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FUSION ALGEBRA AND VERLINDE’S FORMULA

arXiv:hep-th/9212084v1 14 Dec 1992Preliminary versionFUSION ALGEBRA AND VERLINDE’S FORMULAANATOL N. KIRILLOVIsaac Newton Institute for Mathematical Sciences,20 Clarkson Road, Cambridge, CB3 OEH, U.K.andSteklov Mathematical Institute,Fontanka 27, St.Petersburg, 191011, RussiaABSTRACTWe show that the coefficients of a decomposition into an irreducible components ofthe tensor powers of level r symmetric algebra of adjoint representation coincide withthe Verlinde numbers. Also we construct (for sl(2)) the representations of a generallinear group those dimensions are given by corresponding Verlinde’s numbers.In this note we sum up some results dealing with a level r fusion algebra Fr(g)for the semisimple Lie algebra g. In fact we will try to describe in details the fusionalgebra for g = gl(n) and even for g = sl(2) leaving the general case as a collectionof likelihood conjectures.

Our leading idea is to model the Kostant results [Ko1],[Ko2] about the structure of the symmetric algebra of adjoint representation of theLie algebra g for the case of quantum deformation of universal enveloping algebraUq(g) at root of unity: qr = 1.§1. Multiplication in the sl(2) fusion algebra.

The Bethe ansatz ap-proach.We start with a consideration of the Lie algebra g = sl(2). In this case a level rfusion algebra Fr is defined as a finite dimensional algebra over real numbers R withgenerators {vj | j = 0, 1/2, 1, 3/2, .

. .

, r −2/2} and the following multiplicationrules (the level r Clebsch-Gordan series):vj1 b⊗vj2 =min(j1+j2, r−2−j1−j2)Xj=|j1−j2|, j−j1−j2∈Zvj. (1.1)Let us remark that the fusion rules (1.1) correspond to decomposing the tensorproduct Vj1 b⊗Vj2 of restricted representations [Ro] Vj1 and Vj2 of the Hopf algebraUq(sl(2)), when q = exp( 2πir ), into the irreducible parts (see e.g.

[Lu]).It is

2Anatol N. Kirillovalso well-known (see e.g. [Ka])that the fusion algebra F(g) is a commutative andassociative one.Theorem 1.1.

Let us consider a decomposition of a product vj1 b⊗. .

. b⊗vjl inthe fusion algebra Fr:vj1 b⊗vj2 b⊗.

. .

b⊗vjl =Xkaj(k)vk.Thenaj(k) =X{ν}Yn≥1Pn,r(ν, j) + mn(ν)mn(ν),(1.2)where a summation in (1.2) is taken over all partitions ν = (ν1 ≥ν2 ≥· · · ≥0) ofthe number Pls=1 js −k such that for all n ≥1 the following inequalities are validPn,r(ν, j) :=lXs=1min (n, 2js) −2Qn(ν) −max (2k + n −r + 2, 0) ≥0. (1.3)HereQn(ν) :=Xi≥1min (n, νi) =Xi≤nν′i,mn(ν) := ν′n −ν′n+1be the number of parts of the partition ν which are equal to n andm + nn= (m + n)!m!n!be a binomial coefficient.A proof of Theorem 1.1 is based on an investigation of the Bethe equations forRSOS models [BR].

Note that validity of the equalities (1.2) for fixed j1, . .

. jl andfor all 0 ≤k ≤r −2, are equivalent to a combinatorial completeness (e.g.

[Ki]) ofthe Bethe ansatz for RSOS models (for the case g = sl(2)).Now let us consider a simple example: r = 5, j1 = . .

. = jl = 1.

We have two“even” representations in the fusion algebra F5, namely, V0 and V1 and V1 b⊗V1 =V0 + V1. Let us put V b⊗l1= alV0 + blV1.

Then it is easy to see that al+1 = bl andbl+1 = al + bl = bl + bl−1. Consequently, we have al = Fl−1 and bl = Fl, where Flbe the l-th Fibonacci number.

Hence from Theorem 1.1 we obtainCorollary 1.2. Let Fl be the l-th Fibonacci number.

Then we have1)Fl−1 =X{ν}Yn≥1Pn,5(ν, •) + mn(ν)mn(ν),(1.4)

Fusion algebra and Verlinde’s formula3where a summation in (1.4) is taken over all partitions ν such thati)|ν| = lii) Pn,5(ν, •) := l min (n, 2) −2Qn(ν) −max (n −3, 0) ≥0.2)Fl =X{ν}Yn≥1 ePn,5(ν, •) + mn(ν)mn(ν),(1.5)where a summation in (1.5) is taken over all partitions ν such thati)|ν| = l −1ii)ePn,5(ν, •) := l min (n, 2) −2Qn(ν) −max (n −1, 0) ≥0.Let us underline that formulae (1.4) and (1.5) give the different expressionsfor the Fibonacci numbers. For example, a formula (4) gives for F7 the followingexpression in terms of riggid configurations040002005+1+6+1=13,where as the formula (1.5) gives the following one003111+8+4=13.It is possible to give the bijective proofs for identities (1.4) and (1.5).

Further-more, using (1.4) and (1.5) one can construct two Fibonacci lattice (e.g.[St1],[St2]). However, we do not assume to give here any combinatorial details aboutvery interesting combinatorial objects related with fusion algebra: restricted Youngtableaux, restricted Kostka-Foulkes polynomials, restricted Littlewood-Richardsonrule.

All these things deserve a separate publication.Now let us return back to our example concerning with the fusion algebraF5(sl(2)). It seems a very interesting task to find the natural q-analogs for iden-tities (1.4) and (1.5).

We leave for another publication an exact construction ofsuch q-analogs, but here let us consider the following well-known q-analog of theFibonacci numbers Fl(q). Namely, let us defineF0(q) = 0,F1(q) = 1,Fl+1(q) = qFl(q) + Fl−1(q),l ≥1.

(1.6)It is easy to see thatXl≥1Fl(q) · tl =t1 −qt −t2 ,andFl(q) = det | δi,j + δi,j−1 −qδi,j+1| 1≤i,j≤l−1.

4Anatol N. KirillovIn order to understand better the relations between the q-Fibonacci numbers (1.6)and fusion algebra F5, let us introduce an algebra eF5 = {v0, v1 | v1 · v1 = v0 + qv1}.Then it is easy to see thatvl1 = Fl−1(q) · v0 + Fl(q) · v1. (1.7)It is possible to rewrite (1.7) as followsFlFl+1=011q Fl−1Fl.Note that the matrices011xand011ydo not commute (if x ̸= y):011x 011y=1yx1 + xy̸=011y 011x.However, it is easy to check that if we put h(x) =1xx1 + xthen h(x) and h(y)are commute.

Now let us define a vacuum vector |0 >:=10and consider thefollowing vectorshn(x)|0 >:= h(x1)h(x2) . .

. h(xn)|0 >=an(x)bn(x),(1.8)sn(x)|0 >:= 2nhnx2|0 >=cn(x)dn(x).From what it was said above one can deduce that an(x) and bn(x) are to be thesymmetric functions on x = (x1, .

. .

, xn). It is a simple exercise to decompose thesesymmetric functions into a linear combination of the Schur functions sλ(x).Exercise 1.

Let us check:an(x) = 1 +n−1Xk=1Fk · s(1k+1)(x),bn(x) =nXk=1Fk · s(1k)(x),cn(x) = 2n +n−1Xk=12n−1−kFks(1k+1)(x),(1.9)dn(x) =nXk=12n−kFks(1k)(x).

Fusion algebra and Verlinde’s formula5Now let us calculate the values of an(1) and bn(1). For this goal we consider thefollowing elements in the fusion algebra F5(sl(2))H(x) = v0 + xv1 = h(x)|0 >,H := H(1),S(x) = 2v0 + xv1 =2xx2 + x|0 >,S := S(1).We must compute the coefficients of decompositionsHn := anv0 + bnv1,Sn := cnv0 + dnv1.Here we give an answer only for an and cn.From a definition it is clear thatan+1 = an + bn,bn+1 = an + 2bn,cn+1 = 2cn + dn,dn+1 = cn + 3dn.Consequently, if we put ϕ(t) = Pn≥1 antn and ψ(t) = Pn≥1 cntn thenϕ(t) =t −t21 −3t + t2 ,ψ(t) =2t −5t21 −5t + 5t2 .

(1.10)Using the generating functions (1.10), one can easily find (n > 0,k = 3):an = an(1) =2k + 2kXm=0sin2 (m + 1)πk + 22 cos (m + 1)πk + 2−2(n−1),cn = cn(1) = 12kXm=0 r4k + 2 sin (m + 1)πk + 2!−2(n−1).In particular, we have the following equalitiescn = 2n +n−1Xk=12n−1−knk + 1Fk,an = 1 +n−1Xk=1nk + 1Fk,dn =NXk=12n−knkFk.Let us sum up the results of our computations in the fusion algebra F5. First ofall, it was shown (for r = 5) that a multiplicity of a representation V0 in the n-foldrestricted tensor product Sb⊗n of the level r symmetric algebra S(= 2V0 + V1) isequal to the Verlinde number V (r −2, n):[V0 : Sb⊗n] = V (r −2, n) := 12r−2Xm=0(S0m)−2(n−1),

6Anatol N. KirillovwhereSjm =r4r sin (j + 1)(m + 1)πr.After this we proved an existence of a gl(n)-module V(0) such that dim V(0) =V (r −2, n) and computed it character:chV(0) =XλaλWλ,where aλ ∈Z+,l(λ) ≤n,l(λ′) ≤r −32and Wλ be an irreducible representationof gl(n) with the highest weight λ.§2. Verlinde character.In this section we study a decomposition of a product of some distinguish el-ements in the fusion algebra Fr := Fr(sl(2)).

Thus we start with a definition ofthese elements. Let us define a level r symmetric algebra Sr and a module of levelr harmonic polynomials Hr as followSr =r−32Xk=0r −12−kxk · Vk ∈Fr[x],Hr =r−32Xk=0xk · Vk ∈Fr[x].

(2.1)In sequel we will assume that r ≡1(mod 2). Our nearest aim is to find the coeffi-cients in the following decompositionsSb⊗nr=r−32Xk=0ak,n(x) · Vk,(2.2)Hb⊗nr=r−32Xk=0bk,n(x) · Vk.But at the beginning we find the values ak,n(1) and bk,n(1).Theorem 2.1.

We havea0,n(1) = 12r−2Xm=0 r4r sin (m + 1)πr!2−2n,(2.3)b0,n(1) = 2rr−2Xm=0sin2 (m + 1)πr2 cos (m + 1)πr2−2n.

Fusion algebra and Verlinde’s formula7Sketch of a proof. We start with a solution of a local problem, namely, we want tofind a connection matrix M(x) (resp.

N(x)) such thatSb⊗(n+1)r= M(x) · Sb⊗nr,(2.4)Hb⊗(n+1)r= N(x) · Hb⊗nr.Proposition 2.2.The connection matrices M(x) = (mij(x)) and N(x) =(ni,j(x)) have the following matrix elements (1 ≤i, j ≤r−12 )mi,j(x) = xi−jmin(2i−2, r−2j)Xk=0r −12+ i −j −kxk,(2.5)if i ≥j and mij = mji,if i ≥j;nij(x) = xi−jmin(2i−2, r−2j)Xk=0xk,(2.6)if i ≥j and nij = nji,if i ≥j.The statement of Proposition 2.2 may be verified by direct computation using thedefinitions (2.1).Corollary 2.3. We havemij(1) = 12(2i −1)(r + 1 −2j),if 1 ≤i ≤j ≤r −12,and mij(1) = mji(1) if i ≥j.The next step of a proof of Theorem 2.1 is the following observation.Proposition 2.4.

Matrix M(1) admits a decompositionM(1) = T · C−1,where C = (cij), cij = 2δi,j −δi,j+1 −δi,j−1 (1 ≤i, j ≤r−12 ) be the Cartan matrixof tipe A r−12and T = (tij) be a lower triangle matrix with elementst11 = r + 12= det C,tij = r and t1j = −r + 12−j,if 2 ≤j ≤r −12,tij = 0 otherwise.Using Proposition 2.4 we may find the spectra of matrix M(1) and solve a recurrencerelation (2.4) for Sb⊗nr. As a result, we obtain the first formula of (2.3).Now we want to construct the representations V(k) and W(k), 1 ≤k ≤r−12 , ofthe Lie algebra gl(n) such thatdim V(k) = ak,n(1),dim W(k) = bk,n(1).For this aim we use the following observation

8Anatol N. KirillovProposition 2.5. If the matrices M(x) and N(x) are defined by the formulae(2.5) and (2.6), then they satisfy the commutation relationsM(x) · M(y) = M(y) · M(x),N(x) · N(y) = N(y) · N(x),Finally, let us define a vacuum vector |0 >= (1, 0, .

. ., 0)t ∈Rr−12and consider thefollowing vectorsM(x1) .

. .

M(xn) |0 >= (χ1,n(x), . .

. , χ r−12,n(x))t,N(x1) .

. .

N(xn) |0 >= (ϕ1,n(x), . .

. , ϕ r−12,n(x))t.It is clear from our construction thatχk,n(1) = ak,n(1) and ϕk,n(1) = bk,n(1),χk,n(x, .

. .

, x) = ak,n(x) and ϕk,n(x, . .

. , x) = bk,n(x).Theorem 2.6.

The symmetric functions χk,n(x) and ϕk,n(x),1 ≤k ≤r−12 ,may be expressed as the linear combinations of the Schur functions with positiveinteger coefficients, i.e.χk,n(x) =Xλaλsλ(x),where for all partitions λ we have aλ ∈Z+ and if aλ ̸= 0 then l(λ) ≤n andl(λ′) ≤r−32 .It seems an interesting task to define a natural action of the Lie group GL(n)on the space H0(Mn, L⊗(r−2)) such thatchar H0(Mn, L⊗(r−2)) = χ0,n(x). (2.7)Probably, such action may be extracted from [Kh].Acknowledgements.Most of this work was done during my visit to theIsaac Newton Institute for Mathematical Sciences.

I thank B.Feigin, P.Mathieu,J.Weitsman and E.Corrigan for very helpfull discussions.References. [Bo] R.Bott, Int.

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