
5.086233252469905039511029868393067892703782277694912325807871935406635091695721207535723725376811071
2
17
$(13)$:
4.469915502194056556477293727368064540512996468623490515477523342138609871584386442056950778273409234
2
17
$(14)$:
3.731670109218422889986727899401816455872742785294640612492731784689741555302342610634390185089665987
2
17
$(15)$:
2.891634483401269358039331428569883055647623776881406958719626556788741715865585083406568561583409722
2
17
$(16)$:
1.972722606805444747205018389638134221456963006405752633487330270126908132988639203256955994148291016
2
17
$(17)$:
1
2
18
$(0)$:
1
comment: Vacuum representation.
2
18
$(1)$:
1.975376681190275452380080495386874521516813723179760869847809603276145871939425471423926075999773557
comment: Defining representation.
2
18
$(2)$:
2.902113032590307144232878666758764286811397268251500444894611288860306340170387003437585621941622763
2
18
$(3)$:
3.757389729567011177099499638214127147057850761251938360658854172321130420291695920179803241469496513
2
18
$(4)$:
4.520147021340201992437465501124402404531706448057263307030059911565566802989289453144792826131013901
2
18
$(5)$:
5.171603291940106225901188362423825225627522636628886433835533910311862898753802959030190775797138085
2
18
$(6)$:
5.695717525925148250774877410402547941727011323343555289174604873080123751313993368161800925856496573
2
18
$(7)$:
6.079584291419199809022005095139567623593618042726929547806165921955916467017507816539137654241031680
2
18
$(8)$:
6.313751514675043098979464244768186059447320503149318151310053495785384214132895817869008130045887710
2
18
$(9)$:
6.392453221499661547042215734073901408221417826898251127975448614071434053628754975212529399594729056
2
18
$(10)$:
6.313751514675043098979464244768186059447320503149318151310053495785384214132895817869008130045887710
2
18
$(11)$:
6.079584291419199809022005095139567623593618042726929547806165921955916467017507816539137654241031680
2
18
$(12)$:
5.695717525925148250774877410402547941727011323343555289174604873080123751313993368161800925856496573
2
18
$(13)$:
5.171603291940106225901188362423825225627522636628886433835533910311862898753802959030190775797138085
2
18
$(14)$:
4.520147021340201992437465501124402404531706448057263307030059911565566802989289453144792826131013901
2
18
$(15)$:
3.757389729567011177099499638214127147057850761251938360658854172321130420291695920179803241469496513
2
18
$(16)$:
2.902113032590307144232878666758764286811397268251500444894611288860306340170387003437585621941622763
2
18
$(17)$:
1.975376681190275452380080495386874521516813723179760869847809603276145871939425471423926075999773557
2
18
$(18)$:
1
2
19
$(0)$:
1
comment: Vacuum representation.
2
19
$(1)$:
1.977661652450257090139485765868017226130502027148375412798993854320427066197452198498686379936348386
comment: Defining representation.
2
19
$(2)$:
2.911145611572281465622668107534933332879178212693780861343773317976188630836898892453469480793399707
2
19
$(3)$:
3.779599388255095342611690404882907328462340351412089712906118700719291907053332129501409289373373217
2
19
$(4)$:
4.563623160204271209512993255080290128463919026622861273558945396524527651040269901954282456560279372
2
19
$(5)$:
5.245703131914747999656553383424251142408807426576780663625917986900638073833136383805716664950627012
2
19
$(6)$:
5.810602763921738270563003023088769749728468488415665484290044274718234955953244471106167471213450115
2
19
$(7)$:
6.245703131914747999656553383424251142408807426576780663625917986900638073833136383805716664950627012
2
19
$(8)$:
6.541284812654528299652479020948307354594421053771236686357939250844954717237722100452968836496627758
2
19
$(9)$:
6.690744999827376808234358512417840661341518564105870574249892018695480537890231021954878770166772924
2
19
$(10)$:
6.690744999827376808234358512417840661341518564105870574249892018695480537890231021954878770166772924
2
19
$(11)$:
6.541284812654528299652479020948307354594421053771236686357939250844954717237722100452968836496627758
2
19
$(12)$:
6.245703131914747999656553383424251142408807426576780663625917986900638073833136383805716664950627012
2
19
$(13)$:
5.810602763921738270563003023088769749728468488415665484290044274718234955953244471106167471213450115
2
19
$(14)$:
5.245703131914747999656553383424251142408807426576780663625917986900638073833136383805716664950627012
2
19
$(15)$:
4.563623160204271209512993255080290128463919026622861273558945396524527651040269901954282456560279372
2
19
$(16)$:
3.779599388255095342611690404882907328462340351412089712906118700719291907053332129501409289373373217
2
19
$(17)$:
2.911145611572281465622668107534933332879178212693780861343773317976188630836898892453469480793399707
2
19
$(18)$:
1.977661652450257090139485765868017226130502027148375412798993854320427066197452198498686379936348386
2
19
$(19)$:
1
2
20
$(0)$:
1
comment: Vacuum representation.
2
20
$(1)$:
1.979642883761865464752184075553437574753038743897433375677230889847244731450708197208884679676713207
comment: Defining representation.
2
20
$(2)$:
2.918985947228994779780736114132655398124909696844323910088990117990380616592573333941008561347214472
2
20
$(3)$:
3.798906874470902207575614841711494494873520846286721717189439790482279440485644271983716000489265205
2
20
$(4)$:
4.601493012891357117504359411971390833151494693685399707374290353353682830395857880427181606686785395
2
20
$(5)$:
5.310406023179418775123686529656183335232955736625168856349439556586091718259089151902685422955792752
2
20
$(6)$:
5.911214480781927245618209556903977939519077092359256562586881562365473010294567664611523400263725055
2
20
$(7)$:
6.391687658090613939338958438293566726096496952421396536420939335210856463640857660011570803849621547
2
20
$(8)$:
6.742044506785700096676757855363224346567086913288330187439328227122352572184313644054265684095653502
2
20
$(9)$:
6.955152771773473334761794268986800524168052750367930280083099947386554060678915996258506040753394647
2
20
$(10)$:
7.026674183332270377564343192595963684149189635540298844275493800240363551386061051857830658342998740
2
20
$(11)$:
6.955152771773473334761794268986800524168052750367930280083099947386554060678915996258506040753394647
2
20
$(12)$:
6.742044506785700096676757855363224346567086913288330187439328227122352572184313644054265684095653502
2
20
$(13)$:
6.391687658090613939338958438293566726096496952421396536420939335210856463640857660011570803849621547
2
20
$(14)$:
5.911214480781927245618209556903977939519077092359256562586881562365473010294567664611523400263725055
2
20
$(15)$:
5.310406023179418775123686529656183335232955736625168856349439556586091718259089151902685422955792752
2
20
$(16)$:
4.601493012891357117504359411971390833151494693685399707374290353353682830395857880427181606686785395
2
20
$(17)$:
3.798906874470902207575614841711494494873520846286721717189439790482279440485644271983716000489265205
2
20
$(18)$:
2.918985947228994779780736114132655398124909696844323910088990117990380616592573333941008561347214472
2
20
$(19)$:
1.979642883761865464752184075553437574753038743897433375677230889847244731450708197208884679676713207
2
20
$(20)$:
1
3
1
$(0,0)$:
1
comment: Vacuum representation.
3
1
$(0,1)$:
1
comment: Dual of the defining representation.
3
1
$(1,0)$:
1
comment: Defining representation.
3
2
$(0,0)$:
1
comment: Vacuum representation.
3
2
$(0,1)$:
1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
comment: Dual of the defining representation.
3
2
$(1,0)$:
1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
comment: Defining representation.
3
2
$(0,2)$:
1
3
2
$(1,1)$:
1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
3
2
$(2,0)$:
1
3
3
$(0,0)$:
1
comment: Vacuum representation.
3
3
$(0,1)$:
2
comment: Dual of the defining representation.
3
3
$(1,0)$:
2
comment: Defining representation.
3
3
$(0,2)$:
2
3
3
$(1,1)$:
3
3
3
$(2,0)$:
2
3
3
$(0,3)$:
1
3
3
$(1,2)$:
2
3
3
$(2,1)$:
2
3
3
$(3,0)$:
1
3
4
$(0,0)$:
1
comment: Vacuum representation.
3
4
$(0,1)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
comment: Dual of the defining representation.
3
4
$(1,0)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
comment: Defining representation.
3
4
$(0,2)$:
2.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
3
4
$(1,1)$:
4.048917339522305313522214407023369723596387786056518510838223724592572145768854500154607924090195574
3
4
$(2,0)$:
2.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
3
4
$(0,3)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
3
4
$(1,2)$:
4.048917339522305313522214407023369723596387786056518510838223724592572145768854500154607924090195574
3
4
$(2,1)$:
4.048917339522305313522214407023369723596387786056518510838223724592572145768854500154607924090195574
3
4
$(3,0)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
3
4
$(0,4)$:
1
3
4
$(1,3)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
3
4
$(2,2)$:
2.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
3
4
$(3,1)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
3
4
$(4,0)$:
1
3
5
$(0,0)$:
1
comment: Vacuum representation.
3
5
$(0,1)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
comment: Dual of the defining representation.
3
5
$(1,0)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
comment: Defining representation.
3
5
$(0,2)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
3
5
$(1,1)$:
4.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
3
5
$(2,0)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
3
5
$(0,3)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
3
5
$(1,2)$:
5.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
3
5
$(2,1)$:
5.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
3
5
$(3,0)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
3
5
$(0,4)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
3
5
$(1,3)$:
4.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
3
5
$(2,2)$:
5.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
3
5
$(3,1)$:
4.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
3
5
$(4,0)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
3
5
$(0,5)$:
1
3
5
$(1,4)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
3
5
$(2,3)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
3
5
$(3,2)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
3
5
$(4,1)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
3
5
$(5,0)$:
1
3
6
$(0,0)$:
1
comment: Vacuum representation.
3
6
$(0,1)$:
2.532088886237956070404785301110833347871664914160790491708090569284310777713749447056458553361096987
comment: Dual of the defining representation.
3
6
$(1,0)$:
2.532088886237956070404785301110833347871664914160790491708090569284310777713749447056458553361096987
comment: Defining representation.
3
6
$(0,2)$:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
3
6
$(1,1)$:
5.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
3
6
$(2,0)$:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
3
6
$(0,3)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
3
6
$(1,2)$:
7.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
3
6
$(2,1)$:
7.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
3
6
$(3,0)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
3
6
$(0,4)$:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
3
6
$(1,3)$:
7.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
3
6
$(2,2)$:
8.638155724715450304324655663948388819617248805586787798541719976645327265975336753698475339027688405
3
6
$(3,1)$:
7.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
3
6
$(4,0)$:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
3
6
$(0,5)$:
2.532088886237956070404785301110833347871664914160790491708090569284310777713749447056458553361096987
3
6
$(1,4)$:
5.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
3
6
$(2,3)$:
7.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
3
6
$(3,2)$:
7.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
3
6
$(4,1)$:
5.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
3
6
$(5,0)$:
2.532088886237956070404785301110833347871664914160790491708090569284310777713749447056458553361096987
3
6
$(0,6)$:
1
3
6
$(1,5)$:
2.532088886237956070404785301110833347871664914160790491708090569284310777713749447056458553361096987
3
6
$(2,4)$:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
3
6
$(3,3)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
3
6
$(4,2)$:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
3
6
$(5,1)$:
2.532088886237956070404785301110833347871664914160790491708090569284310777713749447056458553361096987
3
6
$(6,0)$:
1
3
7
$(0,0)$:
1
comment: Vacuum representation.
3
7
$(0,1)$:
2.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
comment: Dual of the defining representation.
3
7
$(1,0)$:
2.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
comment: Defining representation.
3
7
$(0,2)$:
4.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
3
7
$(1,1)$:
5.854101966249684544613760503096914353160927539417288586406345868115781388456707349121621612568173412
3
7
$(2,0)$:
4.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
3
7
$(0,3)$:
5.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
3
7
$(1,2)$:
8.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550
3
7
$(2,1)$:
8.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550
3
7
$(3,0)$:
5.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
3
7
$(0,4)$:
5.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
3
7
$(1,3)$:
9.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550
3
7
$(2,2)$:
11.09016994374947424102293417182819058860154589902881431067724311352630231409451224853603602094695569
3
7
$(3,1)$:
9.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550
3
7
$(4,0)$:
5.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
3
7
$(0,5)$:
4.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
3
7
$(1,4)$:
8.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550
3
7
$(2,3)$:
11.09016994374947424102293417182819058860154589902881431067724311352630231409451224853603602094695569
3
7
$(3,2)$:
11.09016994374947424102293417182819058860154589902881431067724311352630231409451224853603602094695569
3
7
$(4,1)$:
8.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550
3
7
$(5,0)$:
4.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
3
7
$(0,6)$:
2.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
3
7
$(1,5)$:
5.854101966249684544613760503096914353160927539417288586406345868115781388456707349121621612568173412
3
7
$(2,4)$:
8.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550
3
7
$(3,3)$:
9.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550
3
7
$(4,2)$:
8.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550
3
7
$(5,1)$:
5.854101966249684544613760503096914353160927539417288586406345868115781388456707349121621612568173412
3
7
$(6,0)$:
2.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
3
7
$(0,7)$:
1
3
7
$(1,6)$:
2.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
3
7
$(2,5)$:
4.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
3
7
$(3,4)$:
5.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
3
7
$(4,3)$:
5.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
3
7
$(5,2)$:
4.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
3
7
$(6,1)$:
2.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
3
7
$(7,0)$:
1
3
8
$(0,0)$:
1
comment: Vacuum representation.
3
8
$(0,1)$:
2.682507065662362337723623297838735435026584996841075797285300235363302213803284546486173045339570923
comment: Dual of the defining representation.
3
8
$(1,0)$:
2.682507065662362337723623297838735435026584996841075797285300235363302213803284546486173045339570923
comment: Defining representation.
3
8
$(0,2)$:
4.513337091666135188782171596297981842074594817770149422137746900120181775693030525928915329171499370
3
8
$(1,1)$:
6.195844157328497526505794894136717277101179814611225219423047135483483989496315072415088374511070293
3
8
$(2,0)$:
4.513337091666135188782171596297981842074594817770149422137746900120181775693030525928915329171499370
3
8
$(0,3)$:
5.911214480781927245618209556903977939519077092359256562586881562365473010294567664611523400263725055
3
8
$(1,2)$:
9.424551572448062434400381153201959781593671910129405984724628462485654785987598190540438729435224425
3
8
$(2,1)$:
9.424551572448062434400381153201959781593671910129405984724628462485654785987598190540438729435224425
3
8
$(3,0)$:
5.911214480781927245618209556903977939519077092359256562586881562365473010294567664611523400263725055
3
8
$(0,4)$:
6.432323038895129968562907710430637240199504514614473332226737018110562392285603859869923890518713842
3
8
$(1,3)$:
11.34353751967705721418111726733461517971858160697372989481361858047603540258017152448144729078243890
3
8
$(2,2)$:
13.17436754568083006523966556579386158676659142790280351966606524523291496446991750392418957461436734
3
8
$(3,1)$:
11.34353751967705721418111726733461517971858160697372989481361858047603540258017152448144729078243890
3
8
$(4,0)$:
6.432323038895129968562907710430637240199504514614473332226737018110562392285603859869923890518713842
3
8
$(0,5)$:
5.911214480781927245618209556903977939519077092359256562586881562365473010294567664611523400263725055
3
8
$(1,4)$:
11.34353751967705721418111726733461517971858160697372989481361858047603540258017152448144729078243890
3
8
$(2,3)$:
14.57224493479662212207570352639985768421107370249191066011519990747820619907145464260679764570659303
3
8
$(3,2)$:
14.57224493479662212207570352639985768421107370249191066011519990747820619907145464260679764570659303
3
8
$(4,1)$:
11.34353751967705721418111726733461517971858160697372989481361858047603540258017152448144729078243890
3
8
$(5,0)$:
5.911214480781927245618209556903977939519077092359256562586881562365473010294567664611523400263725055
3
8
$(0,6)$:
4.513337091666135188782171596297981842074594817770149422137746900120181775693030525928915329171499370
3
8
$(1,5)$:
9.424551572448062434400381153201959781593671910129405984724628462485654785987598190540438729435224425
3
8
$(2,4)$:
13.17436754568083006523966556579386158676659142790280351966606524523291496446991750392418957461436734
3
8
$(3,3)$:
14.57224493479662212207570352639985768421107370249191066011519990747820619907145464260679764570659303
3
8
$(4,2)$:
13.17436754568083006523966556579386158676659142790280351966606524523291496446991750392418957461436734
3
8
$(5,1)$:
9.424551572448062434400381153201959781593671910129405984724628462485654785987598190540438729435224425
3
8
$(6,0)$:
4.513337091666135188782171596297981842074594817770149422137746900120181775693030525928915329171499370
3
8
$(0,7)$:
2.682507065662362337723623297838735435026584996841075797285300235363302213803284546486173045339570923
3
8
$(1,6)$:
6.195844157328497526505794894136717277101179814611225219423047135483483989496315072415088374511070293
3
8
$(2,5)$:
9.424551572448062434400381153201959781593671910129405984724628462485654785987598190540438729435224425
3
8
$(3,4)$:
11.34353751967705721418111726733461517971858160697372989481361858047603540258017152448144729078243890
3
8
$(4,3)$:
11.34353751967705721418111726733461517971858160697372989481361858047603540258017152448144729078243890
3
8
$(5,2)$:
9.424551572448062434400381153201959781593671910129405984724628462485654785987598190540438729435224425
3
8
$(6,1)$:
6.195844157328497526505794894136717277101179814611225219423047135483483989496315072415088374511070293
3
8
$(7,0)$:
2.682507065662362337723623297838735435026584996841075797285300235363302213803284546486173045339570923
3
8
$(0,8)$:
1
3
8
$(1,7)$:
2.682507065662362337723623297838735435026584996841075797285300235363302213803284546486173045339570923
3
8
$(2,6)$:
4.513337091666135188782171596297981842074594817770149422137746900120181775693030525928915329171499370
3
8
$(3,5)$:
5.911214480781927245618209556903977939519077092359256562586881562365473010294567664611523400263725055
3
8
$(4,4)$:
6.432323038895129968562907710430637240199504514614473332226737018110562392285603859869923890518713842
3
8
$(5,3)$:
5.911214480781927245618209556903977939519077092359256562586881562365473010294567664611523400263725055
3
8
$(6,2)$:
4.513337091666135188782171596297981842074594817770149422137746900120181775693030525928915329171499370
3
8
$(7,1)$:
2.682507065662362337723623297838735435026584996841075797285300235363302213803284546486173045339570923
3
8
$(8,0)$:
1
4
1
$(0,0,0)$:
1
comment: Vacuum representation.
4
1
$(0,0,1)$:
1
comment: Dual of the defining representation.
4
1
$(0,1,0)$:
1
4
1
$(1,0,0)$:
1
comment: Defining representation.
4
2
$(0,0,0)$:
1
comment: Vacuum representation.
4
2
$(0,0,1)$:
1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
comment: Dual of the defining representation.
4
2
$(0,1,0)$:
2
4
2
$(1,0,0)$:
1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
comment: Defining representation.
4
2
$(0,0,2)$:
1
4
2
$(0,1,1)$:
1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
4
2
$(0,2,0)$:
1
4
2
$(1,0,1)$:
2
4
2
$(1,1,0)$:
1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
4
2
$(2,0,0)$:
1
4
3
$(0,0,0)$:
1
comment: Vacuum representation.
4
3
$(0,0,1)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
comment: Dual of the defining representation.
4
3
$(0,1,0)$:
2.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
4
3
$(1,0,0)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
comment: Defining representation.
4
3
$(0,0,2)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
4
3
$(0,1,1)$:
4.048917339522305313522214407023369723596387786056518510838223724592572145768854500154607924090195574
4
3
$(0,2,0)$:
2.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
4
3
$(1,0,1)$:
4.048917339522305313522214407023369723596387786056518510838223724592572145768854500154607924090195574
4
3
$(1,1,0)$:
4.048917339522305313522214407023369723596387786056518510838223724592572145768854500154607924090195574
4
3
$(2,0,0)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
4
3
$(0,0,3)$:
1
4
3
$(0,1,2)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
4
3
$(0,2,1)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
4
3
$(0,3,0)$:
1
4
3
$(1,0,2)$:
2.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
4
3
$(1,1,1)$:
4.048917339522305313522214407023369723596387786056518510838223724592572145768854500154607924090195574
4
3
$(1,2,0)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
4
3
$(2,0,1)$:
2.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
4
3
$(2,1,0)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
4
3
$(3,0,0)$:
1
4
4
$(0,0,0)$:
1
comment: Vacuum representation.
4
4
$(0,0,1)$:
2.613125929752753055713286346854374307167522376698539055097796733816162082922384101903707440385256286
comment: Dual of the defining representation.
4
4
$(0,1,0)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
4
4
$(1,0,0)$:
2.613125929752753055713286346854374307167522376698539055097796733816162082922384101903707440385256286
comment: Defining representation.
4
4
$(0,0,2)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
4
4
$(0,1,1)$:
6.308644059797900080226019104441527454457188880153108999558187658938302112926793536763067179786634671
4
4
$(0,2,0)$:
4.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
4
4
$(1,0,1)$:
5.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
4
4
$(1,1,0)$:
6.308644059797900080226019104441527454457188880153108999558187658938302112926793536763067179786634671
4
4
$(2,0,0)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
4
4
$(0,0,3)$:
2.613125929752753055713286346854374307167522376698539055097796733816162082922384101903707440385256286
4
4
$(0,1,2)$:
5.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
4
4
$(0,2,1)$:
6.308644059797900080226019104441527454457188880153108999558187658938302112926793536763067179786634671
4
4
$(0,3,0)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
4
4
$(1,0,2)$:
6.308644059797900080226019104441527454457188880153108999558187658938302112926793536763067179786634671
4
4
$(1,1,1)$:
9.656854249492380195206754896838792314278687501507792292706718951962929913848428155401550137310566291
4
4
$(1,2,0)$:
6.308644059797900080226019104441527454457188880153108999558187658938302112926793536763067179786634671
4
4
$(2,0,1)$:
6.308644059797900080226019104441527454457188880153108999558187658938302112926793536763067179786634671
4
4
$(2,1,0)$:
5.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
4
4
$(3,0,0)$:
2.613125929752753055713286346854374307167522376698539055097796733816162082922384101903707440385256286
4
4
$(0,0,4)$:
1
4
4
$(0,1,3)$:
2.613125929752753055713286346854374307167522376698539055097796733816162082922384101903707440385256286
4
4
$(0,2,2)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
4
4
$(0,3,1)$:
2.613125929752753055713286346854374307167522376698539055097796733816162082922384101903707440385256286
4
4
$(0,4,0)$:
1
4
4
$(1,0,3)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
4
4
$(1,1,2)$:
6.308644059797900080226019104441527454457188880153108999558187658938302112926793536763067179786634671
4
4
$(1,2,1)$:
5.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
4
4
$(1,3,0)$:
2.613125929752753055713286346854374307167522376698539055097796733816162082922384101903707440385256286
4
4
$(2,0,2)$:
4.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
4
4
$(2,1,1)$:
6.308644059797900080226019104441527454457188880153108999558187658938302112926793536763067179786634671
4
4
$(2,2,0)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
4
4
$(3,0,1)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
4
4
$(3,1,0)$:
2.613125929752753055713286346854374307167522376698539055097796733816162082922384101903707440385256286
4
4
$(4,0,0)$:
1
4
5
$(0,0,0)$:
1
comment: Vacuum representation.
4
5
$(0,0,1)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
comment: Dual of the defining representation.
4
5
$(0,1,0)$:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
4
5
$(1,0,0)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
comment: Defining representation.
4
5
$(0,0,2)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
4
5
$(0,1,1)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
4
5
$(0,2,0)$:
6.758770483143633536216437109298925879744832537057858532361146651096884843983557835798983559351792270
4
5
$(1,0,1)$:
7.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
4
5
$(1,1,0)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
4
5
$(2,0,0)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
4
5
$(0,0,3)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
4
5
$(0,1,2)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
4
5
$(0,2,1)$:
11.17024461095340637472944096505922216748891371974757829024981054592963804368908620075493389238878539
4
5
$(0,3,0)$:
6.758770483143633536216437109298925879744832537057858532361146651096884843983557835798983559351792270
4
5
$(1,0,2)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
4
5
$(1,1,1)$:
15.58171873876317921324244482081951845523299490243729804813847444076239124339461456571088422542577851
4
5
$(1,2,0)$:
11.17024461095340637472944096505922216748891371974757829024981054592963804368908620075493389238878539
4
5
$(2,0,1)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
4
5
$(2,1,0)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
4
5
$(3,0,0)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
4
5
$(0,0,4)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
4
5
$(0,1,3)$:
7.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
4
5
$(0,2,2)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
4
5
$(0,3,1)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
4
5
$(0,4,0)$:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
4
5
$(1,0,3)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
4
5
$(1,1,2)$:
15.58171873876317921324244482081951845523299490243729804813847444076239124339461456571088422542577851
4
5
$(1,2,1)$:
15.58171873876317921324244482081951845523299490243729804813847444076239124339461456571088422542577851
4
5
$(1,3,0)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
4
5
$(2,0,2)$:
11.17024461095340637472944096505922216748891371974757829024981054592963804368908620075493389238878539
4
5
$(2,1,1)$:
15.58171873876317921324244482081951845523299490243729804813847444076239124339461456571088422542577851
4
5
$(2,2,0)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
4
5
$(3,0,1)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
4
5
$(3,1,0)$:
7.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
4
5
$(4,0,0)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
4
5
$(0,0,5)$:
1
4
5
$(0,1,4)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
4
5
$(0,2,3)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
4
5
$(0,3,2)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
4
5
$(0,4,1)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
4
5
$(0,5,0)$:
1
4
5
$(1,0,4)$:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
4
5
$(1,1,3)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
4
5
$(1,2,2)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
4
5
$(1,3,1)$:
7.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
4
5
$(1,4,0)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
4
5
$(2,0,3)$:
6.758770483143633536216437109298925879744832537057858532361146651096884843983557835798983559351792270
4
5
$(2,1,2)$:
11.17024461095340637472944096505922216748891371974757829024981054592963804368908620075493389238878539
4
5
$(2,2,1)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
4
5
$(2,3,0)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
4
5
$(3,0,2)$:
6.758770483143633536216437109298925879744832537057858532361146651096884843983557835798983559351792270
4
5
$(3,1,1)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
4
5
$(3,2,0)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
4
5
$(4,0,1)$:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
4
5
$(4,1,0)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
4
5
$(5,0,0)$:
1
5
1
$(0,0,0,0)$:
1
comment: Vacuum representation.
5
1
$(0,0,0,1)$:
1
comment: Dual of the defining representation.
5
1
$(0,0,1,0)$:
1
5
1
$(0,1,0,0)$:
1
5
1
$(1,0,0,0)$:
1
comment: Defining representation.
5
2
$(0,0,0,0)$:
1
comment: Vacuum representation.
5
2
$(0,0,0,1)$:
1.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
comment: Dual of the defining representation.
5
2
$(0,0,1,0)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
5
2
$(0,1,0,0)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
5
2
$(1,0,0,0)$:
1.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
comment: Defining representation.
5
2
$(0,0,0,2)$:
1
5
2
$(0,0,1,1)$:
1.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
5
2
$(0,0,2,0)$:
1
5
2
$(0,1,0,1)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
5
2
$(0,1,1,0)$:
1.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
5
2
$(0,2,0,0)$:
1
5
2
$(1,0,0,1)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
5
2
$(1,0,1,0)$:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
5
2
$(1,1,0,0)$:
1.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
5
2
$(2,0,0,0)$:
1
5
3
$(0,0,0,0)$:
1
comment: Vacuum representation.
5
3
$(0,0,0,1)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
comment: Dual of the defining representation.
5
3
$(0,0,1,0)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(0,1,0,0)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(1,0,0,0)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
comment: Defining representation.
5
3
$(0,0,0,2)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(0,0,1,1)$:
4.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
5
3
$(0,0,2,0)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(0,1,0,1)$:
5.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
5
3
$(0,1,1,0)$:
5.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
5
3
$(0,2,0,0)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(1,0,0,1)$:
4.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
5
3
$(1,0,1,0)$:
5.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
5
3
$(1,1,0,0)$:
4.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
5
3
$(2,0,0,0)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(0,0,0,3)$:
1
5
3
$(0,0,1,2)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(0,0,2,1)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(0,0,3,0)$:
1
5
3
$(0,1,0,2)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(0,1,1,1)$:
4.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
5
3
$(0,1,2,0)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(0,2,0,1)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(0,2,1,0)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(0,3,0,0)$:
1
5
3
$(1,0,0,2)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(1,0,1,1)$:
5.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
5
3
$(1,0,2,0)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(1,1,0,1)$:
5.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
5
3
$(1,1,1,0)$:
4.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
5
3
$(1,2,0,0)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(2,0,0,1)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(2,0,1,0)$:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(2,1,0,0)$:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
5
3
$(3,0,0,0)$:
1
5
4
$(0,0,0,0)$:
1
comment: Vacuum representation.
5
4
$(0,0,0,1)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
comment: Dual of the defining representation.
5
4
$(0,0,1,0)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
5
4
$(0,1,0,0)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
5
4
$(1,0,0,0)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
comment: Defining representation.
5
4
$(0,0,0,2)$:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
5
4
$(0,0,1,1)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
5
4
$(0,0,2,0)$:
6.758770483143633536216437109298925879744832537057858532361146651096884843983557835798983559351792270
5
4
$(0,1,0,1)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
5
4
$(0,1,1,0)$:
11.17024461095340637472944096505922216748891371974757829024981054592963804368908620075493389238878539
5
4
$(0,2,0,0)$:
6.758770483143633536216437109298925879744832537057858532361146651096884843983557835798983559351792270
5
4
$(1,0,0,1)$:
7.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
5
4
$(1,0,1,0)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
5
4
$(1,1,0,0)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
5
4
$(2,0,0,0)$:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
5
4
$(0,0,0,3)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
5
4
$(0,0,1,2)$:
7.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
5
4
$(0,0,2,1)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
5
4
$(0,0,3,0)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
5
4
$(0,1,0,2)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
5
4
$(0,1,1,1)$:
15.58171873876317921324244482081951845523299490243729804813847444076239124339461456571088422542577851
5
4
$(0,1,2,0)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
5
4
$(0,2,0,1)$:
11.17024461095340637472944096505922216748891371974757829024981054592963804368908620075493389238878539
5
4
$(0,2,1,0)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
5
4
$(0,3,0,0)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
5
4
$(1,0,0,2)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
5
4
$(1,0,1,1)$:
15.58171873876317921324244482081951845523299490243729804813847444076239124339461456571088422542577851
5
4
$(1,0,2,0)$:
11.17024461095340637472944096505922216748891371974757829024981054592963804368908620075493389238878539
5
4
$(1,1,0,1)$:
15.58171873876317921324244482081951845523299490243729804813847444076239124339461456571088422542577851
5
4
$(1,1,1,0)$:
15.58171873876317921324244482081951845523299490243729804813847444076239124339461456571088422542577851
5
4
$(1,2,0,0)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
5
4
$(2,0,0,1)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
5
4
$(2,0,1,0)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
5
4
$(2,1,0,0)$:
7.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
5
4
$(3,0,0,0)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
5
4
$(0,0,0,4)$:
1
5
4
$(0,0,1,3)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
5
4
$(0,0,2,2)$:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
5
4
$(0,0,3,1)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
5
4
$(0,0,4,0)$:
1
5
4
$(0,1,0,3)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
5
4
$(0,1,1,2)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
5
4
$(0,1,2,1)$:
7.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
5
4
$(0,1,3,0)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
5
4
$(0,2,0,2)$:
6.758770483143633536216437109298925879744832537057858532361146651096884843983557835798983559351792270
5
4
$(0,2,1,1)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
5
4
$(0,2,2,0)$:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
5
4
$(0,3,0,1)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
5
4
$(0,3,1,0)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
5
4
$(0,4,0,0)$:
1
5
4
$(1,0,0,3)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
5
4
$(1,0,1,2)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
5
4
$(1,0,2,1)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
5
4
$(1,0,3,0)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
5
4
$(1,1,0,2)$:
11.17024461095340637472944096505922216748891371974757829024981054592963804368908620075493389238878539
5
4
$(1,1,1,1)$:
15.58171873876317921324244482081951845523299490243729804813847444076239124339461456571088422542577851
5
4
$(1,1,2,0)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
5
4
$(1,2,0,1)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
5
4
$(1,2,1,0)$:
7.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
5
4
$(1,3,0,0)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
5
4
$(2,0,0,2)$:
6.758770483143633536216437109298925879744832537057858532361146651096884843983557835798983559351792270
5
4
$(2,0,1,1)$:
11.17024461095340637472944096505922216748891371974757829024981054592963804368908620075493389238878539
5
4
$(2,0,2,0)$:
6.758770483143633536216437109298925879744832537057858532361146651096884843983557835798983559351792270
5
4
$(2,1,0,1)$:
9.822948255619545677026007711520592575488162365379439515777327789665506399411056729911900666073986244
5
4
$(2,1,1,0)$:
8.290859369381589606621222410409759227616497451218649024069237220381195621697307282855442112712889257
5
4
$(2,2,0,0)$:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
5
4
$(3,0,0,1)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
5
4
$(3,0,1,0)$:
4.411474127809772838513003855760296287744081182689719757888663894832753199705528364955950333036993122
5
4
$(3,1,0,0)$:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
5
4
$(4,0,0,0)$:
1
Definition
The quantum dimension $d_{N,k}(\lambda)$ of an integrable highest weight $\lambda$ in the unitary $SU(N)$ Wess–Zumino–Witten theory at level $k$ is $S_{\lambda,0}/S_{0,0}$, where $S$ is its modular matrix and $0$ denotes the vacuum [2]. The positive real embedding is used.
Parameters
$N$
—   matrix size ($N\geq2$)
$k$
—   level ($k\geq1$)
$(a_1,\ldots,a_{N-1})$
—   Dynkin labels (The tuple $(a_1,\ldots,a_{N-1})$ consists of nonnegative integers with $\sum_i a_i\leq k$ and specifies $\lambda=\sum_i a_i\omega_i$, where $\omega_i$ are the fundamental weights along the $A_{N-1}$ chain.)
Formulas
(1)
$\displaystyle d_{N,k}(\lambda)=\prod_{1\leq i<j\leq N}\frac{[\ell_i-\ell_j+j-i]_q}{[j-i]_q}=\prod_{1\leq i<j\leq N}\frac{\sin(\pi(\ell_i-\ell_j+j-i)/m)}{\sin(\pi(j-i)/m)}$, the type-$A$ quantum Weyl formula [1]. [3].
(2)
$\displaystyle d_{N,k}(\lambda)=s_{\ell}(q^{N-1},q^{N-3},\ldots,q^{1-N})$, where $s_\ell$ is the Schur polynomial [2].
(3)
$\displaystyle d_{2,k}(a\omega_1)=\frac{\sin((a+1)\pi/(k+2))}{\sin(\pi/(k+2))}=U_a(\cos(\pi/(k+2)))$, where $U_a$ is the Chebyshev polynomial of the second kind. In particular, $d_{2,k}(\omega_1)=2\cos(\pi/(k+2))$, twice a cosine at a rational multiple of $\pi$.
(4)
$\displaystyle d_{N,k}(a_1,\ldots,a_{N-1})=d_{N,k}(a_{N-1},\ldots,a_1)$.
(5)
$\displaystyle d_{N,k}(a_1,\ldots,a_{N-1})=d_{N,k}(k-\sum_i a_i,a_1,\ldots,a_{N-2})$, the affine Dynkin rotation.
(6)
$\displaystyle d_{N,k}(0)=d_{N,k}(k\omega_j)=1$ for $1\leq j\leq N-1$.
Comments
(7)
$N$ is the matrix size, so the Lie rank is $N-1$. Set $m=N+k$ and $q=\exp(\pi i/m)$, a primitive $2m$-th root of unity. We use symmetric quantum integers $[b]_q=(q^b-q^{-b})/(q-q^{-1})$ and normalize long roots to squared length $2$ [1]. A convention that calls $\exp(2\pi i/m)$ its deformation parameter therefore uses $q^2$ in this notation.
(8)
The Dynkin labels are ordered along the $A_{N-1}$ chain, with $\omega_1$ the defining representation. The character partition is $\ell=(\ell_1,\ldots,\ell_N)$, where $\ell_i=\sum_{u=i}^{N-1}a_u$ and $\ell_N=0$. Distinct highest weights remain separate rows even when their dimensions agree. This includes the vacuum and the invertible representations of dimension $1$.
(9)
The theory is the standard unitary $SU(N)$ theory, without a quotient of the group, conformal extension or Galois conjugation.
(10)
For the defining $SU(2)$ representation, $d_{2,k}(\omega_1)$ equals the spectral radius of the path on $k+1$ vertices. That table contains these paths through $k=6$.
(11)
The global dimension $\mathcal G_{N,k}$ of the same fusion category is the sum of $d_{N,k}(\lambda)^2$ over the entire integrable alcove, including every representation of dimension $1$ [1].
Programs
(P1)
Sage
def qdim(N, k, a, prec=429):
    assert N >= 2 and k >= 1 and len(a) == N-1
    assert all(t in ZZ and t >= 0 for t in a) and sum(a) <= k
    m = N+k
    K = CyclotomicField(2*m)
    q = K.gen()
    R = RealBallField(prec)
    exact, enclosure = K.one(), R.one()
    for i in range(N-1):
        for j in range(i+1, N):
            u, v = j-i+sum(a[i:j]), j-i
            exact *= (q**u-q**(-u))/(q**v-q**(-v))
            enclosure *= (R.pi()*u/m).sin()/(R.pi()*v/m).sin()
    return QQ(exact) if exact in QQ else enclosure

qdim(3, 3, [1, 1])  # the SU(3) adjoint at level 3 has dimension 3
References
[1]
R. Coquereaux, Quantum McKay correspondence and global dimensions for fusion and module-categories associated with Lie groups, Journal of Algebra 398 (2014), 258–283, Sections 2.1–2.2. (arXiv)
[2]
R. Coquereaux, Global dimensions for Lie groups at level k and their conformally exceptional quantum subgroups, preprint (2010), Section 2.2. (arXiv)
Links
Similar tables
Roots of unity —   contains the cyclotomic parameters $q=\exp(\pi i/(N+k))$
Chebyshev polynomials of the second kind —   evaluate to all $SU(2)$ quantum dimensions at $\cos(\pi/(k+2))$
Schur polynomials —   give type-$A$ characters whose principal specializations are the quantum dimensions
Cosines at rational multiples of pi —   give half the defining $SU(2)$ quantum dimension at $x=1/(k+2)$
Spectral radii of connected graphs —   agree on paths with the defining $SU(2)$ dimensions through $k=6$
Data properties
Entries are of type: real number
digits: 100 (Unknown key)
How well the digits are known: proven
Table is complete: no (it holds every integrable highest weight for $N=2$ at $1\leq k\leq20$, $N=3$ at $1\leq k\leq8$, $N=4$ at $1\leq k\leq5$, and $N=5$ at $1\leq k\leq4$, including the vacuum and all dimension-one rows; these 37 complete alcoves provide reference coverage across the first four type-$A$ ranks before extending rank-one depth, with noncanonical level cutoffs that balance the growth of $\binom{k+N-1}{N-1}$ weights per alcove; the attached argument-selection.md explains each discrete index and the practical bounds)
How they were obtained:

Exact arithmetic in $\mathbb Q(\zeta_{2(N+k)})$ determines every rational result algebraically; these values are written exactly. All other values are enclosed by Arb arithmetic throughout the sine product, using 429 working bits, retaining at least 421 measured bits of accuracy, and writing 100 significant decimal digits. The positive sine arguments fix the unitary embedding. The attached numerical-evidence.json records the measured accuracy and serialized sizes.

more

The attached checks.py compares every quantum Weyl product exactly with an independently computed Schur character, using coefficient convolution and a Jacobi–Trudi determinant. A separate 600-bit complex-ball evaluation of each exact character certifies containment in its written decimal interval. Known-answer and deliberate-error controls precede the comparisons. Conjugation and affine Dynkin rotation, all $SU(2)$ Chebyshev rows and fusion identities, and all 37 global-dimension normalizations are checked, with the same interval checks repeated on the database readback.



API Python package Skill for AI Privacy Legal notice Source
Data licensed CC BY-SA 4.0. No tracking, no analytics, essential cookies only.