2 Abstract.Weproposeanalgorithmcomputingexplicitgeneratingfunctionsofthe stablemultiplicitiesofirreduciblerepresentations(n jj;)ofsnarisinginthe

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1 coecientsofu(n)#sn,o(n)#snando(n 1)#Sn Generatingfunctionsforstablebranching 93166Noisy-le-Grandcedex,France ylehrstuhliifurmathematik,universitatbayreuth,95440bayreuth,germany zinstitutgaspardmonge,universitedemarne-la-vallee,2ruedelabutte-verte, ThomasScharfy,Jean-YvesThibonz,andBrianG.Wybournex Poland xinstytutfizyki,uniwersytetmiko lajakopernika,ul.grudziadzka5/7,87-100torun, ysupportedbyprocope zsupportedbyprocope xsupportedbyapolishkbngrant

2 2 Abstract.Weproposeanalgorithmcomputingexplicitgeneratingfunctionsofthe stablemultiplicitiesofirreduciblerepresentations(n jj;)ofsnarisinginthe irrep. oftheunitaryororthogonalgroup;i.e.,wecomputethemultiplicitiesinawaywhich isindependentofnandm,theweightofthelabel(m jj;)ofthecorresponding restrictionfromu(n),o(n)oro(n 1)toSnofanirreducibletensorrepresentation

3 1.Introduction ThesymmetricgroupSnplaysanimportantroleinthoseareasofphysicsandchemistry involvingpermutationalsymmetrysuchasintheimplementationofthepauliexclusion principleinconstructingtotallyantisymmetricwavefunctionsforidenticalfermions. 3 coecientsinvolvetheinnerplethysms[11,13]: subgrouprestrictionsu(n)#sn,o(n)#snando(n 1)#Sn.Inthesecasesthe Thissituationarisesinthecaseofevaluatingthebranchingcoecientsforthegroup- Suchapplicationsarise,forexample,intheclassicationofn electronstatesandin symplecticmodelsofnucleiandquantumdots[4].theseapplicationsfrequentlyrequire theresolutionofsymmetrisedpowersoftheirreducibletensorrepresentationsofsn. O(n)#Sn O(n 1)#Sn[]#h1if=Cg fg#h1if=mg []#h1if=gg (1) (2) thespecialcases=(n);(1n),forobtainingthecompletecontentoftheplethysm Sn[10,7,8,14]isexploited. Theevaluationoftheaboveinnerplethysmsisthekeyproblemconsideredherein.Most previousformulationshaveinvolvedlittlewood'smethods[6,7,12],seealso[1,2]for wherem;candgareinniteseriesofs functions[6,5]andthereducednotationfor (3) generatingfunctionsusingtheformalismofvertexoperatorsinthesamewayasin ofthegreatestpart1of.weconstructexplicitalgorithmsforobtainingtherelevant showthatitisalsopossibletoobtainbranchingformulaewhicharealsoindependent n-independentresults,e.g.fortherestrictionofanirrepfgofu(n)tosn.herewe ofirreduciblerepresentations(n jj;)ofsn.theclassicalmethodsallowtogive highlydesirabletobeabletoconstructgeneratingfunctionstoyieldthemultiplicities whereasinterestoftenliesinthecomputationofspeciccoecients.tothatenditis [14,3,13]anddemonstratetheprocedurewithseveralillustrativeexamples. dierenceofvariablesetsaredenedbypk(x Y)=pk(X) pk(y)andthattheset 2.NotationsandBackground denotedeitherby^s(s)orbys^s.werecallthatsymmetricfunctionsofaformal Ourstandardreferenceforsymmetricfunctionswillbe[9]andwewilladoptitsnotations asin[13].aschurfunctionisdenotedbyfgorbys,dependingonwhetheritis f1;z;z2;:::gisidentiedwiththepowerseries(1 z) 1. interpretedasacharacterofu(n)orasanoperator.theinnerplethysm()fgis

4 4AnalgebraicformulationofLittlewood'sreducednotationforsymmetricfunctionscan details(seealso[9],inparticularex.29,p.95.,ex.3,p.75f.,ex.25,p.91). Thatis,Littlewood'sreducednotationhihastobeinterpretedastheinniteseries 1s.Wesketchsomeofitsbasicpropertiesandreferthereaderto[14,3,13],e.g.,for begivenbyconsideringaparticularcaseofavertexoperator Theadjoints(w.r.t.Hall'sinnerproduct)ofmultiplicationofsymmetricfunctionsby zs:=xn2zs(n;)zn: thegeneratingfunctionsforthecompletesymmetricandelementarysymmetric functions,arealgebraautomorphismsdz;dzandwehave z(x):=xn0hn(x)zn;andz(x):=xn0en(x)zn; foranysymmetricfunctionf.thisamountstosaythatforthepowersumswehave InLambda-ringnotationtheseoperatorscanbedescribedas DzF(X)=F(X+z);D zf(x)=f(x z); DzD z=id: andforschurfunctions Finally,thevertexoperatorhasafactorization Dzs(X)=Xi0s=(i)(X)zi;D zs(x)=xi0s=(1i)(x)( z)i: Dzpk(X)=pk(X)+zk;D zpk(x)=pk(x) zk; aformula,whichwewilluseinthesequel. 3.Stableinnerplethysmmultiplicities zf(x)=z(x)d 1=zF(X)=z(X)F(X 1=z) (4) themultiplicityofanirreduciblerepresentation(n jj;)ofsnintheinnerplethysm 3.1.Theresult Inthissection,weaddresstheproblemofcomputinginannandmindependentway

5 h1ifm jj;g.thiswillbeappliedtothecalculationofthebranchingcoecients of(1)-(3)intheforthcomingsections. WeknowthatLittlewood'sreducednotationallowsustoexpandtheinnerplethysm h1ifgasalinearcombinationofstablecharacters: h1ifg=xdhi (5) 5 willalwaysbeequaltod. weset=(1;),andweconsiderthegeneratingfunction Here,wegoonestepfurther,andallowtherstpart1oftobearbitrary.Thatis, sothatforanynsuchthatn jj1,themultiplicityoftheirrep(n jj;)ofsn Then,thec(z)'(z),where'(z)=Yk1(1 zk),arerationalfunctionswhichcanbe explicitlycomputedbytheproceduredescribedbelow. Forexample,with=(2;1),weobtainc(z)=(1 z)'(z) 1a(z),with F(z):=Xn2Zh1ifn;gzn:=Xn2Zzn^s(n;)h1i:=Xc(z)hi: (6) a21;2(z)=(1+z z3)z2 a21;1(z)=1 3z+2z2+3z4 2z5 z6+z7 a21;0(z)= (1 z3)(1 z)2 (1 z)4(1 z3)(1+z) z4 z2(1 z)3(1 z3) (8) (7) andfor=(4;2;2),a21;422(z)isequalto z6(z16 z14 2z13 2z12+z11+z10+3z9+z8+z7+2z6 z4 3z3 z2 1) a21;11(z)=(1+z z3 z4+z5)z2 (1 z)3(1 z2)(1 z3) (10) (9) to of(30;4;2;2)intheinnerplethysm(37;1)f10;2;1gisequalto125. TakingtheTaylorexpansionofthelatteruptoorder10,wegetthatthemultiplicity Similarlywendthatthemultiplicityof(31;3;3;2;1)in(39;1)f30;4;3;2;1gisequal (z6 1)(z+z2+1)(z 1)6(z2 1)3(z3 1)(z4+z3+z2+z+1) (11) 3.2.Derivationofthegeneratingfunctions Usingthevertexoperatorformula(4) Xn2Zzns(n;)= zs=z(x)sx 1z (12)

6 wehave 6andtakingintoaccounttheinnerplethysmseries(cf.[1]) F(z)=z(X)sX 1z^h1i ^zh1i=xn0^hnh1izn=(1 z)1x =^zh1isx 1z^h1i1 z=(1 Forexample,with=(2;1)onewouldhave whereu:=[s(x 1z)]^h1i. U=s21 1zs21=1+1z2s21=12^h1i=^s21h1i 1z(h1ih1i)+1z2h1i: =(1 z)1x 1 zu; (14) Toevaluatesuchanexpression,weuseLittlewood'sformula,whichgivestheresultas acombinationofstablecharactershi=1d 1s=1s(X 1),andwekeepapart thefactor1,writingu=1h. Inourexample,weobtain ^s21=1h1i=h1ih1i=h2i+h11i+h1i+h0i=1(s2+s11 s1+1) ^s21h1i=h1if21g=h21i+h2i+h11i+h1i=1s21 ^s21=11h1i=h1i=1(s1 1) (15) (16) Next,weexpandtheinternalproduct(14),takingintoaccounttheproperty and,nally,u=1s21 1z(s2+s11 s1+1)+1z2(s1 1)=1H (18) (17) whichgivesf(z)=(1 z)1x 1 zf(x)=fx 1 zu(x) 1 z =(1 z)ux 1 zhx 1 z: (19)

7 Nowweextractavertexoperatorbywriting and(notethatd 1D1=id) 1X 1 z=1(x)1zx 1 z 7 Thus, 1zX F(z)=(1 z)1z 1 zhx 1 z=d 11 =1z 1 zd 11zX z(x+1) 1 z!hx+1 1 zhx+1 =Yi21 1 zi 11zX 1 z 11zX 1 zhx+1 1 zhx+1 1 z: 1 z1 z: Ifwewrite then 1 zhx+1 1 z=xa(z)s; (20) Wecannowcomputea(z)usingpropertiesofthescalarproductandadjointoperators: with c(z)=a(z)yi21 F(z)=Xc(z)hi a(z)=s;1zx =*s;yk1zk(x)hx+1 1 zhx+1 1 z 1 z+ 1 zi: NowweexpandH(X+1 (pk+1)=(1 zk).lettherationalfunctionsd(z)bedenedby HX+1 1 z=xd(z)s(x): =sx+z 1 z)byreplacingeachpowersumpkintheexpansionofhby 1 z;hx+1 1 z: (21) Now, sx+z 1 z=xs(x)s=z 1 z; (22)

8 For=(2;1),thecoecientsd21;(z)aregivenby 8sothat HX+1 1 z= (1 z3)(z 1)2s3+1+z2 za(z)=xd(z)s=z (1 z3)(z 1)2s21+ 1 z Fromtheseexpressions,weobtaintherequiredgeneratingfunctions. z(1 z)3s2+2z 1 z(1 z)3s11+1 3z+2z2+z3 z2(1 z)3s1+ (1 z3)(z 1)2s111 z(1 z3)(z 1)2s0 z4 3.3.Summaryofthealgorithm (23) Tocomputethegeneratingfunctionc(z): (i)evaluatef=s(x 1z),eitherbyexpandingsonthebasispandreplacingeach (ii)computeu=^fh1iasalinearcombinationofstablecharactershibymeansof Littlewood'sformula,andwriteitintheformU=1H,takingintoaccountthe powersumpkbypk z k,orbythemoreecientformula factthathi=1s(x 1). sx 1z=`() Xr=0 1zrs=1r(X): (iv)takethescalarproductofthepreviousexpressionwith (iii)evaluatehx+1 replacingeachpkby(pk+1)=(1 zk). 1 z,forexamplebyexpandinghintermsofpowersums,and (v)c(z)=(1 z)'(z) 1a(z). yieldsa(z). (aclosedformulaforsz sx+z 1 zcanbefoundforexamplein[9],ex.2p.45).this 1 z=xsz 1 zs=(x) Themultiplicityoftheirreduciblerepresentation(n jj;)ofsn(foranyn)inthe ofc(z). innerplethysmh1ifm;gisthenequaltothecoecientofzminthetaylorexpansion

9 Thewell-knownstablebranchingruleforU(n)#Snis 4.TherestrictionU(n)#Snfg#h1if=Mg 9 where Tocomputethebranchingcoecientsweconsiderthegeneratingseries M=1=Yi1 andrewriteitasbeforeasf(z)=hd1zd 1=zsi^h1i F(z):=Xn2Zh1if(n;)=Mgzn:=Xc(z)hi; 1 xi: Thus,ifwedeneHby1H=sX 1z+1^h1i; =(1 z) 1z(X)sX 1z+1^h1i =1X =z(x+1)sx 1z+1^h1i wearriveatc(z)=yk111 zsx 1z+1^h1i: For=(2;1),weget where a(z)=sx+z 1 zka(z); 1 z;hx+1 1 z: (25) (24) HX+1 1 z=(2z5 3z4 2z2+3z 1) + +(z2 3z+1) (z2+z+1)(1 z)3s111+(3z3 8z2+5z 1) (z 1)3zs11+ (z2+z+1)(z 1)3z2s0+(z2 3z+1) z (z2+z+1)(1 z)3s3+ z(z 1)3z2s1 (z 1)3zs2 (z2+z+1)(1 z)3s21 (z2+1) (26)

10 10 Fromthiswemaycompute c21;11(z)= 1+3z2+15z3+42z4+102z5+215z6+425z7+785z8+1391z9 c21;211(z)=2z2+10z3+36z4+104z5+260z6+587z7+1229z8+2425z9 Hence,themultiplicityof(11;2;1;1)intherestrictionoftheirrepf12;2;1gofU(15) tos15isequalto14366,thecoecientofz12intheexpansionofc21;211(z) z z z z z14+Oz z z z z z14+Oz15 ItisknownthatthestablebranchingruleforO(n)#Snisgivenby[11] 5.TherestrictionO(n)#Sn where G=1 1[h2]=MC=Yi1 []#h1if=gg Wehave Wewanttocomputethecoecientsc(z)ofthegeneratingseries F(z):=Xn2Zh1if(n;)=Ggzn:=Xc(z)hi: 1 xiyij(1 xixj): Wenowusethefollowingproperties: =(1 z) 1D 1[h2]z(X)sX 1z+1^h1i: =D 1[h2]z(X+1)sX 1z+1^h1i F(z)=hD1D 1[h2]zD 1=zsi^h1i whereisthecomultiplication(pk)=pkpk(i.e.(f)=f(xy),denotedbyin Lemma5.1.Foranysymmetricfunctionsf;g, [9],p.128)andthemultiplicationoperator(fg)=fg. D 1[h2](fg)=D 1[h2]D 1[h2]D 1(fg);

11 (Heremeanstensorproduct,notplethysm). ofthescalarproduct Proof:Lethbeanarbitrarysymmetricfunction.Usingthedualitybetween multiplicationandcomultiplication,wehavethefollowingsequenceoftransformations DD 1[h2](fg);hE=hfg;( 1[h2]h)i=hfg;( 1[h2])(h)i11 =hfg; 1[h21+h1h1+1h2](h)i Lemma5.2.Foranysymmetricfunctionf, =DD 1(fg); 1[h2] 1[h2](h)E =DD 1[h2]D 1[h2]D 1(fg);hE: have D 1(X)z(X)f(X)=Xr0( 1)rDsrz(X)Ds1rf(X) Proof:TakingintoaccountthefactthatDshn=0if`()>1andDsrhn=hn r,we =z(x)d z(x)f(x)=z(x)f(x z): D 1zf=zf(X z): Also,fromthewell-knownexpansionof 1[h2](cf.[9],ex.9,p.78)weget 1[h2]= where 1 h2+schurfunctionsindexedbypartitionswithmorethanonepart,hence whichimpliesthat D 1[h2]z=(1 z2)z; U:=D 1[h2]sX 1z+1 z^h1i=1h: F(z)=(1 z2) (1 z)^zh1iu (27) where Thus,asintheprevioussection, a(z)=sx+z c(z)=(1 z2)yk11 1 z;hx+1 1 zka(z) 1 z: (28) (29)

12 c21;11(z)= 1+z2+6z3+17z4+41z5+84z6+163z7+294z8+510z9+850z10 12 Ontheexample=(2;1),weget c21;2(z)=2z2+6z3+18z4+41z5+86z6+165z7+301z8+522z9+876z10 c21;221(z)=z2+3z3+12z4+36z5+95z6+221z7+478z8+966z9+1857z10 c21;5211(z)=z7+6z8+25z9+86z10+252z11+663z z z14+o(z15) +1378z z z z14+O(z15) +1422z z z z14+O(z15) O(18)toS18isequalto1599. sothatforexample,themultiplicityof(95211)intherestrictionoftheirrep[13;2;1]of +3416z z z z14+O(z15) Theseriestobecomputedhereis 6.TherestrictionO(n 1)#Sn andwritingasaboveu=d 1[h2]sX 1z z^h1i Acalculationsimilartotheoneoftheprecedingsectionshowsthat F(z)=Xn2Zh1if(n;)=Cg=hD 1[h2]zD 1=zsi^h1i F(z)=(1 z2)zd 1[h2]sX 1z z^h1i (31) (30) wherea(z)isonceagaingivenby intheformu=1hwehave Forexample,with=(2;1) a(z)=sx+z c(z)=(1 z)(1 z2)yk11 1 z;hx+1 1 zka(z) 1 z: (32) c21;11(z)=z3+3z4+7z5+15z6+29z7+52z8+89z9+147z10+235z z12+558z13+834z14+O(z15) (33)

13 c21;211(z)=z2+4z3+11z4+26z5+56z6+111z7+208z8+372z9+641z10 c21;421(z)=z4+3z5+12z6+37z7+98z8+231z9+507z z z z z z14+O(z15) +2022z z z14+O(z15) 13 sothat,forexample,themultiplicityof(5211)intherestrictionoftheirrep[521]of O(8)toS(9)isequalto26. 7.Astabilitypropertyofthecoecientsc(z) h1ihn21i106hn 3iforn10.Weobserveasimilarstabilitywhenincreasing acertainnaturalway.letusstartbylookingatsomeconcreteexamplesforinner plethysm.theremarksaredirectlyapplicabletotherestrictions(1)-(3),astheprevious sectionshaveshown. Therstthreeexamplesintable1suggestthatthesequenceofmultiplicitiesbecomes constantwhenincreasingtherstrowofandtherstrowof.forexample,onehas Themultiplicitiesofirrepsexhibitacertainstabilityproperty,ifgroupedtogetherin showninthefourthexample. therstcolumn,butthenhavetomakeshiftsonthelefthandsideby1;2;3;4;:::as Toseethisxandandset:=(2;3;:::).Thenitisimmediatefromthe h1ifm;gcmp()hp;iandh1ifm+q;gcm+q Tostatethepropertyinapreciseway,suppose denitionsthatc(z)=xzcm1()zm: forallqqs(andqsdependsonandonly). Then,the(Cm+q Thispropertyissharedbyh1ifm;=Mg;h1ifm;=Ggandh1ifm;=Cg. p+q())qformanitesequenceofintegerssuchthatcm+q p+q()hp+q;i: p+q()=cm+qs p+qs() (34) itcanbeprovedbyusingthevertexoperatormethod. z 1c(z)andwillshowthatthisexpressionconvergesasaformalLaurentserieswhen 1goestoinnity.Thispropertyistrueforalltherestrictionsconsideredbefore,and Henceweonlyneedtoshowastabilitypropertyofthec(z)(or,equivalently,ofthe a(z)),when1increases.togetcontrolonthecorrespondingshiftsweratherconsider (35) Letubeanotherindeterminate,commutingwithz.Then usx+z 1 z=ux+z =uz 1 zu(x)sx 1u+z 1 zsx 1u+z 1 z;(36)

14 14 Table1.Somemultiplicitiesofhiintheinnerplethysmh1ifg Multiplicity ,2,111,2, Multiplicity Multiplicity ,2,17,2,110,2,114,2,119,2,125,2, whence Xia(i;)(z)ui= usx+z Multiplicity zu(x)sx 1u+z 1 z;hx+1 1 z =uz 1 zsx 1u+z 1 zp(z;u); 1 z;hx+1+u 1 z (37)

15 isalaurentpolynomialinuwithrationalfunctionsinzascoecients. where P(z;u):=tXi=bri(z)ui:=sX 1u+z 1 z;hx+1+u 1 z(38) 15 Recall(from[9],ex.5p.27,e.g.)that Therefore,togeta(1;);1t,i.e.thecoecientofu1in uz 1 z=xi uz 1 zp(z;u); Qij=1(1 zj)ui: zi ofu1in itsucestoconsiderthecoecientofu1in Ontheotherhand,asaLaurentseriesinz,thiscoecientconvergestothecoecient 0@1 b Q1 j=1(1 zj)u10@ b l=1 t z1 XQlj=1(1 zj)ul1ap(z;u): zl when1goestoinnity.hencez 1c;(z)convergestotheconstantterm(w.r.t.u)in Yj2(1 zj) 1Yj1(1 zj) 10@ b l= ttlul1ap(z;u); P(z;u)= Ifwechoose=(2;1)and:=(),theemptypartition,then (1 z3)(z 1)2u3+2z 1 z(1 z)3u2+1 3z+2z2+z3 l= ttlul1ap(z;u): Xz2(1 z)3u+ (1 z3)(z 1)2; z4(39) andtheconstanttermof equalsz 3 (1 z3)(z 1)2+2z 1 =z 1+2+3z+z2+z3 z(1 z)2(1 z3): z 3u 31+uz+u2z2+u3z3P;(z;u) (1 z)3+1 3z+2z2+z3 (1 z3)(z 1)2! z4

16 16 Finallyfor1largeenoughtheexpansionofz 1c(z)uptoorder7is Thecoecientofzqintheaboveexpansionisthe\stablemultiplicity"ofh1iin oftable1;106,thecoecientofz3,appearsasthelimitinthesecondexampleof (thelaurentexpansionalwaysstartswithz `()).Thiscanbeinterpretedasfollows. h1if1+q;2;1g.thecoecientofz0is5inaccordancewiththerstexample table1. z z+45z2+106z3+230z4+467z5+901z6+O(z7) :=(02;03;:::)0,i.e.withoutitsrstcolumn.Astheshiftsproceedbystepsof Inthefourthexampleoftable1therstcolumnofincreases.Wethereforeset 1;2;3;:::,wewillhavetoconsiderthelimitz (01 t+1 Heretisanon-negativeintegerdependingonandonly(cf.equation(41)below). Theneverythingcanbeproveninthesamewayasbeforebyusingthedualnotions. Thedualversionofthevertexoperatoris Xi( 1)i+1+jjs(i;0)0ui= u(x)sx+1u: 2)c(z),when01tendstoinnity. andrecalling(e.g.,from[9],ex.5p.27)that Setting Q(z;u):=tXi=b~ri(z)ui:=sX+1u+z 1 z;hx+1 u 1 z(41) (40) uz 1 z=xiz(i+1 dominatesandthedesiredlimitz (01 t+1 fori!1onlyoneterm,namely Qi t j=1(1 zj)( u)i t~rt(z); z(i t+1 2) Qij=1(1 zj)( u)i; 2) Forexample,with:=(2;1)and:=(),wegett:=3, Yj2(1 zj) 1Yj1(1 zj) 1~rt(z): 2)c(z)is (43) (42) andthedesiredexpansionis h101iinh1if01 2 Wecaninterpretthisasfollows.Thecoecientofzqgivesthestablemultiplicityof z+3z2+8z3+19z4+41z5+82z6+158z7+290z8+516z9+o(z10): 2+q;2;1g(recallthatt=3).Thecoecientofz4is19inaccordance ~rt(z):= (1 z3)(z 1)2; z withtable1above.

17 playsakeyroleinsymplecticmodelsofnucleiwhichisacombinatoriallyexplosive 8.Concludingremarks ThecomputationofbranchingcoecientsofU(n)#Sn,O(n)#SnandO(n 1)#Sn problem.previousalgorithmsbecomecomputationallyimpossiblewhenthenumberof 17 ofearlieralgorithmsandrequirenouseofmodicationrules.theyhavetheconsiderable nucleonsbecomeslarge.thealgorithmsoutlinedinthispaperovercomethelimitations advantageoverothermethodsofbeingabletoyieldspeciccoecientsratherthanthe Acknowledgements completesetofcoecients,mostofwhich,inpracticalcalculations,areredundant. Thepreparationofthispaperhasbeenfacilitatedbyseveralcomputerprograms.The Thestabilitypropertiesofthebranchingcoecientsandinnerplethysmmultiplicities examplespresentedherehavebeencomputedwiththehelpofj.stembridge'smaple ndanaturalexplanationintermsofgeneratingfunctions. SCHUR. propertiesprovedinsection7wereconjecturedonthebasisoftablescomputedwith packagesfforsymmetricfunctionsandbys.veigneau'ssymfoface3.0.the

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