INFINITE-DIMENSIONAL LIE SUPERALGEBRAS AND HOOK SCHUR FUNCTIONS 3 0 0 SHUN-JEN CHENG∗ AND NGAU LAM∗∗ 2 n a Abstract. Making use of a Howe duality involving the infinite-dimensional J Lie superalgebra gl∞|∞ and the finite-dimensional group GLl of [CW3] we 7 derive a character formula for a certain class of irreducible quasi-finite repre- 1 b sentations of gl in terms of hook Schur functions. We use the reduction ∞|∞ ] procedure of gl to gˆl to derive a character formula for a certain class T b∞|∞ n|n R of level 1 highbest weight irreducible representations of gˆln|n, the affine Lie su- . peralgebra associated to the finite-dimensional Lie superalgebra gln|n. These th modules turn out to form the complete set of integrablegˆln|n-modules of level a 1. We also show that the characters of all integrable level 1 highest weight m irreducible gˆlm|n-modules may be written as a sum of products of hook Schur [ functions. 2 v 4 3 0 1. Introduction 6 0 Symmetric functions have been playing an important role in relating combina- 2 toricsandrepresentationtheoryofLiegroups/algebras. Interesting combinatorial 0 / identities involving symmetric functions, more often than not, have remarkable h t underlying representation-theoreticexplanation. Asanexampleconsider theclas- a m sical Cauchy identity : 1 v (1.1) = s (x ,x ,···)s (y ,y ,···), λ 1 2 λ 1 2 i (1−x y ) X Yi,j i j Xλ r a where x1,x2,··· and y1,y2,··· are indeterminates, and sλ(x1,x2,···) stands for theSchurfunctionassociatedtothepartitionλ. Herethesummationofλaboveis over all partitions. Now the underlying representation-theoretic interpretation of (1.1) is of course the so-called (GL,GL) Howe duality [H1] [H2]. Namely, let Cm andCn bethem-andn-dimensionalcomplexvectorspaces, respectively. Wehave an action of the respective general linear groups GL and GL on Cm and Cn. m n This induces a joint action of GL ×GL on Cm⊗Cn, which in turn induces an m n action on the symmetric tensor S(Cm⊗Cn). As partitions of appropriate length may be regarded as highest weights of irreducible representations of a general ∗Partially supported by NSC-grant 91-2115-M-002-007of the R.O.C. ∗∗Partially supported by NSC-grant 90-2115-M-006-015of the R.O.C. 1 2 SHUN-JENCHENG AND NGAULAM linear group, (1.1) simply gives an identity of characters of the decomposition of S(Cm ⊗Cn) with respect to this joint action. It was observed in [BR] that a generalization of Schur functions, the so-called the hook Schur functions (see (3.1) for definition), play a similar role in the representation theory of a certain class of finite-dimensional irreducible mod- ules over the general linear Lie superalgebra. To be more precise, consider the general linear Lie superalgebra gl acting on the complex superspace Cm|n of m|n (super)dimension (m|n). We may consider its induced action on the k-th ten- sor power Tk(Cm|n) = k(Cm|n). It turns out [BR] that the tensor algebra T(Cm|n) = ∞k=0Tk(CmN|n) is completely reducible as a glm|n-module and the characters ofLthe irreducible representations appearing in this decomposition are given by hook Schur functions associated to partitions lying in a certain hook whose shape is determined by the integers m and n. Now, as in the classical case, one may consider the joint action of two general linear Lie superalgebras gl ×gl on the symmetric tensor S(Cm|n ⊗Cp|q). This action again is com- m|n p|q pletely reducible and its decomposition with respect to the joint action, in a sim- ilar fashion, gives rise to a combinatorial identity involving hook Schur functions [CW1]. So here we have an interplay between combinatorics and representation theory of finite-dimensional Lie superalgebras as well. For another interplay in- volving Schur Q-functions and the queer Lie superalgebra see [CW2]. For further articles related to Howe duality in the Lie superalgebra settings we refer to [S1], [S2], [N] and [OP]. The purpose of the present paper is to demonstrate that symmetric functions may play a similarly prominent role relating combinatorics and representation theory of infinite-dimensional Lie superalgebras as well. It was shown in [CW3] that on the infinite-dimensional Fock space generated by n pairs of free bosons and n pairs of free fermions we have a natural commuting action of the finite- dimensional group GL and the infinite-dimensional Lie superalgebra gl of n ∞|∞ central charge n. It can be shown [CW3] that the pair (GL ,gl ) forms a n ∞|∞ b dual pair in the sense of Howe. The irreducible representations of GL appearing n b in this decomposition ranges over all rational representations, so that they are parameterizedbygeneralizedpartitionsoflengthnotexceedingn. Theirreducible representations of gl appearing in the same decomposition are certain quasi- ∞|∞ finitehighest weight irreducible representations. Inparticularthisrelates rational b representations of the finite-dimensional group GL and a certain class of quasi- n finite representations of the infinite-dimensional Lie superalgebra gl . ∞|∞ A natural question that arises is the computation of the character of these b quasi-finite highest weight irreducible representations of gl . This is solved in ∞|∞ the present paper by combining the Howe duality of [CW3] and a combinatorial b identity involving hook Schur functions (Proposition 3.1). It turns out that the charactersoftheserepresentations canbewrittenasaninfinitesumofproductsof INFINITE-DIMENSIONAL LIE SUPERALGEBRAS AND HOOK SCHUR FUNCTIONS 3 two hook Schur functions. Each coefficient of these products can be determined by decomposing a certain tensor product of two finite-dimensional irreducible representations of GL . The same method applied to the classical Lie algebra n ˆ gl (using Lemma 3.1 now together with the dual Cauchy identity instead of ∞ ˆ Proposition3.1)givesrisetoacharacterformulaforgl involving Schurfunctions ∞ instead of hook Schur functions. This formula has been discovered earlier by Kac andRadul[KR2]andevenearlier inthesimplest casebyAwata, Fukuma, Matsuo and Odake [AFMO2]. However, our approach in these classical cases appears to be simpler. It is indeed remarkable that the same character identity obtained in [KR2] with Schur functions replaced by hook Schur functions associated to the same partitions gives rise to our character identity for gl . That is, the ∞|∞ coefficients remain unchanged! b Making use of the same combinatorial identity we then proceed to compute the corresponding q-character formula for this class of highest weight irreducible representations of gl . We remark that when computing just the q-characters ∞|∞ we can obtain a simpler formula, which involves a sum of just hook Schur func- b tions, instead of a product of hook Schur functions. We note that a q-character formula in the case when the central charge is 1 has been obtained earlier by Kac and van de Leur [KL]. Our formula in this case looks rather different from theirs, thus giving rise to another combinatorial identity. It would be interesting to find a purely combinatorial proof of this identity. ˆ We use the reduction procedure from gl to the gl [KL] in the level 1 ∞|∞ n|n case to obtain a character formula for certain highest weight irreducible repre- b ˆ sentations of the affine Lie superalgebra gl at level 1. However, the Borel n|n subalgebra coming from the reduction procedure is different from the standard Borel subalgebra, and hence the corresponding highest weights in general are different. However, we show that by a sequence of odd reflections [PS] our high- est weights may be transformed into highest weights corresponding to integrable ˆ gl -modules (in the sense of [KW]) so that we obtain a character formula for all n|n ˆ integrable highest weight gl -modules of level 1. In [KW] a character formula n|n ˆ has been obtained for level 1 integrable highest weight irreducible gl -modules. m|n Our formula looks rather different. We also show that by applying our method together with [KW] the charac- ˆ ters of all level 1 integrable representations of gl , m ≥ 2, may be written m|n in terms of hook Schur functions as well. This seems to indicate the relevance of these generalized symmetric functions in the representation theory of affine superalgebras. As we have obtained a character formula for certain representations of gl ∞|∞ at arbitrary positive integral level, it is our hope that our formula may provide b 4 SHUN-JENCHENG AND NGAULAM ˆ some direction in finding a character formula for integrable gl -, or maybe even n|n ˆ gl -modules, at higher positive integral levels. m|n The paper is organized as follows. In Section 2 we collect the definitions and notation to be used throughout. In Section 3 we first prove the combinatorial identity mentioned above and then use it to write the characters of certain gl - ∞|∞ modules in terms of hook Schur functions. In Section 4 we calculate a q-character b formula for these modules, while in Section 5 we calculate the characters of the ˆ associated affine gl -modules. In Section 6 we compute the tensor product de- n|n composition of two gl -modules. It turns out that even though such a decom- ∞|∞ position involves an infinite number of irreducible components, each irreducible b component appears with a finite multiplicity. This multiplicity can be computed via the usual Littlewood-Richardson rule. 2. Preliminaries Let Cm|n = Cm|0 ⊕C0|n denote the m|n-dimensional superspace. Let gl be m|n the Lie superalgebra of general linear transformations on the superspace Cm|n. Choosing a basis {e ,··· ,e } forthe even subspace Cm|0 anda basis {f ,··· ,f } 1 m 1 n for the odd subspace C0|n, we may regard gl as (m+n)×(m+n) matrices of m|n the form E a (2.1) , (cid:18)b e(cid:19) where the complex matrices E, a, b and e are respectively m×m, m×n, n×m and n × n. Let X denote the corresponding elementary matrix with 1 in the ij i-th row and j-th column and zero elsewehre, where X = E,b,a,e. Then h = m CE + n Ce is a Cartan subalgebra of gl . i=1 ii j=1 jj m|n PIt is clear tPhat any ordering of the basis {e ,··· ,e ,f ,··· ,f } that preserves 1 m 1 n the order among the even and odd basis elements themselves gives rise to a Borel subalgebra of gl containing h. In particular the ordering e < ··· < e < f < m|n 1 m 1 ··· < f gives rise to the standard Borel subalgebra. In the case when m = n the n ordering f < e < f < e < ··· < f < e gives rise to a Borel subalgebra that 1 1 2 2 n n we will refer to as non-standard from now on. Fixing the standard Borel subalgebra, we let Vλ denote the finite-dimensional m|n highest weight irreducible module with highest weight λ. Let ǫ ∈ h∗ be defined by ǫ (E ) = δ and ǫ (e ) = 0. Furthermore let δ i i jj ij i jj j be defined by δ (E ) = 0 and δ (e ) = δ . Then ǫ and δ are the fundamental j ii j ii ij i j weights of gl . m|n Let C[t,t−1] be the ring of Laurent polynomials in the indeterminate t. Let gˆl ≡ gl ⊗C[t,t−1]+CC+Cdbethe affine Lie superalgebra associated to the m|n m|n Lie superalgebra gl . Writing A(k) for A⊗tk, A ∈ gl , the Lie (super)bracket m|n m|n INFINITE-DIMENSIONAL LIE SUPERALGEBRAS AND HOOK SCHUR FUNCTIONS 5 is given by [A(k),B(l)] = [A,B](k +l)+δ kStr(AB)C, k+l,0 [d,A(k)] = kA(k), A,B ∈ gl , k,l ∈ Z, m|n Here C is a central element, d is the scaling element and Str denote the super trace operator of a matrix, which for a matrix of the form (2.1) takes the form Tr(E)−Tr(e). A Cartan subalgebra of gˆl is given by hˆ = h + CC + Cd. We may extend m|n respectively ǫ and δ to elements ǫ˜ and δ˜ in hˆ∗ in a trivial way. Furthermore we i j i j define Λ˜ ∈ hˆ∗ and δ˜∈ hˆ∗ by Λ˜ (h) = Λ˜ (d) = 0, Λ˜ (C) = 1 and δ˜(h) = δ˜(C) = 0, 0 0 0 0 ˜ δ(d) = 1, respectively. Let B ⊆ gl be a Borel subalgebra containing h. Then B + CC + Cd + m|n gl ⊗tC[t] is a Borel subalgebra of gˆl . We define highest weight irreducible m|n m|n ˆ modules of gl in the usual way. It is clear that any highest weight irreducible m|n gˆl -module is completely determined by an element Λ ∈ hˆ∗. We will denote m|n ˆ this module by L(gl ,Λ). m|n Consider now the infinite-dimensional complex superspace C∞|∞ with even ba- sis elements labelled by integers and odd basis elements labelled half-integers. Arranging the basis elements in strictly increasing order any linear transforma- tion may be written as an infinite-sized square matrix with coefficients in C. This associative algebra is naturally Z -graded, so that it is an associative superalge- 2 bra, which we denote by M˜ . Let ∞|∞ 1 M := {A = (a ) ∈ M˜ ,i,j ∈ Z| a = 0 for |j −i| >> 0}. ∞|∞ ij ∞|∞ ij 2 That is, M consists of those matrices in M˜ with finitely many non-zero ∞|∞ ∞|∞ diagonals. We denote the corresponding Lie superalgebra by gl . Furthermore ∞|∞ let us denote by e , i,j ∈ 1Z the elementary matrices with 1 at the i-th row and ij 2 j-th column and 0 elsewhere. Then the subalgebra generated by {e |i,j ∈ 1Z} ij 2 is a dense subalgebra inside gl . ∞|∞ The Lie superalgebra gl has a central extension (by an even central element ∞|∞ C), denoted from now on by gl , corresponding to the following two-cocycle ∞|∞ b α(A,B) = Str([J,A]B), A,B ∈ gl , ∞|∞ where J denotes the matrix e , and for a matrix D = (d ) ∈ gl , r≤0 rr ij ∞|∞ Str(D) stands for the supertraPce of the matrix D and which here is given by (−1)2rd . We note that the expression α(A,B) is well-defined for A,B ∈ r∈1Z rr 2 Pgl . ∞|∞ 6 SHUN-JENCHENG AND NGAULAM The Lie superalgebra gˆl has a natural 1Z-gradation by setting degE = ∞|∞ 2 ij j −i, for i,j ∈ 1Z. Thus we have the triangular decomposition 2 ˆ ˆ ˆ ˆ gl = (gl ) ⊕(gl ) ⊕(gl ) , ∞|∞ ∞|∞ − ∞|∞ 0 ∞|∞ + where the subscripts +, 0 and − respectively denote the positive, zero-th and negative graded components. Thus we have a notion of a highest weight Verma module, which contains a unique irreducible quotient, which is determined by an element Λ ∈ (gl )∗. We will denote this module by L(gl ,Λ). Let ω , ∞|∞ 0 ∞|∞ s s ∈ 1Z, denote the fundamental weights of gl . That is, ω (e ) = 0, r ∈ 1Z, 2 b ∞|∞ sb rr 2 and ω (C) = 0. Furthermore let Λ ∈ (gl )∗ with Λ (e ) = 0 and Λ (C) = 1. s 0 ∞|∞b0 0 rr 0 Note that by declaring the highest weight vectors to be of degree zero, the b module L(gl ,Λ) is naturally 1Z-graded, i.e. ∞|∞ 2 b L(gl ,Λ) = ⊕ L(gl ,Λ) . ∞|∞ r∈12Z+ ∞|∞ r The module L(gl ,Λ) ibs said to be quasi-finitbe [KR1] if dimL(gl ,Λ) < ∞, ∞|∞ ∞|∞ r for all r ∈ 1Z . 2 + b b 3. A Character formula for gˆl -modules ∞|∞ First we recall the notion of the hook Schur function of Berele-Regev [BR]. Let x = {x ,x ,···} be a countable set of variables. To a partition λ of non- 1 2 negative integers we may associate the Schur function s (x ,x ,···). We will λ 1 2 write s (x) for s (x ,x ,···). For a partition µ ⊂ λ we let s (x) denote the λ λ 1 2 λ/µ corresponding skew Schur function. Denoting by µ′ the conjugate partition of a partition µ the hook Schur function corresponding to a partition λ is defined by (3.1) HS (x;y) := s (x)s (y), λ µ λ′/µ′ X µ⊂λ where as usual y = {y ,y ,···}. 1 2 Let λ be a partition and µ ⊆ λ. We fill the boxes in µ with entries from the linearly ordered set {x < x < ···} so that the resulting tableau is semi- 1 2 standard. Recall that this means that the rows are non-decreasing, while the columns are strictly increasing. Next we fill the skew partition λ/µ with entries from the linearly ordered set {y < y < ···} so that it is conjugate semi- 1 2 standard, which means that the rows are strictly increasing, while its columns are non-decreasing. We will refer to such a tableau as an (∞|∞)-semi-standard tableau (cf. [BR]). To each such tableau T we may associate a polynomial (xy)T, which is obtained by taking the products of all the entries in T. Then we have [BR] (3.2) HS (x;y) = (xy)T, λ X T INFINITE-DIMENSIONAL LIE SUPERALGEBRAS AND HOOK SCHUR FUNCTIONS 7 where the summation is over all (∞|∞)-semi-standard tableaux of shape λ. We have the following combinatorial identity involving hook Schur functions that is crucial in the sequel. Proposition 3.1. Let x = {x ,x ,···}, y = {y ,y ,···} be two infinite count- 1 2 1 2 able sets of variables and z = {z ,z ,··· ,z } be m variables. Then 1 2 m (3.3) (1−x z )−1(1+y z ) = HS (x;y)s (z), i k j k λ λ Y X i,j,k λ where 1 ≤ i,j < ∞, 1 ≤ k ≤ m and λ is summed over all partitions λ with length not exceeding m. Proof. Consider the classical Cauchy identity (3.4) (1−x z )−1(1−y z )−1 = s (x,y)s (z), i k j k λ λ Yi,j Xλ where λ is summed over all partitions of length not exceeding m. Recall that for any partition λ one has (cf. [M] (I.5.9)) (3.5) s (x,y) = s (x)s (y). λ µ λ/µ X µ⊂λ Let ω denote the involution of the ring of symmetric functions, which sends the elementary symmetric functions to the complete symmetric functions, so that we have ω(s (x)) = s (x). Now applying ω to the set of variables y in (3.4) we λ λ′ obtain together with (3.5) (1−x z )−1(1+y z ) = ( s (x)s (y))s (z) i k j k µ λ′/µ′ λ Y X X i,j,k λ µ⊂λ = HS (x;y)s (z), λ λ X λ (cid:3) as required. We note that Proposition 3.1 in the case when the sets of variables are all finite sets follows from the Howe duality ([H1], [H2]) involving a general linear Lie superalgebra and a general linear Lie algebra described in [CW1]. Since we will need this result in the case when both algebras involved are Lie algebras later on we will recall it here. Proposition 3.2. [H2] The Lie algebras gl and gl with their natural actions d m on S(Cd ⊗Cm) form a dual pair. With respect to their joint action we have the following decomposition. S(Cd ⊗Cm) ∼= Vλ ⊗Vλ, d m X λ where the summation is over all partitions with length not exceeding min(l,m). 8 SHUN-JENCHENG AND NGAULAM ˆ Below we will recall the gl ×gl duality of [CW3]. Consider l pairs of free ∞|∞ l fermions ψ±,i(z) and l pairs of free bosons γ±,i(z) with i = 1,··· ,l. That is we have ψ+,i(z) = ψ+,iz−n−1, ψ−,i(z) = ψ−,iz−n, n n Xn∈Z Xn∈Z γ+,i(z) = γ+,iz−r−1/2, γ−,i(z) = γ−,iz−r−1/2 r r r∈X1+Z r∈X1+Z 2 2 with non-trivial commutation relations [ψ+,i,ψ−,j] = δ δ and [γ+,i,γ−,j] = m n ij m+n,0 r s δ δ . ij r+s,0 Let F denote the corresponding Fock space generated by the vaccum vector |0 >. That is ψ+,i|0 >= ψ−,i|0 >= γ±,i|0 >= 0, for n ≥ 0, m > 0 and r > 0. n m r These operators are called annihilation operators. Explicitly we have an action of gˆl of central charge l on F given by (i,j ∈ Z ∞|∞ and r,s ∈ 1 +Z) 2 l e = : ψ+,pψ−,p :, ij −i j Xp=1 l e = − : γ+,pγ−,p :, rs −r s Xp=1 l e = : ψ+,pγ−,p :, is −i s Xp=1 l e = − : γ+,pψ−,p : . rj −r j Xp=1 An action of gl on F is given by the formula l E = : ψ+,iψ−,j : − : γ+,iγ−,j : . ij −n n −r r Xn∈Z r∈X1/2+Z Here and further :: denotes the normal ordering of operators. That is, if A and B are two operators, then : AB := AB, if B is an annihilation operator, while : AB := (−1)p(A)p(B)BA, otherwise. As usual, p(X) denotes the parity of the operator X. INFINITE-DIMENSIONAL LIE SUPERALGEBRAS AND HOOK SCHUR FUNCTIONS 9 Before stating the duality of [CW3] we need some more notation. For j ∈ Z + we define the matrices X−j as follows: ψ−,l ψ−,l−1 ··· ψ−,1 0 0 0 ψ−,l ψ−,l−1 ··· ψ−,1 X0 = 0. 0. 0. , . . .  . . ··· .    ψ−,l ψ−,l−1 ··· ψ−,1  0 0 0  γ−,l γ−,l−1 ··· γ−,1 −1 −1 −1 2 2 2 ψ−,l ψ−,l−1 ··· ψ−,1 X−1 = −1 −1 −1 , . . . . . .  . . ··· .    ψ−,l ψ−,l−1 ··· ψ−,1  −1 −1 −1  γ−,l γ−,l−1 ··· γ−,1 −1 −1 −1 2 2 2 γ−,l γ−,l−1 ··· γ−,1 −3 −3 −3 2 2 2 X−2 = ψ−,l ψ−,l−1 ··· ψ−,1,  −2 −2 −2   .. .. ..   . . ··· .    ψ−,l ψ−,l−1 ··· ψ−,1  −2 −2 −2  . . . . . . γ−,l γ−,l−1 ··· γ−,1 −1 −1 −1 2 2 2  γ−,l γ−,l−1 ··· γ−,1  −3 −3 −3 2 2 2 X−k ≡ X−l =  ... ... ··· ... , k ≥ l.   γ−,l γ−,l−1 ··· γ−,1   −l+1 −l+1 −l+1 2 2 2 γ−,l γ−,l−1 ··· γ−,1   −l−21 −l−21 −l−21 The matrices Xj, for j ∈ N, are defined similarly. Namely, Xj is obtained from X−j by replacing ψ−,k by ψ+,l−k+1 and γ−,k by γ+,l−k+1. i i r r For 0 ≤ r ≤ l, we let Xi(i ≥ 0) denote the first r ×r minor of the matrix Xi r and let Xi (i < 0) denote the first r×r minor of the matrix Xi. −r Consider a generalized partition λ = (λ ,λ ,··· ,λ ) of length not exceeding l 1 2 p with λ ≥ λ ≥ ··· ≥ λ > λ = 0 = ··· = λ > λ ≥ ··· ≥ λ . 1 2 i i+1 j−1 j l Nowtheirreducible rationalrepresentations ofGL areparameterizedbygener- l alized partitions, hence these may be interpreted as highest weights of irreducible representations of GL. We denote the corresponding finite-dimensional highest l weight irreducible GL - (or gl-) module by Vλ. Let λ′ be the length of the j-th l l l j column of λ. We use the convention that the first column of λ is the first column 10 SHUN-JENCHENG AND NGAULAM of the partition λ ≥ λ ≥ ··· ≥ λ . The column to the right is the second 1 2 i column of λ, while the column to the left of it is the zeroth column and the column to the left of the zeroth column is the −1-st column. We also use the convention that a non-positive column has non-positive length. As an example consider λ = (5,3,2,1,−1,−2) with l(λ) = 6. We have λ′ = −1, λ′ = −2, −1 0 λ′ = 4 etc. (see (3.6)). 1 (3.6) ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ For Λ ∈ (gl )∗, we set Λ = Λ(e ), for s ∈ 1Z. Given a generalized partition ∞|∞ 0 s ss 2 λ with l(λ) ≤ l, we define Λ(λ) ∈ (gl )∗ by: b ∞|∞ 0 b Λ(λ) = hλ′ −ii, i ∈ N, i i Λ(λ) = −h−λ′ +ji, j ∈ −Z , j j + 1 Λ(λ) = hλ −(r −1/2)i, r ∈ +Z , r r+1/2 + 2 1 Λ(λ) = −h−λ +(s−1/2)i, s ∈ − −Z , s p+(s+1/2) + 2 Λ(λ)(C) = l. Hereforanintegerk theexpression < k >≡ k, ifk > 0, and< k >≡ 0, otherwise. We have the following theorem. Theorem 3.1. [CW3] The Lie superalgebra gˆl and gl form a dual pair on ∞|∞ l F in the sense of Howe. Furthermore we have the following (multiplicity-free) decomposition of F with respect to their joint action F ∼= L(gˆl ,Λ(λ))⊗Vλ, ∞|∞ l X λ where the summation is over all generalized partitions of length not exceeding l. Furthermore, the joint highest weight vector of the λ-component is given by detXλl+1 ···detX−1 ·detX0 ·detX1 ·detX2 ···detXλ1 |0i. λ′λl+1 λ′−1 λ′0 λ′1 λ′2 λ′λ1