REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 4, Pages 46–63 (February 7, 2000) S 1088-4165(00)00065-0
b SYMMETRIC POLYNOMIALS AND Uq(sl2)
NAIHUAN JING
Abstract. We study the explicit formula of Lusztig’s integral forms of the b level one quantum affine algebra Uq(sl2) in the endomorphism ring of sym- metric functions in infinitely many variables tensored with the group algebra of Z. Schur functions are realized as certain orthonormal basis vectors in the vertex representation associated to the standard Heisenberg algebra. In this picture the Littlewood-Richardson rule is expressed by integral formulas, and −1 b is used to define the action of Lusztig’s Z[q, q ]-form of Uq(sl2)onSchur polynomials. As a result the Z[q, q−1]-lattice of Schur functions tensored with the group algebra contains Lusztig’s integral lattice.
1. Introduction The relation between vertex representations and symmetric functions is one of the interesting aspects of affine Kac-Moody algebras and the quantum affine alge- bras. In the late 1970’s to the early 1980’s the Kyoto school [DJKM] found that the polynomial solutions of KP hierarchies are obtained by Schur polynomials. This breakthrough was achieved in formulating the KP and KdV hierarchies in terms of affine Lie algebras. On the other hand, I. Frenkel [F1] identified the two construc- tions of the affine Lie algebras via vertex operators, which put the boson-fermion correspondence in a rigorous formulation. I. Frenkel [F2] further showed that the boson-fermion correspondence can give the Frobenius formula of the irreducible characters for the symmetric group Sn (see also [J1]). Schur functions also played a key role in Lepowsky and Primc construction [LP] of certain bases for higher level representations of the affine Lie algebra slb(2). In [J1, J2] the vertex operator approach to classical symmetric functions was de- veloped to study Schur’s Q-functions and more generally Hall-Littlewood symmetric functions. These families of symmetric polynomials appear naturally as orthogonal bases in the vertex representation. The formulation of Hall-Littlewood polynomials in terms of the boson-fermion correspondence was found in [J4] afterwards. Since then the vertex operator approach to symmetric functions is used to study both old and new problems in symmetric functions (see [CT] and [Ga]). In 1988 Macdonald introduced more general (orthogonal) symmetric polynomi- als, which are a two-parameter deformation of the Schur polynomials (cf. [M]). At the same time the vertex representation of the quantum affine algebra was
Received by the editors February 17, 1999 and, in revised form, December 10, 1999. 2000 Mathematics Subject Classification. Primary 17B; Secondary 5E. Key words and phrases. Symmetric functions, vertex operators, quantum affine algebras, Littlewood-Richardson rule. Research supported in part by NSA grant MDA904-97-1-0062 and Mathematical Sciences Re- search Institute.
c 2000 American Mathematical Society 46 b SYMMETRIC POLYNOMIALS AND Uq (sl2)47 constructed in [FJ]. The success of the vertex operator interpretation of Hall- Littlewood polynomials suggests a possible formulation for the Macdonald polyno- mial in the quantum vertex representation. Recently J. Beck, I. Frenkel and the author [BFJ] have used the q-vertex op- erators to study the canonical bases for the level one irreducible modules for the b quantum affine algebra Uq(sl2). The zonal Macdonald polynomials are shown as some “canonical” bases of the basic representation sitting between Kashiwara and Lusztig’s canonical and dual canonical bases [L, K]. This essentially answered the question about the vertex realization of (zonal) Macdonald polynomials. The Mac- donald basis constructed in [BFJ] also satisfy the characteristic properties of bar invariance and orthogonality under the Kashiwara form. The transition matrix from the canonical basis to the Macdonald basis is triangular, integral and bar-invariant and was conjectured to be positive. Since the transition matrix from Macdonald polynomials to (modified) Schur polynomials is also triangular, a natural problem is to determine the action of the quantum affine algebra on the Schur polynomials. The main goal of this paper is to explicitly realize the quantum affine algebra b Uq(sl2) by Schur functions with the help of the Littlewood-Richardson rule. We first modify the vertex operator realization of Schur functions and express all products of Schur and dual Schur vertex operators in terms of the Schur basis using the symmetry of Clifford algebras, which is an analog of the linkage symmetry for the weights of the Lie algebra sl(n). We then use the idea [J3] of expressing Schur b functions inside the basic representation of Uq(sl2) to realize the action of divided b powers of Drinfeld generators of Uq(sl2). b Moreover, this enables us to extend the action to Lusztig’s integral form UA(sl2), −1 where A = Z[q, q ]. Let ΛF (x1,x2, ···) be the ring of symmetric functions in the xn (n ∈ N) over the ring F . We show that the lattice M mα iα/2 VA(Λi)= ΛA(x1,x2, ···) ⊗ e e ,i=0, 1 m∈Z
b is invariant under Lusztig’s A-form UA(sl2) of divided powers, thus it contains the b lattice UA(sl2)vΛi . From another direction in [CP, BCP] Beck, Chari and Pressley construct a PBW basis for the algebra UA, which partly generalizes Garland’s work [G]. In the forthcoming paper with Chari [CJ], we will combine the two directions to study, among other things, the level one representations of Lusztig’s integral form inside the basic representation. We also find a vertex operator approach to the Littlewood-Richardson rule. In particular, an integral formula for the Littlewood-Richardson rule is found, and we use this to give a combinatorial description of divided powers of the Cheval- ley generators. The special case of only Chevalley generators corresponds to the deletion and insertion procedure on Young tableaux in the fermionic construction b of Uq(sl2) used by Misra and Miwa in [MM] (see also [H]). Our explicit formulas of the divided powers of Chevalley generators suggest that there are corresponding formulas in the fermionic case. The fermionic picture together with our formulas will explain the meaning of the Littlewood-Richardson rule in the boson-fermion correspondence. The method in this paper can be generalized to quantum affine algebras of ADE types, and this will provide more information about Schur functions as crystal bases 48 NAIHUAN JING
[BCP]. Our formulas will also be helpful in understanding the positivity conjecture of [BFJ]. The paper is organized as follows. In Section 2 we redevelop the vertex oper- ator approach to Schur polynomials and express all mixed products of Schur and dual Schur vertex operators in terms of Schur functions, and we derive an integral formula for multiplication of Schur polynomials (Littlewood-Richardson rule). In Section 3 we first construct a Schur basis for the Frenkel-Jing vertex representation b of Uq(sl2). We then use the Littlewood-Richardson rule to give explicit formulas for the action of the divided powers of the Drinfeld generators in terms of the Schur basis. In the last section (Sec. 4) we show that the A-lattice of Schur functions tensored with the group algebra A[Zα] is a sublattice of a Lusztig’s A-lattice in- side the vertex representation, which provides a simple combinatorial model for the b homogeneous picture of the basic module for UA(sl2).
2. Schur functions and vertex operators
Let ΛF be the ring of symmetric functions in infinitely many variables x1,x2, ··· over the ring F . In this section we take F = Q, and later we will take F = Q(q) and Z[q, q−1]. A partition λ =(λ1,λ2, ··· ,λl)ofn, denoted λ ` n, is a special decomposition of n: n = λ1 + ···+ λl with λ1 ≥···≥λl ≥ 1. The number l is called the length of λ. We will identify (λ1, ··· ,λl)with(λ1, ··· ,λl, 0, ··· , 0) if we want to view λ in Zn when n ≥ l(λ). Sometime we prefer to use another notation for λ:(1m1 2m2 ···) where mi is the number of times that i appears among the parts of λ.Thesetof partitions will be denoted by P. There are several well-known bases in ΛF parameterized by partitions: the power sum symmetric functions ··· pλ = pλ1 pλl P n Q with pn = xi ( -basis); the monomial symmetric functions X X σ(λ ) ··· σ(λ) 1 ··· σ(λn ) mλ(x1, ,xn)= x = x1 xn , σ σ where σ runs through distinct permutations of λ as tuples; and the Schur functions sλ form a basis over Z. In terms of finitely many variables the Schur function is given by the Weyl character formula P sgn(σ)xσ(λ+δ) σ∈Sn (2.1) sλ(x1, ··· ,xn)= , Πi (2.2) (sλ,sµ)=δλ,µ. b SYMMETRIC POLYNOMIALS AND Uq (sl2)49 It can be shown [M] that under this inner product (2.3) (pλ,pµ)=zλδλ,µ, Q mi m1 m2 ··· where zλ = i≥1 i mi!forλ =(1 2 ). We recall the vertex operator approach to Schur functions [J1]. Let {bn|n =06 } ∪{c} be the set of generators of the Heisenberg algebra with defining relations (2.4) [bm,bn]=mδm,−nc, [c, bm]=0. The Heisenberg algebra has a canonical natural representation in the Q-space 0 V = Sym(b−n s), the symmetric algebra generated by the b−n, n ∈ N. The action is given by ∂v (2.5) b−n.v = b−nv, bn.v = n , ∂b−n (2.6) c.v = v. It is clear that 1 is the highest weight vector in V . Let us introduce two vertex operators (cf. t =0in[J2]): X∞ X∞ b− b (2.7) S(z)=exp( n zn)exp(− n z−n) n n X n=1 n=1 −n = Snz , n∈Z X∞ X∞ b− b (2.8) S∗(z)=exp(− n zn)exp( n zn) n n X n=1 n=1 ∗ n = Snz . n∈Z If we view z as a complex variable, then I I dz dz S = S(z)zn ,S∗ = S∗(z)z−n , n z n z H dz where the contour integral is normalized so that z =1. It follows that for n ≥ 0 ∗ (2.9) Sn.1=δn,0,S−n.1=δn,0. Lemma 2.1 ([J2]). The components of S(z) and S∗(z) satisfy the following com- mutation relations. ∗ ∗ ∗ ∗ SmSn + Sn+1Sm−1 =0,SmSn + Sn−1Sm+1 =0, ∗ ∗ SmSn + Sn+1Sm+1 = δm,n. The following result will be useful in our discussion. Proposition 2.1. Let δ =(n − 1,n − 2, ··· , 1, 0). The operator products δ ∗ ∗ ∗ δ S(z1)S(z2) ···S(zn)z and S (z1)S (z2) ···S (zn)z are skew-symmetric under the action of Sn. For any w ∈ Sn we have w(δ) l(w) δ S(zw(1))S(zw(2)) ···S(zw(n))z =(−1) S(z1)S(z2) ···S(zn)z , ∗ ∗ ∗ w(δ) l(w) ∗ ∗ ∗ δ S (zw(1))S (zw(2)) ···S (zw(n))z =(−1) S (z1)S (z2) ···S (zn)z . 50 NAIHUAN JING Proof. The equations follow by repeatedly using the commutation relations satisfied by the Schur vertex and the dual vertex operators (see Lemma 2.1). The space V has a natural Hermitian product given by ∗ (2.10) bn = b−n. ··· ` Under the inner product, the elements b−λ = b−λ1 b−λl (λ n) span an orthog- onal basis in V and (2.11) (b−λ,b−µ)=δλ,µzλ. Definition 2.2. The characteristic map ch from the vertex space V to the ring ΛQ of symmetric functions is the Q-linear map given by ··· −→ ··· b−λ = b−λ1 b−λl pλ = pλ1 pλl . It is clear that ch is an isomorphism of vector spaces. In particular, the vector S−n.1 corresponds to the homogeneous symmetric polynomial sn. For this reason we will simply write sn = S−n.1,n∈ Z+. The generating function of sn is given by X∞ X∞ b− (2.12) s zn = exp( n zn). n n n=0 n=1 The following result appeared in [J1, J4]. For completeness we include a modified proof. Theorem 2.3. The space V is isometrically isomorphic to ΛQ under the map ch. { ··· ` ∈ Z } { ··· ` The sets h−λ = sλ1 sλ2 sλl : λ n, n + and S−λ1 S−λ2 S−λl .1:λ ∈ Z } Q { ··· } n, n + are both -basis. Moreover the basis S−λ1 S−λ2 S−λl .1 is orthonor- mal and expressed explicitly by ··· (2.13) S−λ.1:=S−λ1 S−λ2 S−λl .1=det(sλi−i+j ), which corresponds to the Schur function sλ in ΛQ. Proof. To show the Sn-symmetry we consider the modified vertex operators asso- ciated with the root lattice Zα with (α|α)=1.LetV˜ = V ⊗ Q[Zα], where Q[Zα] is the group algebra generated by emα,m∈ Z over Q. Define X α ∂ −n−1/2 (2.14) S(z)=S(z)e z = Snz , n∈Z+1/2 ∗ X ∗ ∗ −α −∂ n−1/2 (2.15) S (z)=S (z)e z = Snz , n∈Z+1/2 where the operators eα and z∂ act on Q(q)[Zα] as follows: eαemα = e(m+1)α,z∂emα = zmemα, The components satisfy the Clifford algebra relations { } { ∗ ∗ } (2.16) Sm, Sn = Sm, Sn =0, { ∗ } (2.17) Sm, Sn = δm,n, where m, n ∈ Z +1/2. b SYMMETRIC POLYNOMIALS AND Uq (sl2)51 For an l-tuple λ =(λ1,λ2, ··· ,λl) satisfying λi = λj − j + i for some i and j we have ··· ··· −lα S−λ1 S−λl .1=S−λ1+1/2S−λ2−1/2 S−λl−l+1/2e = S−λ+δ−(m−1/2)1 =0 due to λ + δ =(i, j)(λ + δ) and (2.16–2.17). Here 1 denotes the integer vector (1, 1, ··· , 1), and similarly m1 =(m, m, ··· ,m). In general for any other l-tuple µ we have ··· ··· (2.18) S−µ1 S−µl .1=sgn(µ)S−λ1 S−λl .1, where λ is the partition related to µ: λ = σ(µ + δ) − δ,andthesignsgn(µ)= (−1)l(σ). See [J1] for details. We shall use vertex operators to give another proof that the Schur functions form a Z-linear basis. Definition 2.4. For an l-tuple µ we define the Schur function sµ to be the sym- ··· metric function corresponding to S−µ1 S−µl .1 under the characteristic map. We will simply write ··· sµ = S−µ.1=S−µ1 S−µl .1 . We say that two integral l-tuples µ and λ are related if there is a permutation σ such that µ + δ = σ(λ + δ). If there exists an odd permutation σ such that µ + δ = σ(µ + δ), then we say that µ is degenerate. An integral tuple µ is said to be non-increasing if µ1 ≥ µ2 ≥ ··· ≥ µl. In particular, a non-increasing positive integral tuple is a partition. If there are no two parts µi,µj (i Theorem 2.7. The vectors {S∗ ···S∗ .1 | λ ` n, n ∈ Z } are also orthonormal, λ1 λl + and we have ∗ ∗ ∗ |λ| |λ| (2.20) S S ···S .1=(−1) s 0 =(−1) det(s 0 − ), λ1 λ2 λl λ λi i+j where |λ| = λ1 + λ2 + ···+ λl. Moreover, we have ∗ ∗ |ν| S− ···S− S ···S .1=(−1) sgn(λ, µ)s 0 , µ1 µl ν1 νk π(µ,π(ν) ) ∗ ∗ |µ| ··· − ··· − − 0 0 Sµ1 Sµk S ν1 S νl .1=( 1) sgn(µ, ν)sπ(µ,π(ν) ) , where (µ, ν) is the juxtaposition of µ and ν, the associated partition π(µ, ν) is 0 0 obtained by π(µ, ν)=σ((µ, ν)+δ) − δ for some σ ∈ Sl+k,andπ(µ, π(ν) ) denotes the dual partition of π(µ, π(ν)0). As a consequence of Theorem 2.7 it follows that − |λ| ∗ S−λ.1=( 1) Sλ.1. Example 2.8. ∗ ∗ S−1S2 S2 .1=S−1S−2S−2.1=0, ∗ ∗ ∗ ∗ − S−1S2 S1 S1 S1 .1= S−1S−4S−1.1=S−3S−2S−1.1=s(3,2,1), − where (1, 4, 1) + δ =(3, 5, 1) ∼ (5, 3, 1) = (3, 2, 1) + δ. To close this section we derive the Littlewood-Richardson rule in our picture, b which will be used later to realize the action of the quantum affine algebra Uq(sl2) on ΛQ[q,q−1]. In particular, we give a new proof of the integrality of Schur functions using vertex operators. PropositionP 2.2. Let λ and µ be two partitions of lengths m and n respectively. Then sλsµ = CMNs(λ+M,µ−N),whereCMN is the number of integral matrices (kij ) such that (k11 + k12 + ···+ k1n, ··· ,km1 + ...+ kmn)=M, (k11 + k21 + ...+ km1, ··· ,k1n + k2n + ···+ kmn)=N, where kij ≥ 0. In particular, the Schur functions form a Z-lattice in ΛQ. Proof. It follows from the definition that Z dz dw s s = S(z ) ···S(z ).1S(w ) ···S(w ).1z−λw−µ . λ µ 1 m 1 n z w Observe that S(z1) ···S(zm).1S(w1) ···S(wn).1 Y w = (1 − j )−1S(z ) ···S(z )S(w ) ···S(w ).1 . z 1 m 1 n i,j i As an infinite series in |wj| < |zi| we have Y w Y w w (1 − j )−1 = (1 + j +( j )2 + ···) zi zi zi i,j Yi,j X kij −kij = ( wj zi ) i,j k Y ij − − k·1 ··· k·n k1· ··· km· = w1 wn z1 zm , k=(kij ) with k·j = k1j + ···+ kmj , ki· = ki1 + ···+ kin,kij ≥ 0. b SYMMETRIC POLYNOMIALS AND Uq (sl2)53 Plugging the expansion into the integral and invoking Theorem (2.3), we prove the proposition. Remark 2.9. The number CMN is equal to the index of SλπSµ in Sn [JK]. We can also write Y −1 sλsµ = (1 − Rij ) s(λ,µ), i,j where Rij is the raising operator defined on the parts of the Schur functions Rij s(··· ,λi,··· ,λj ,··· ) = s(··· ,λi+1,··· ,λj −1,··· ). We remark that the characteristic map ch is actually defined over Z if we let ch(S−λ.1) = sλ. Then the space VZ is isometrically isomorphic to ΛZ. b 3. Vertex representations of UA(sl2) Let A be the ring Z[q, q−1] of Laurent polynomials in q over Z. ∈ Z qn−q−n − For n + we define [n]= q−q−1 .Theq-factorial [n]! is equal to [n][n 1] ··· n [n]! [2][1] and then the q-Gaussian numbers are defined naturally by [ m ]= [m]![n−m]! for n ≥ m ≥ 0. By convention [0] = [1] = 1. For an element a in an algebra over Q (n) an (q)weusea to denote the divided power [n]! . b The quantum affine algebra Uq(sl2) is the associative algebra over Q(q) generated 1 d by the Chevalley generators ei,fi,Ki (i =0, 1) and q subject to the following defining relations: −1 −1 d −d −d d KiKi = Ki Ki =1,qq = q q =1, d 1 1 d KiKj = KjKi,qKi = Ki q , −1 aij −1 −aij KiejKi = q ej,KifjKi = q fj, d −d δi,0 d −d −δi,0 q eiq = q ei,qfiq = q fi, K − K−1 [e ,f ]=δ i i , i j ij q − q−1 − 1Xaij − − − r (r) (1 aij r) 6 ( 1) ei ejei =0 ifi = j, r=0 − 1Xaij − − − r (r) (1 aij r) 6 ( 1) fi fjfi =0 ifi = j, r=0 2 −2 c where (aij )= −22 is the extended Cartan matrix [Ka]. The element K0K1 = q b is a central element of Uq(sl2). b A b (n) (n) d Let UA(sl2)bethe -subalgebra [L] of Uq(sl2) generated by ei , fi , Ki , q b b for i =0, 1andn ∈ N.ThenUA(sl2) ⊗A Q(q) ' Uq(sl2). For m ∈ Z,r ∈ N we define Yr − −1 − − K ; m K qm+1 s − K qs 1 m i = i i , r q − q−1 s=1 b which belong to the Cartan subalgebra of UA(sl2). 54 NAIHUAN JING b The module V (Λi)(i =0, 1) is the simple highest weight module of Uq(sl2) generated by the highest weight vector vΛi such that δij d eivΛi =0,KjvΛi = q vΛi ,qvΛi = vΛi ,i=0, 1. b b An A-subspace W of an Uq(sl2)-module V is called an A-lattice for UA(sl2)ifW b is invariant under UA(sl2)andW ⊗A Q(q) ' V . The (level one) modules V (Λi) had been realized by vertex operators in [FJ] for b b Uq(sl2). Since we will construct the vertex representation of UA(sl2) we need to modify the original construction. b Let Uq(h) be the Heisenberg algebra with the generators {an|n =06 }∪{C} and the defining relations m (3.1) [a ,a ]=δ − C, [C, a ]=0. m n m, n 1+q2|m| m The level one irreducible representation V (Λi) is realized on the vertex repre- sentation space 0 iα/2 V (Λi)=SymQ(q)(a−n s) ⊗ Q(q)[Zα]e ,i=0, 1, 0 where SymQ(q)(a−n s) denotes the Q(q)-symmetric algebra generated by the Heisen- iα/2 berg generators a−n,n ∈ N. The element e (the highest weight vector) is for- mally adjoined to Q(q)[Zα]=Q(q)hemα|m ∈ Zi. We define two kinds of operators on the vector space Q(q)[Zα]eiα/2: (3.2) enα.emαeiα/2 = e(m+n)αeiα/2,n∈ Z, (3.3) ∂.emαeiα/2 =(2m + i)emαeiα/2. In particular, Q(q)[Zα]eiα/2 is a Q(q)[Zα]-module. b 0 The Heisenberg algebra Uq(h)actsontheQ(q)-space SymQ(q)(a−n s)via n ∂v − − (3.4) a n.v = a nv, an.v = 2n , 1+q ∂a−n (3.5) C.v = v, b where v ∈ Uq(h). b We define the vertex operators associated to Uq(sl2)by X∞ 2n −n X∞ 2n −n + (1 + q )q n (1 + q )q −n α ∂ (3.6) X (z)=exp( a− z )exp(− a z )e z n n n n X n=1 n=1 + −n−1 = Xn z , n∈Z X∞ 2n X∞ 2n − 1+q n 1+q −n −α −∂ (3.7) X (z)=exp(− a− z )exp( a z )e z n n n n X n=1 n=1 − −n−1 = Xn z . n∈Z b SYMMETRIC POLYNOMIALS AND Uq (sl2)55 The normal order of vertex operator products is defined by rearranging the exponential factors. For example, : X+(z)X+(w): ∞ ∞ X 2n −n X 2n −n (1 + q )q n n (1 + q )q −n −n = exp( a− (z + w ))exp(− a (z + w )) n n n n n=1 n=1 × e2αz∂w∂ . The components of the Drinfeld generators satisfy some quadratic relations. Lemma 3.1 ([FJ]). The components of X(z) satisfy the following commutation relations − 2 2 − XmXn q Xn Xm = q Xm−1Xn+1 Xn+1Xm−1, 1 X+X− − X−X+ = (ψ − φ ) , m n n m q − q−1 m+n m+n where the polynomials ψn and φ−n in the an are defined by X X (q3n − q−n)a (3.8) Ψ(z)= ψ z−n = exp( n z−n)q∂, n n n≥0 n∈N X X −n − 3n n (q q )a−n n −∂ (3.9) Φ(z)= φ− z = exp( z )q . n n n≥0 n∈N b The following map defines the irreducible Uq(sl2)-module structure for V (Λi). → + → − → ∂ e1 X0 ,f1 X0 ,K1 q , → − −∂ → ∂ + → 1−∂ e0 X1 q ,f0 q X−1,K0 q . The vertex space V (Λi) is endowed with the standard inner product via ∗ α ∗ −α an = a−n, (e ) = e , (z∂)∗ = z−∂. It follows from the commutation relations (3.1) that Y mα iα/2 nα iα/2 1 (3.10) (a−λe e ,a−µe e )=δmnδλµzλ , 1+q2λj j≥1 where zλ is as in Section 2. By Section 2 there are two special bases in V (Λi): the power sum basis mα iα/2 mα iα/2 {a−λe e } and the Schur basis {sλe e }. However, the Schur basis is no longer orthogonal with respect to the inner product (3.10). 2n Let b−n = a−n,bn =(1+q )an, n ∈ N,then{b−n} generate a standard Heisenberg algebra as in Section 2. In terms of the new Heisenberg generators we have X∞ X∞ 2n X 1 n 1+q −n −n S(z)=exp( a− z )exp(− a z )= S z , n n n n n n=1 n=1 n∈Z X∞ X∞ 2n X ∗ 1 n 1+q −n ∗ n S (z)=exp(− a− z )exp( a z )= S z , n n n n n n=1 n=1 n∈Z 56 NAIHUAN JING which generate the Schur function basis. We call S(z)(orS∗(z)) the Schur (or dual Schur) vertex operator. For a partition λ and m ∈ Z we define the Schur symmetric polynomial in V (Λi) (cf. Theorem 2.3): mα iα/2 mα iα/2 ··· mα iα/2 (3.11) sλe e := S−λe e = S−λ1 S−λl e e . The element sλ is a polynomial over Q in terms of the power sum aµ,where|µ| = |λ|. Note that S(z) acts trivially on the lattice vector emαeiα/2. We need to recall some further terminology about partitions. Let λ and µ be two partitions; we write λ ⊃ µ if the Young diagram of λ contains that of µ.The set difference λ \ µ is called a skew diagram. The conjugate of a skew diagram θ = λ \ µ is θ0 = λ0 \ µ0 and we define X (3.12) |θ| = θj = |λ|−|µ|. A skew diagram θ is a horizontal n-strip (resp. a vertical n-strip) if |θ| = n and 0 ≤ ≤ θj 1(resp. θj 1) for each j. Thus a horizontal (resp. vertical) strip has at most one column (resp. rows) in its diagram. For a partition µ and an integer m we let Vn = Vn(m, µ) be the set of the partitions of λ such that the skew diagram λ \ π(m, π(µ)0)0 is a vertical n-strip. We also let Hn = Hn(m, µ) be the set of partitions λ such that the skew diagram λ \ π(m, µ)isahorizontaln-strip. Note that Vn may be described as the set of partitions λ such that the skew diagram λ \ π(1m,µ)isaverticaln-strip. The following is called the Pieri rule [M]: X (3.13) snS−mS−µ.1= sgn(m, µ)sλ, ∈H λ Xi(m,µ) n 0 0 (3.14) s1n S−mS−µ.1= (−1) sgn(m, π(µ) ) sλ. λ∈Vn(m,µ) We will use λ − µ to denote the difference of two integral tuples in Zn. b Theorem 3.2. The quantum affine algebra Uq(sl2) is realized on the Fock space Λ ⊗ Q(q)[Zα]eiα/2 of symmetric functions by the following action: X+s emαeiα/2 = e(m+1)αeiα/2 nµ l(µ)−2Xm−n−1−i X −2m−n−1−i−2j × q sgn(−2m − n − 1 − i − j, µ) sλ , j=0 λ∈Hj where Hj = Hj(−2m−n−1−i−j, µ),thesignreferstosgn(−2m−n−1−i−j, µ)= (−1)l(σ) such that (−2m − n − 1 − i − j, µ)+δ = σ(λ + δ) and λ is the partition of length at most l(µ)+1. Also we have X−s emαeiα/2 =(−1)n+1+ie(m−1)αeiα/2 nµ − − µ1+2mXn 1+i X 2j 0 × q sgn(2m − n − 1 − j + i, µ ) sλ , j=0 λ∈Vj 0 where Vj = Vj(2m − n − 1 − j + i, µ ),thesignreferstosgn(2m − n − 1 − i − j, µ)= (−1)l(σ) such that (2m − n − 1 − j + i, µ)+δ = σ(λ + δ) and λ is the partition of length at most l(µ)+2m − n − 1 − j + i. b SYMMETRIC POLYNOMIALS AND Uq (sl2)57 This result will be proved in more generality later in Theorem 3.5. We can reformulate the result in terms of the standard inner product in Section ··· ··· Zn 2. Let uj =(0, , 0, 1, 0, , 0) be the jth unit vector in .Let1(l1,··· ,lj ) be the ··· sum of the unit vectors ul1 , ,ulj . Proposition 3.1. For n ∈ Z and a partition µ we have − mα iα/2 Xn sµe e X Xl(λ) X (m−1)α iα/2 − 2 j ∗ = sλe e ( q ) (S − − − S−µ,S− −1 ··· ), 2m+i n 1 j (λ (l1, ,lj )) λ j=0 l1<··· VA(Λi) ⊗A Q(q) ' V (Λi). Theorem 3.5. The A-linear spaces VA(Λi) are invariant under the action of the (r) divided powers Xn . More precisely, we have +(r) mα iα/2 X sµe e n X −3(r)−r(n+1+2m+i) −2|λ| (m+r)α iα/2 = q 2 q sλ · s(−λ−2δ−(n+1+2m+i)1,µ)e e , l(λ)≤r −(r) mα iα/2 X sµ0 e e n X r(n+1+i) (r) 2|λ| (m−r)α iα/2 =(−1) q 2 q sλ0 · sπ(−λ−2δ−(n+1−2m−i)1,µ)0 e e , l(λ)≤r 58 NAIHUAN JING where the summations run through all partitions λ of length ≤ r, δ =(r − 1,r− 2, ··· , 0),and1 =(1, ··· , 1) ∈ Zr. r Proof. Let z =(z1, ··· ,zr), w =(w1, ··· ,wl), and 1 =(1, ··· , 1) ∈ Z in the following computation: r x+ s emαeiα/2 n I µ dzdw = X+(z ) ···X+(z )S(w ) ···S(w )z(n+1)1w−µ emαeiα/2 1 r 1 l zw I ∞ X n −n (q + q ) n n = exp a− (z + ···+ z ) : S(w ) ···S(w ): n n 1 r 1 l n=1 Y w Y w × (z − z )(z − q−2z )(1 − j ) (1 − q−1 j ) i j i j w z i I X X∞ n −n +(r) mα iα/2 1 (q + q ) n n x s e e = exp a− (z + ···+ z ) n µ [r]! n n 1 r w∈Sr n=1 Y Y w Y w × : S(w ) ···S(w ): (z − z ) (1 − j ) (1 − q−1 j ) 1 l i j w z i X −2`(w) − r q = q (2)[r]!. w∈Sr From the orthogonality of Schur functions [M] it follows that X∞ X a−n n n exp (z + ···+ z ) = s (a− )s (z ), n 1 r λ k λ i n=1 l(λ)≤r b SYMMETRIC POLYNOMIALS AND Uq (sl2)59 where sλ(a−k) is the Schur function in terms of the power sum a−µ and sλ(zi)is −1 the Schur polynomial in the variables z1, ··· ,zr. Replacing zj by q zj we get +(r) mα iα/2 xn sµe e X I −3(r)−r(n+1+2m+i) −2|λ| 2δ+(n+1+2m+i)1 = q 2 q sλ sλ(z)z l(λ)≤r (m+r)α iα/2 −µ dz dw × S(z1) ···S(zr)S(w1) ···S(wl)e e w I z w X X l(w) w(λ+δ) − r − (−1) z 3(2) r(n+1+2m+i) Q δ+(n+1+2m+i)1 = q sλ − z j −(r) mα iα/2 X sµ0 e e n I (−1)|µ| dzdw = X−(z ) ···X−(z )S∗(w ) ···S∗(w )zn+1w−µe(m−r)αeiα/2 [r]! 1 r 1 l zw I (−1)|µ| = : S∗(q2z ) ···S∗(q2z ):S∗(z ) ···S∗(z )S∗(w ) ···S∗(w ) [r]! 1 r 1 r 1 l Y dzdw × (z − q2z )zδ+(n+1−2m−i)1w−µ e(m−r)αeiα/2 i j zw Ii (r) mα iα/2 −3 r −r(2m+i+1) e s e e = q (2) 1 µ X −2|λ| (m+r)α iα/2 (4.1) × q sλs(−λ−2δ−(2m+1+i)1,µ) e e , l(λ)≤r (r) mα iα/2 r(1+i) r f s 0 e e =(−1) q(2) 1 µ X 2|λ| (m−r)α iα/2 (4.2) × q sλ0 sπ(−λ−2δ+(2m+i−1)1,µ)0 e e , l(λ)≤r f (r)s emαeiα/2 = qr(5−r)/2 0 µ X −2|λ| (m+r)α iα/2 (4.3) × q sλs(−λ−2δ−(2m+i)1,µ) e e , l(λ)≤r (r) mα iα/2 ri − r −r(2m+i) e s 0 e e =(−1) q (2) 0 µ X 2|λ| (m−r)α iα/2 (4.4) × q sλ0 sπ(−λ−2δ+(2m+i−2)1,µ)0 e e , l(λ)≤r mα iα/2 1−2m−i mα iα/2 (4.5) K0sµe e = q sµe e , (4.6) K s emαeiα/2 = q2m+is emαeiα/2, 1 µ µ K ; l 1 − 2m − i + l (4.7) 0 s emαeiα/2 = s emαeiα/2, r µ r µ K ; l 2m + i + l (4.8) 1 s emαeiα/2 = s emαeiα/2. r µ r µ As a consequence of these formulas and the Littlewood-Richardson rule (2.2) we get the following theorem. b Theorem 4.1. The A-linear space VA(Λi) is an UA(sl2)-lattice in V (Λi).Inpar- b iα/2 ticular, UA(sl2)e is a sublattice of VA(Λi). b SYMMETRIC POLYNOMIALS AND Uq (sl2)61 Proposition 4.1. For m ≥ 0 we have (2m) mα − m m(2m−1) −mα f1 e =( 1) q e , (2m+1) −mα − m −(2m+1)(m−2) −(m+1)α f0 e =( 1) q e , (2m) −mα α/2 − m −m(2m−5) mα f0 e e =( 1) q e , (2m+1) mα α/2 − m m(2m+1) −(m+1)α f1 e e =( 1) q e . (2m) mα − Proof. The four formulas are proved similarly. Take f1 e .Observethat 2δ + (2m−1)1 =(−2m+1, −2m+3, ···, 2m−3, 2m−1) is of weight zero, thus only λ =0 (2m) mα contributes to the summation in f1 e (see (4.2)). Since the longest element in m(2m−1) S2m has inversion number m(2m − 1), the sign of s−2δ+(2m−1)1 is (−1) = (−1)m. Corollary 4.1. For m ≥ 0 we have (2m) (2m−1) ··· (2) − m 3m2 −mα f1 f0 f1 f0.1=( 1) q e , (2m+1) (2m) ··· (2) (m+1)(m+2) (m+1)α f0 f1 f1 f0.1=q e , (2m) (2m−1) ··· (2) α/2 − m m(m+2) mα α/2 f0 f1 f0 f1e =( 1) q e e , (2m+1) (2m) ··· (2) α/2 3m(m+1) −(m+1)α α/2 f1 f0 f0 f1e = q e e . Example 4.2. In the following we abbreviate f (n1) ···f (nr).1=f (n1) ···f (nr) in i1 ir i1 ir the basic representation V (Λ0). 2 α f0 = q e , 2 2 f1f0 = −q (1 + q )s1, (2) − 3 −α f1 f0 = q e , 2 2 α f0f1f0 = q (q +1)s1e , 4 2 4 f1f0f1f0 = −(q + q )(s2 − q s12 ), (2) − 3 f0f1 f0 = q (s12 +[3]s2), 2 2 2 α f0f1f0f1f0 = q (1 + q ) (s2 + s12 )e , (2) (2) 4 α f0 f1 f0 = q (s2 +[3]s12 )e , (3) (2) 6 2α f0 f1 f0 = q e , (2) 5 2 −α f1 f0f1f0 = q (1 + q )s1e . 62 NAIHUAN JING Example 4.3. As in the last example we use f (n1) ···f (nr) to denote i1 ir f (n1) ···f (nr)eα/2 in the basic representation V (Λ ). i1 ir 1 −α f1 = e , −2 −2 f0f1 = q (1 + q )s1, (2) − 3 α f0 f1 = q e , −2 −α f1f0f1 = −(q +1)s1e , 2 f0f1f0f1 =(1+q )(s2 − s12 ), (2) − 5 f1f0 f1 = q ([3]s12 + s2), 2 2 2 −α f1f0f1f0f1 =(1+q ) (s2 + q s12 )e , (2) (2) 5 −α f1 f0 f1 = q ([3]s2 + s12 )e , (3) (2) 6 −2α f1 f0 f1 = q e , (2) − α f0 f1f0f1 = [2]s1e . It would be interesting to see the relation between our formulas and the fermionic picture [LLT]. Finally we would like to remark on generalizing the result of this paper to quan- tum affine algebras of ADE type. Using the realization [FJ] it is clear that the action of divided powers of Drinfeld generators on the Schur symmetric functions in multi-sets of variables are definable, but not as explicit as in the sl2-case due to the complexity of the interaction between adjacent simple roots. We do not know how to deal with the hypergeometric functions appearing in the interaction. Another reason for this difficulty is as follows. In [BFJ] we studied two lattices as- b sociated with the canonical and dual canonical basis of Uq(sl2). Both are equivalent under the Macdonald inner product but are not equivalent under the Schur inner product. The lattice associated with the canonical basis is easier to be formulated to the ADE cases as shown in [CJ], but the relation to the Littlewood-Richardson rule is sacrificed. We choose to study the lattice of dual canonical basis in order to use Littlewood-Richardson rule, and are also motivated by the transition relation with the Macdonald polynomial. Acknowledgments The author wishes to thank I. Frenkel for helpful discussions. He also thanks M. L. Ge for the warm hospitality in the summers of 1994 and 1997 at Nankai Insti- tute of Mathematics, where the author had the privilege to stay in the S. S. Chern Villa (Home of Geometer) to work out the main computations. The author is also grateful to the Mathematical Sciences Research Institute in Berkeley for providing a lovely environment in the final stage of this work. We also thank V. Chari and S. Leidwanger for their careful reading of the paper. References [BCP] J. Beck, V. Chari and A. Pressley, An algebraic characterization of the affine canonical basis, Duke Math. J. 99 (1999), 455-487. CMP 2000:01 [BFJ] J. Beck, I. B. Frenkel and N. Jing, Canonical basis and Macdonald polynomials, Adv. in Math. 140 (1998), 95-127. CMP 99:05 b SYMMETRIC POLYNOMIALS AND Uq (sl2)63 [CJ] V. Chari and N. Jing, Realization of level one representations of Uq(ˆg)atarootofunity, math.QA/9909118. [CT] C. Carr´e and J.-Y. Thibon, Plethysm and vertex operators, Adv. in Appl. Math. 13 (1992), 390-403. MR 94c:05070 [CP] V. Chari and A. Pressley, Quantum affine algebras at roots of unity, Representation Theory 1 (1997), 280–328. MR 98e:17018 [DJKM] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Transformation groups for soliton equations. Nonlinear integrable systems—classical theory and quantum theory (Kyoto, 1981), pp. 39–119, World Sci. Publishing, Singapore, 1983. MR 86a:58093 [F1] I. B. Frenkel, Two constructions of affine Lie algebra representations and boson- fermion correspondence in quantum field theory, J. Funct. Anal. 44 (1981), 259–327. MR 83b:17012 [F2] I. B. Frenkel, Lectures at Yale University, 1986. [FJ] I. B. Frenkel and N. Jing, Vertex representations of quantum affine algebras, Proc. Natl. Acad. Sci. USA 85 (1988), 9373-9377. MR 90e:17028 [G] H. Garland, The arithmetic theory of loop algebras, J. Algebra 53 (1978), 480-551; Erra- tum: The arithmetic theory of loop algebras, J. Algebra 63 (1980), 285. MR 80a:17012; MR 81d:17008 [Ga] A. M. Garsia, Orthogonality of Milne’s polynomials and raising operators, Discrete Math. 99 (1992), 247-264. MR 93m:05201 [H] T. Hayashi, Q-analogues of Clifford and Weyl algebras-spinor and oscillator represen- tations of quantum enveloping algebras, Commun. Math. Phys. 127 (1990), 129-144. MR 91a:17015 [JK] G. James and A. Kerber, The representation theory of the symmetric group,Encyclope- dia of Math. and its Appl. 16, Addison-Wesley, Reading, MA, 1981. MR 83k:20003 [J1] N. Jing, Vertex operators, symmetric functions and the spin groups Γn,J.Algebra138 (1991), 340-398. MR 92e:17033 [J2] N. Jing, Vertex operators and Hall-Littlewood functions, Adv. in Math. 87 (1991), 226- 248. MR 93c:17039 [J3] N. Jing, Vertex operators and generalized symmetric functions, in: Proc. of Conf. on Quantum Topology (KSU, March 1993), ed. D. Yetter, World Scientific, Singapore, 1994, pp. 111-126. MR 96e:17062 [J4] N. Jing, Boson-fermion correspondence for Hall-Littlewood polynomials, J. Math. Phys. 36 (1995), 7073-7080. MR 96m:17049 [J5] N. Jing, Quantum Kac-Moody algebras and vertex representations, Lett. Math. Phys. 44 (1998), no. 4, 261–271. MR 99j:17043 [K] M. Kashiwara, Global crystal bases of quantum groups, Duke Math. J. 73 (1993), 383– 413. [Ka] V. G. Kac, Infinite dimensional Lie algebras, 3rd. ed., Cambridge Univ. Press, Cam- bridge, 1990. MR 92k:17038 [LLT] A. Lascoux, B. Leclerc, J.-Y. Thibon, Ribbon tableaux, Hall-Littlewood functions, quan- tum affine algebras and unipotent varieties, S´em. Lothar. Combin. 34 (1995), 23 pp. MR 98m:05195 (1) [LP] J. Lepowsky and P. Primc, Structure of the standard modules for affine Lie algebra A1 , Contemp. Math. 46, Amer. Math. Soc., Providence, RI, 1985. MR 87g:17021 [L] G. Lusztig, Introduction to quantum groups, Progress in Mathematics 110,Birkh¨auser, Boston, 1993. MR 94m:17016 [M] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, New York, 1995. MR 96h:05207 [MM] K. C. Misra and T. Miwa, Crystal bases for the basic representations of Uq(slˆ (n)), Commun. Math. Phys. 134 (1990), 79-88. MR 91j:17021 [Z] A. Zelevinsky, Representations of finite classical groups, A Hopf algebra approach. Lec- ture Notes in Mathematics, 869, Springer-Verlag, Berlin-New York, 1981. MR 83k:20017 Department of Mathematics, North Carolina State University, Raleigh, North Car- olina 27695-8205 E-mail address: [email protected]