Symmetric functions and the Fock space

Transcription

1 Symmetric functions and the Fock space representation of Í Õ Ð Ò µ (Lectures at the Isaac Newton Institute, Cambridge) Bernard LECLERC June 00 Dedicated to Denis UGLOV ( ) Introduction Throughout these lectures, Ò ¾ will be a fixed integer and Õ an indeterminate. (Virtually all objects that we shall consider will depend on this Ò, but we shall not repeat it all the time neither record it explicitely in our notation.) Using the representation theory of the quantum affine algebra Í Õ ÐÒ µ one can introduce for each pair of partitions and of the same integer Ñ two polynomials Õµ and Õµ with integer coefficients. (They depend on Ò). For example, for Ò ¾ and Ñ, one has: Õµ µ ½µ ¾ ¾ µ ¾½ ¾ µ ½ µ µ ½ ¼ ¼ ¼ ¼ ½µ Õ ½ ¼ ¼ ¼ ¾ ¾ µ ¼ Õ ½ ¼ ¼ ¾½ ¾ µ Õ Õ ¾ Õ ½ ¼ ½ µ Õ ¾ ¼ ¼ Õ ½ Õµ µ ½µ ¾ ¾ µ ¾½ ¾ µ ½ µ µ ½ Õ Õ ¾ ¼ ¼ ½µ ¼ ½ Õ ¼ Õ ¾ ¾ µ ¼ ¼ ½ Õ Õ ¾ ¾½ ¾ µ ¼ ¼ ¼ ½ Õ ½ µ ¼ ¼ ¼ ¼ ½ These polynomials are defined as the coefficients of two canonical bases of the Fock space representation of Í Õ ÐÒ µ. They were studied in [LLT, LT] motivated by the work of James on modular representations of symmetric groups and finite general linear groups [J]. It was conjectured in [LLT, LT] that the ½µ are some decomposition numbers for Hecke algebras and quantized Schur algebras at roots of unity. These conjectures were proved by Ariki [A] (for Hecke algebras) and by Varagnolo-Vasserot [VV] (for Schur algebras). The Õµ have been shown to coincide with the coefficients of Lusztig s character formula for the simple modules of Í Ú Ð Ö µ, where Ö Ñ and Ú ¾ is a primitive Òth root of [VV]. Similarly, the Õµ are the coefficients of Soergel s formula for the indecomposable tilting Í Ú Ð Ö µ- modules [VV, GW, LT]. In particular, both families are parabolic Kazhdan-Lusztig polynomials for the Coxeter groups of type ½µ Ö ½. Recently, some of them have been explicitely described in terms of the Littlewood-Richardson coefficients [LM] (see also [CT]).

2 The polynomials Õµ contain as special cases the Kostka-Foulkes polynomials and more generally the Õ-Littlewood-Richardson coefficients of [CL, LLT, LT] defined combinatorially in terms of Ò-ribbon tableaux. The aim of these lectures is to review the definition, the calculation and the main properties of Õµ and Õµ. We shall emphasize the close connections between the Fock space representation of Í Õ ÐÒ µ and the ring of symmetric functions. In the last section, we shall indicate some connections with the Macdonald polynomials. The affine Lie algebra Ð Ò Let ¼Ò ½ be the Ò Ò matrix ¾ ¾ ¾ ¾ Ò ¾µ ¾ ¾ ½ ¼ ¼ ¼ ½ ½ ¾ ½ ¼ ¼ ¼ ¼ ½ ¾ ¼ ¼ ¼ ¼ ¼ ¼ ¾ ½ ¼ ¼ ¼ ¼ ½ ¾ ½ ½ ¼ ¼ ¼ ½ ¾ Ò µ The complex Lie algebra with generators ¼ Ò ½µ submitted to the relations ¼ () ¼ ¼ ¼ ¼ ¼ () ¼ ¼ ¼µ () Æ () µ ½ ¼ µ ½ ¼ µ (5) is the Kac-Moody algebra of type ½µ Ò ½, also denoted ÐÒ [Ka]. (Here, for Ü ¾, Ü means the endomorphism Ý Ü Ý.) The subalgebra of obtained by omitting the generator is denoted by ¼. The universal enveloping algebra of is denoted by Í µ. The Fock space representation of Ð Ò. A concrete realization of can be obtained by letting it act on the Fock space. This goes back to [JiMi]. Let È be the set of all partitions. By convention È contains the empty partition. As usual, we identify a partition with its Young diagram. A node of whose content is congruent to modulo Ò is called an -node. Let be an infinite-dimensional complex vector space with a distinguished -basis µ indexed by ¾ È. We define endomorphisms ¼ Ò ½µ of, by µ µ µ µ µ Æ ¼ µ µ (6) where the first sum is over all partitions obtained by removing from an -node, the second sum is over all partitions obtained from by adding an -node, and Æ ¼ µ denotes the number of

3 Figure : The Young diagram of ½µ with its ¼-nodes and its ½-nodes ¼-nodes of. It is easy to see that,, and defines a representation of, that is, setting, one can check that satisfy relations () () () () (5). In the sequel we shall write for simplicity in place of. Example Let Ò ¾. We have (see Figure ) ¼ ½µ ¾µ ½ ½µ ½ ½µ ½µ ¼ ½µ ¾ ½µ ½ ½µ µ By construction, µ is a highest weight vector, that is, a common eigenvector of ¼ Ò ½ annihilated by all s. The subrepresentation Í µ µ generated by µ is irreducible. It is the simplest infinite-dimensional representation of, and for this reason it is often called the basic representation. One may regard it as an affine analogue of the vector representation of the finitedimensional Lie algebra Ð Ò.. Let ËÝÑ denote the -algebra of symmetric functions in a infinite set of variables [Mcd]. For ¾ È, let be the Schur function and Ô the product of power sums corresponding to. We denote by the scalar product for which the basis is orthonormal. To ¾ ËÝÑ we associate the endomorphism defined as the adjoint of the operator of multiplication by, that is, ¾ ËÝѵ We have a natural isomorphism of vector spaces from ËÝÑ to given by µ. From now on, we shall identify these two spaces via this isomorphism. For ¾, let ¾ Ò be defined by Ô if ¼, multiplication by Ô if ¼. One can show that the operators Ò ¾ µ commute with the action of ¼. This implies that for any ¾ È, the power sum Ô Ò is a highest weight vector of. On the other hand, the with not divisible by Ò belong to the image of in Ò. It follows that as a -module, decomposes into the direct sum Å ¾È Í µ Ô Ò Å ¾È Ô ¼ ÑÓ Ò Ô Ò Here, all summands are irreducible, and isomorphic as ¼ -modules. Moreover, the degree generator acts on the highest weight vectors by Ô Ò Ô Ò Hence it separates the irreducible components generated by power sums of different degrees. Note however that at this stage there is no canonical choice of highest weight vectors in a given degree, and in fact the basis Ô Ò ¾ È of the space of highest weight vectors could well be replaced by any -basis of Ô Ò ¾ Æ consisting of homogeneous elements. We are now going to see that by Õ-deforming this picture, we are lead to a distinguished basis of highest weight vectors.

4 The quantum affine algebra Í Õ Ð Ò µ Let à յ. The algebra Í Õ Í Õ ÐÒ µ is defined [Dr, Ji] as the associative algebra over à with generators à à ½ ¼ Ò ½µ ½ submitted to the relations ½ ¼ ½ ¼ à à ½ à ½ à ½ à à à à (7) ½ Ã Ã Õ Ã Ã ½ Õ (8) ½ ½ ½ à à (9) ¼ ½ Õ ½ ¼ ¼ ½ Õ ¼ (0) ½ ¼ ½ ¼ ¼µ () ½ Ã Ã Æ Õ Õ ½ () ½µ ½ ½µ ½ ½ ¼ µ () ½ ¼ µ () Here, we have used the Õ-integers and Õ-binomial coefficients given by Õ Õ Õ Õ ½ ½ ½ Ñ The subalgebra obtained by omitting the generator is denoted by Í ¼ Õ. Ñ Ñ 5 The Fock space representation of Í Õ Ð Ò µ Let Õ Å Ã be the Fock space over the field Ã. Following [H, MM] one has an action of Í Õ on Õ defined as follows. Let and be two Young diagrams such that is obtained from by adding an -node. Such a node is called a removable -node of, or an indent -node of. Let Á Ö µ (resp. ÊÖ µ) be the number of indent -nodes of (resp. of removable -nodes of ) situated to the right of ( not included). Set Æ Ö µ ÁÖ µ ÊÖ µ. Then µ Õ Æ Ö µ µ (5) where the sum is over all partitions such that is an -node. Similarly µ Õ Æ Ð µ µ (6) where the sum is over all partitions such that is an -node, and Æ Ð µ is defined as µ but replacing right by left. Finally, Æ Ö These equations make Õ into an integrable representation of Í Õ. µ Õ Æ ¼ µ µ (7)

5 Example Let Ò ¾. We have (see Figure ) ¼ ½µ Õ ½ ¾µ ½ ½µ ¼ ½µ Õ ¾ ¾ ½µ ½ ½µ ½µ ½ ½µ µ As in the classical case, the subrepresentation Í Õ µ is the simplest highest weight irreducible representation and is called the basic representation of Í Õ. In order to work out the decomposition of Õ into irreducible components, Kashiwara, Miwa and Stern have introduced some Õ-analogues of the operators Ò [KMS]. (Apparently, there is no natural Õ-analogue of for non divisible by Ò.) Unfortunately, the action of these Õ-bosons is difficult to describe combinatorially. To get around this problem, a different family of operators has been defined in [LLT], whose action on Õ has a simple expression in terms of ribbons. h(r) Figure : An -ribbon of height ʵ Figure : A skew diagram with its subdiagram shaded Figure : A horizontal 5-ribbon strip of weight and spin 7 A ribbon is a connected skew Young diagram Ê of width, i.e. which does not contain any ¾ ¾ square (see Figure ). The rightmost and bottommost cell is called the origin of the ribbon. An Ò-ribbon is a ribbon made of Ò square cells. Let be a skew Young diagram, and let be the horizontal strip made of the bottom cells of the columns of (see Figure ). We say that is a horizontal Ò-ribbon strip of weight if it can be tiled by Ò-ribbons the origins of which lie in. One can check that if such a tiling exists, it is unique (see Figure ). Define the spin of Ê as ÔÒ Êµ ʵ ½ 5

6 Figure 5: Calculation of Ú ¾µ and Ú ½½µ for Ò ¾ where ʵ is the height of Ê, and the spin of an Ò-ribbon strip as the sum of the spins of the ribbons which tile it. For ¾ Æ, let Î be the linear operator acting on by Î µ Õµ ÔÒ µ µ (8) the sum being over all such that is a horizontal Ò-ribbon strip of weight. These operators pairwise commute and they also commute with the action of the subalgebra ÍÕ ¼ [LLT, LT]. Hence for each ½ Ö µ ¾ È, the vector Ú Î ½ Î Ö µ (9) is a highest weight vector of the Í Õ -module Õ, and one has the decomposition Õ Å ¾È where all summands are irreducible, and isomorphic as Í ¼ Õ -modules. Note that it follows easily from (8) that at Õ ½, Î Ô Ò Í Õ Ú (0) that is, Î is a Õ-analogue of the multiplication by the plethysm Ô Ò Ü Ò ½ ÜÒ ¾ µ. Hence Ú is a Õ-analogue of the symmetric function Ô Ò. Example Let Ò ¾. The highest weight vectors in degree are as follows (see Figure 5) Ú ¾µ Î ¾ µ µ Õ ½ ½µ Õ ¾ ¾ ¾µ Ú ½½µ Î ¾ ½ µ µ Õ ½ ½µ Õ ¾ ½µ ¾ ¾µ Õ ½ ¾ ½ ½µ Õ ¾ ½ ½ ½ ½µ 6 The bar involution of Õ 6. There is a unique map Ü Ü from Õ to itself satisfying µ µ () ³ ÕµÜ ÕµÝ ³ Õ ½ µü Õ ½ µý Ü Ý ¾ Õ ³ ¾ à µ () Ü Ü ¼ Ò ½ Ü ¾ Õ µ () 6

7 0 0 Figure 6: The ladder decomposition of ¾µ for Ò Î Ü Î Ü ½ Ü ¾ Õ µ () Indeed, by (9) (0) () () and () this map is determined on the space of highest weight vectors of Õ, and by () it can be uniquely extended to the whole space Õ. Write µ Õµ µ Ñ Õµ Ñ Theorem ([LT, LT]) The matrix Ñ has entries in Õ Õ ½ and is lower unitriangular if its rows and columns are labelled by partitions arranged in decreasing lexicographic order. More precisely, Õµ can be nonzero only if is less or equal to for the dominance order on partitions. The triangularity property is not obvious. To prove it, one uses the fermionic realization of Õ given in [KMS] to obtain a description of the bar-involution in terms of the straightening rules of Õ-wedge products. However, this other description is not appropriate for practical calculations. We shall now explain a different algorithm. 6. A partition is said to be Ò-regular if it has no part repeated more than Ò ½ times, that is, if we write in multiplicative form ½ Ñ ½ ¾ Ñ ¾ µ all Ñ s are strictly less than Ò. We shall use the ladder decomposition of an Ò-regular partition ([JK], 6..5, p.8). The ladders of are the intersections of its Young diagram with the straight lines of equation Ý ½ ÒµÜ ¼ ½ ¾ µ (Here we take the origin of coordinates to be the center of the leftmost bottom box of ). By construction, the nodes of lying on a given ladder Ý ½ ÒµÜ all have the same residue Ò ÑÓ Ò. Let be the number of ladders containing at least one node of. Denote these ladders from left to right by Ä ½ Ä, so that Ä ½ is the ladder through the origin and Ä is the rightmost ladder intersecting. Let be the number of nodes of lying on Ä, and let Ö be their common Ò-residue. We define where Öµ Ö Ö. Ø µ µ Ö ½µ Ö ½ ½µ Ö ½ µ (5) Example 5 Let Ò and ¾µ. The ladder decomposition of is shown in Figure 6. This yields Ø ¾µ ½ ¾µ ¾ ¼ ¾µ ½ ¾ ¼ µ ¾µ Õ ½ ½ ½ ½ ½µ 7

8 More generally, if ½ Ñ ½ Ö ÑÖ µ is not Ò-regular we write «½ ½ Ö Ö µ and ½ ½ Ö Ö µ where Ñ Ò and ¼ Ò. Then is Ò-regular and has a ladder decomposition Ä ½ Ä with corresponding integers Ö ½ Ö ½. We set where Ú «is given by (9). Ø µ µ Ö ½µ Ö ½ ½µ Ö ½ Ú «(6) Example 6 Let Ò ¾. The vectors Ø µ for all partitions of are Ø µ ½ ¼ ½ ¼ µ µ Õ ½µ Õ ¾ ½ ½µ Õ ¾ ½ ½ ½ ½µ Ø ½µ ¼ ¾µ ½ ¼ µ ½µ Õ ¾ ¾µ Õ ¾ ¾ ½ ½µ Ø ¾ ¾µ Ú ¾µ µ Õ ½ ½µ Õ ¾ ¾ ¾µ Ø ¾ ½ ½µ ½ ¼ Ú ½µ µ Õ ½µ Õ ½ ¾ ½ ½µ ½ ½ ½ ½µ Ø ½ ½ ½ ½µ Ú ½½µ µ Õ ½ ½µ Õ ¾ ½µ ¾ ¾µ Õ ½ ¾ ½ ½µ Õ ¾ ½ ½ ½ ½µ Because of the combinatorial simplicity of (5) and (8), the expansion of Ø µ on the basis µ is easily computable. Clearly, it only involves vectors µ with. Write Ø µ By construction Ø Õµ ¾ Õ Õ ½. Ø Õµ µ Ì Ñ Ø Õµ Ñ Lemma 7 Ø µ ¾ È is a à -basis of Õ invariant under the bar-involution Ü Ü. The fact that Ø µ is bar-invariant follows immediately from () and (). The proof that the Ø µ s with Ò-regular form a basis of the basic representation was given in [LLT]. Since the Î s and the s commute with each other, we see that the Ø µ s with the same Ò-singular part «form a à -basis of the subrepresentation Í Õ Ú «, and the result follows. The bar-invariance of Ø µ yields Ñ Ì Ñ Ì Ñ µ ½ where Ì Ñ Ø Õ ½ µ Ñ. Hence, from the calculation of the Ø µ s we deduce the matrix of the bar-involution. Note however that this involves inverting the matrix Ì Ñ of size the number of partitions of Ñ. This is the time and space consuming step of the algorithm. Note also that Ì Ñ is not triangular, so using this approach, the triangularity of Ñ remains rather mysterious. Example 8 Let us calculate for Ò ¾. The partitions of are arranged in the order µ ½µ ¾ ¾µ ¾ ½ ½µ ½ ½ ½ ½µ From Example 6 we get Ì ¾ ½ ¼ ½ ½ ½ Õ ½ Õ ½ Õ Õ ½ ¼ Õ Õ ¾ ¼ Õ ¾ ½ Õ Õ ¾ ¼ Õ ½ Õ ½ Õ ¾ ¼ ¼ ½ Õ ¾ 8

9 It follows that Ì Ì µ ½ ¾ ½ ¼ ¼ ¼ ¼ Õ Õ ½ ½ ¼ ¼ ¼ Õ ¾ ½ Õ Õ ½ ½ ¼ ¼ ¼ Õ ¾ ½ Õ Õ ½ ½ ¼ Õ ¾ ½ ¼ Õ ¾ ½ Õ Õ ½ ½ 7 The canonical bases of Õ 7. Let Ä (resp. Ä ) be the free Õ -module (resp. the free Õ ½ -module) with basis µ. Theorem 9 There exist two bases µ ¾ È and characterized by the following properties µ µ µ µ mod ÕÄ µ µ µ µ mod Õ ½ Ä µ ¾ È of Õ Proof Let us prove the existence of. Fix Ñ ¾ Æ and let ½ ¾ be the list of partitions of Ñ arranged in decreasing lexicographic order. By Theorem, µ µ, so we can take µ µ. We now argue by induction and suppose that for a certain Ö we have constructed vectors Ö ½ µ Ö ¾ µ µ satisfying the conditions of the theorem. Moreover, we assume that Ö µ Ö µ Ö «Õµ Ö µ ½ Öµ (7) that is, the expansion of Ö µ only involves vectors µ with Ö. We can therefore write, by solving a linear system with unitriangular matrix, Ö µ Ö µ ½ Ö Õµ Ö µ where the coefficients Õµ belong to Õ Õ ½. By applying the bar involution to this equation we get that Õ ½ µ Õµ, hence Õµ Õµ Õ ½ µ with Õµ ¾ ÕÕ. Now set Ö µ Ö µ ½ Ö Õµ Ö µ We have Ö µ Ö µ ÑÓ ÕÄ, Ö µ Ö µ, and the µ-expansion of Ö µ is of the form (7) as required, hence the existence of follows by induction. The proof of the existence of is similar. To prove unicity, we show that if Ü ¾ ÕÄ is bar invariant then Ü ¼. Otherwise write Ü È Õµ µ, and let be the largest partition for which Õµ ¼. Then µ occurs in Ü with coefficient Õ ½ µ, hence Õµ Õ ½ µ. But since Õµ ¾ ÕÕ this is impossible. ¾ It is easy to see that the µ which belong to the basic representation are precisely those for which is Ò-regular, and that they coincide with Kashiwara s lower global base of this irreducible module. In fact, is a lower global base of Õ (see e.g. [LM]). Uglov [U] has generalized this construction to the higher level Fock space representations of Í Õ, thereby giving a simple algorithm for computing the canonical basis of any integrable simple Í Õ -module. Recently, Kashiwara [K] has obtained in a similar way a global base for all the Õ-deformed Fock spaces of [KMPY] associated with a perfect crystal of an affine Lie algebra. 9

10 7. We can now define Õµ and Õµ by and µ µ Õµ µ (8) Õ ½ µ µ (9) Note that the proof of Theorem 9 shows how to compute the Õµ and Õµ once the Õµ are known. This only involves solving a unitriangular system, so it is rather cheap. Example 0 Let Ò ¾. Using the matrix computed in Example 8 it is easy to calculate the matrices of and given in the introduction. By construction Õµ Õµ ½, and Õµ and Õµ belong to ÕÕ if. Recall that to ¾ È is associated an Ò-core Òµ ¾ È and a È Ò-quotient ¼µ Ò ½µ µ ¾ È Ò (see [JK] or [Mcd] I., ex. 8). They satisfy Òµ Ò ¼Ò µ ½, and one can reconstruct from its Ò-core and Ò-quotient. Proposition (i) Õµ and Õµ can be nonzero only if and Òµ Òµ. (ii) Õµ can be nonzero only if is less or equal to for the dominance order on È. (iii) Õµ can be nonzero only if is less or equal to for the dominance order on È. Here, (i) follows from the fact that µ and µ are weight vectors for Í Õ, that is, eigenvectors for and all Ã. On the other hand µ is also a weight vector, and µ and µ have the same weight if and only if and Òµ Òµ. Properties (ii) and (iii) follow from Theorem. The polynomials Õµ and Õµ enjoy a less obvious duality property. Write Ñ Õµ Ñ Â Ñ ¼ ¼ Õµ Ñ where ¼ denotes the partition conjugate to. Theorem ([LT, LT]) For all Ñ ¾ Æ, one has Ñ Â ½ Ñ. The proof relies on the following symmetry of the bar involution ¼ ¼ Õµ Õµ ¾ ȵ (0) Theorem has been generalized by Uglov to the higher level Fock spaces [U]. 8 Relations with the Kazhdan-Lusztig polynomials 8. In this section, we fix Ö ¾ Æ and we consider only partitions with at most Ö parts. We can regard them as elements of Ö by adding at the end an appropriate string of ¼ s. The symmetric group Ë Ö acts on Ö by permuting coordinates. We denote by ËÖ the group of transformations of Ö generated by Ë Ö and the group of translations with respect to the sublattice Ò Ö. We choose as fundamental domain for this action of ËÖ on Ö the set ½ Ö µ ¾ Ö Ò ½ ¾ Ö ¼ The group ËÖ is isomorphic to the (extended) affine Weyl group of type ½µ Ö ½. Therefore, it is endowed with a length function Û Ûµ and a Bruhat order. For each Ù ¾ Ö there is a 0

11 unique Û Ù ¾ ËÖ of minimal length such that Û ½ Ùµ ¾. Given a pair Ü Ûµ of elements of Ù Ë Ö with Ü Û, we also have a Kazhdan-Lusztig polynomial È ÜÛ Õµ defined as the coefficient of Ì Ü in the expansion of the Kazhdan-Lusztig base element ¼ Û of the Hecke algebra associated with ËÖ. (Here we follow Soergel s convention [So] and normalize the generators Ì of the Hecke algebra so that Ì Õµ Ì Õ ½ µ ¼ and therefore ¼ Ì Õ.) Define Ö ½ Ö ¾ ½ ¼µ and consider two partitions and such that and Òµ Òµ. This implies that Ù and Ú belong to the same ËÖ -orbit. Let be the intersection of with this orbit, and let Ë µ be the subgroup of ËÖ consisting of the Û such that Û µ. This is in fact a Young subgroup of Ë Ö, and we denote by Û ¼ its longest element. Put Ù Û ¼ Ù, where Û ¼ is the longest element of Ë Ö, and similarly Ú Û ¼ Ú. Finally, set Û Ù Û Ù Û ¼ and Û Ú Û Ú Û ¼. Theorem ([VV]) With the above notation, we have Õµ Õµ Ü¾Ë µ Õµ ܵ È ÛÚÜÛ Ù Õµ () Ý¾Ë Ö Õµ ݵ È Ý ÛÙ Û Ú Õµ () The proof is based on the Õ-wedge realization of Õ and on the quantum affine Schur-Weyl duality of Cherednik, Ginzburg-Vasserot and Chari-Pressley. Uglov has obtained similar results for the higher level Fock spaces [U]. A formula equivalent to () was also found independently by Goodman and Wenzl [GW] in the case where is an Ò-regular partition. The theorem shows that both Õµ and Õµ are parabolic Kazhdan-Lusztig polynomials, as defined by Deodhar [D]. Using the geometric interpretation of such polynomials in terms of Schubert varieties of finite codimension in some affine flag manifold [KT], one obtains Corollary The polynomials Õµ and Õµ have non-negative coefficients. In [VV], Varagnolo and Vasserot have constructed a basis of the Fock space by projecting the canonical basis of the Hall algebra of the cyclic quiver of type ½µ Ò ½, and they conjectured that. This conjecture has been proved by Schiffmann [Sc], and it provides another proof of the positivity of Õµ. 8. As an application of Theorem, we can deduce some interesting relations between the polynomials Õµ. Given two partitions and of Ñ as in 8., we define two partitions and of Ñ ¼ Ñ Ò ½µÖ Ö ½µ in the following way. There is a unique decomposition Ò«with the partition ¼ being Ò-regular. Put ¾ Ò ½µ Û ¼ µ Ò«() Ò ½µ Ö ½µ Ò ½µ Ö ½µµ () It is easy to check that the parts of are pairwise distinct, and thus is always Ò-regular. Theorem 5 ([L]) With the notation above, we have ¼ ¼ Õµ Õ Û ¼µ Û ¼ µ Õ ½ µ (5) The proof uses Theorem and a theorem of Soergel on Kazhdan-Lusztig polynomials [So].

12 Example 6 Let Ò and ¾ ½µ. The non-zero ¼ ¼ Õµ are obtained for ¾ ½µ ½ ½µ µ ½µ We can take Ö, so that Ñ ¼ ¾ ¾ ¾½. We have ¾ ½µ ½ ¼ ¼µ ½ ¾ µ ¾ ½ ¼µ ½ ¼ ¼µ ½¾ µ Moreover, Ö ½µ is regular, hence Û ¼ µ ¼ in this case. Therefore, taking ½¼ µ ½½ µ ½¼ µ ½¾ µ respectively, we have ¼ ¾½½½½µ Õµ Õ ½¾ µ Õ ½ µ This formula shows that any vector ¼ µ of can be easily computed from the corresponding vector µ, which belongs to the basic representation of Í Õ because is Ò-regular. Hence, in a sense, Kashiwara s global basis of this irreducible submodule contains all the information about the whole basis of Õ. By taking Õ ½, we see that the decomposition matrix for the quantized Schur algebra of rank Ñ is a submatrix of the decomposition matrix for the Hecke algebra of rank Ñ ¼, a result which can also be checked directly (see [L]). This is a kind of quantization of a result of Erdmann [E] about symmetric groups and general linear groups. 9 Relations with the Kostka-Foulkes polynomials and some Õ-analogues of the Littlewood-Richardson coefficients 9. Let and be two partitions of Ñ with at most Ö parts. Lusztig has shown that the Kostka- Foulkes polynomial à յ is a Kazhdan-Lusztig polynomial of type ½µ Ö ½. With our notation, this result can be stated as Theorem 7 ([Lu, Lu]) If Ò Ö then ÒÒ Õµ Ã Õ ¾ µ. Note that Ò and Ò belong to the ËÖ -orbit of, which is regular if Ò Ö. Therefore the sum in () reduces to the single term È ÛÒ Û Ò Õµ. More generally, one can study the polynomials Ò Õµ for any and remove the restriction Ò Ö. This leads to some Õ-analogues of the Littlewood-Richardson coefficients. Recall È the operator Î Î ½ Î Ö. We have seen that at Õ ½, Î «µ Ô Ò «Write, where the are the inverse Kostka numbers. We introduce the operators Ë ¾ Ò Í ¼ Õ Õ defined by Ë Î (6) At Õ ½, Ë is the operator of multiplication by Ô Ò. By Theorem 5, the vectors µ of can be easily computed in terms of those µ for which is Ò-regular. The following theorem, which may be regarded as a Õ-analogue of the Steinberg-Lusztig tensor product theorem for Í Ú Ð Ö µ, shows that similarly the vectors «µ of can be obtained in a simple way from those µ for which the conjugate ¼ of is Ò-regular.

13 Figure 7: A -ribbon tableau of shape ½µ, weight ¾ ½ ½µ and spin Theorem 8 ([LT, LT]) Let «be a partition such that «¼ is Ò-singular. Write «Ò where ¼ is Ò-regular. Then «µ Ë µ. In particular, taking «we get Òµ Ë µ (7) Hence we see that the vectors Òµ form a canonical basis of the space of highest weight vectors of Õ, which reduces at Õ ½ to the basis of plethysms Ô Ò. Therefore, the polynomial Ò Õ ½ µ is a Õ-analogue of the scalar product Ô Ò. Now, by a theorem of Littlewood [Li], this scalar product is ¼ if the Ò-core Òµ is not empty, and if Òµ then Ô Ò Ò µ ¼µ Ò ½µ (8) where ¼µ Ò ½µ µ is the Ò-quotient of and Ò µ is its Ò-sign. It follows that Ò Õµ is a Õ-analogue of the Littlewood-Richardson coefficient ¼µ Ò ½µ ¼µ Ò ½µ We shall therefore write ¼µ Ò ½µ Õµ Ò Õµ 9. An Ò-ribbon tableau Ì of shape and weight ½ Ö µ is defined as a chain of partitions «¼ «½ «Ö such that ««½ is a horizontal Ò-ribbon strip of weight. Graphically, Ì may be described by numbering each Ò-ribbon of ««½ with the number (see Figure 7). We denote by Ì Ò µ the set of Ò-ribbon tableaux of shape and weight. The spin of a ribbon tableau Ì is defined as the sum of the spins of its ribbons. Let Ä Õµ Ì ¾Ì Ò µ Õ ÔÒ Ì µ (9) be the polynomial obtained by enumerating Ò-ribbon tableaux of shape and weight according to spin. It follows immediately from (8) and (9) that By (6) and (7) we obtain Î µ Ò Õµ ¼µ Ò ½µ Ä Õ ½ µ µ (0) Õµ Ä Õµ ()

14 Figure 8: The -ribbon tableaux of shape ¾ ½µ and dominant weight This can be seen as a combinatorial description of Ò Õµ. Unfortunately, the inverse Kostka numbers have alternating signs, and this is not a positive combinatorial description as we would like. Example 9 Let Ò, and ¾ ½µ, so that ¼µ ½µ ¾µ µ ½µ ½ ¾ µ ½µµ. The polynomials Ä ¾½µ Õµ are as follows (see Figure 8) Ä ¾½µ µ Õµ ¼ Ä ¾½µ ½µ Õµ Õ Ä ¾½µ ¾ ¾ µ Õµ Õ Õ Ä ¾½µ ¾½ ¾ µ Õµ ¾Õ ¾Õ Õ Ä ¾½µ ½ µ Õµ Õ Õ Õ Õ It follows that µ ¾½µ Õµ ½µ ½µ ½ ¾ µ ½µ Ä ¾½µ ½µ Õµ Ä ¾½µ µ Õµ Õ ¾ µ ¾½µ Õµ ¾¾ µ ½µ ½ ¾ µ ½µ Ä ¾½µ ¾ ¾ µ Õµ Ä ¾½µ ½µ Õµ Õ ¾ µ ¾½µ Õµ ¾½¾ µ ½µ ½ ¾ µ ½µ Ä ¾½µ ¾½ ¾ µ Õµ Ä ¾½µ ¾ ¾ µ Õµ Ä ¾½µ ½µ Õµ Ä ¾½µ µ Õµ Õ Õ µ ¾½µ Õµ ½ µ ½µ ½ ¾ µ ½µ Ä ¾½µ ½ µ Õµ Ä ¾½µ ¾½ ¾ µ Õµ Ä ¾½µ ¾¾ µ Õµ ¾ Ä ¾½µ ½µ Õµ Ä ¾½µ µ Õµ Õ However in the case Ò ¾, the combinatorics of domino tableaux being easier than that of general Ò-ribbon tableaux, it is possible to reduce () to a positive sum. In [CL] a special class of domino tableaux called Yamanouchi domino tableaux was introduced. The set of Yamanouchi domino tableaux of weight and shape, denoted by Ñ ¾ µ, is in one to one correspondence with the set of Littlewood-Richardson tableaux counting the multiplicity of in ¼µ ½µ, but this bijection is not straightforward (see [vl]). In terms of these, one has ¾ Õµ ¼µ ½µ Õµ ¾Ñ ¾ µ Õ ÔÒ µ ()

15 Note also that the equality Ã Õ ¾ µ Ä Ò Õµ Ò µµ () expressing the Kostka-Foulkes polynomial in terms of ribbon tableaux was first proved in [LLT] using a different method. 0 Formulas for Õµ and Õµ in some good weight spaces In Section 9 we have calculated Õµ and Õµ for some partitions and with empty Ò-core. Quite opposite to this case, one can also compute explicitely Õµ and Õµ for partitions and with certain large Ò-cores. The expression is in terms of the classical Littlewood- Richardson coefficients. For ¾, let µ be the partition containing the following parts: the first ½ integers with multiplicity Ò ½, the next ½ integers going ¾ by ¾ with multiplicity Ò ¾, the next ½ integers going by with multiplicity Ò, the next ½ integers going Ò ½ by Ò ½ with multiplicity ½. For example, if Ò and, µ ½ ¾ ½¾ ¾ ½ ¾ ½ ¾ ¾¾ ¾ ¼µ For Ò ¾, µ ½ ¾ ½µ is a staircase partition, that is, a ¾-core partition. It is easy to check that more generally, for any Ò, µ is an Ò-core partition. Theorem 0 ([LM]) Let and be partitions such that Then Òµ Òµ µ Õµ Õ Æ µ and Ò ½ ¼ «¼ «Ò ¼Ò ¼ Ò ½ µ Ò ½ where «¼ «Ò ¼ Ò ½ run through È subject to the conditions and We also have «¼ ½ µ µ µ Æ µ ¼Ò Õµ Õ µ «¾ ½ ¼ µ () µ «µ «½ µ ¼ (5) ¼ Ò ½ µ µ µ µ ¼Ò ½ µ ««½ «Ò ½ ½ µ µ µ «(6) ¼ «½ «5

16 where the sum is over all families of partitions ««¾ È ¼ Ò conditions «µ and µ ¼Ò ½ «µ µ µ µ ½µ subject to the Example Let Ò ¾ and. We have µ ¾ ½µ. The list of partitions with ¾-core ¾ ½µ and with weight µ ¾ is ½µ µ ½ µ ¾µ ½ ¾ µ ¾ ¾ ½µ ½ µ ¾ ¾ ½µ ¾ ½ µ ¾ ½ µ and the corresponding Õµ are ½µ ½ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ µ ¼ ½ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ½ µ Õ Õ ½ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¾µ ¼ ¼ ¼ ½ ¼ ¼ ¼ ¼ ¼ ¼ ½ ¾ µ ¼ Õ ¼ Õ ½ ¼ ¼ ¼ ¼ ¼ ¾ ¾ ½µ ¼ Õ ¾ Õ Õ ¾ Õ ½ ¼ ¼ ¼ ¼ ½ µ Õ ¾ Õ ¾ Õ ¼ Õ ¼ ½ ¼ ¼ ¼ ¾ ¾½µ ¼ ¼ ¼ Õ Õ ¾ Õ ¼ ½ ¼ ¼ ¾ ½ µ ¼ Õ Õ ¾ ¼ Õ ¾ Õ Õ ¼ ½ ¼ ¾½ µ Õ ¼ Õ ¾ ¼ ¼ ¼ Õ ¼ ¼ ½ These results were inspired by some Morita equivalences arising in the modular representation theory of the symmetric groups and of the finite general linear groups [CK, HM]. Note the remarkable fact that all polynomials Õµ and Õµ are just monomials in this case. Note also that they depend only on the Ò-quotients of and and not on the Ò-core µ as long as it is big enough. It turns out that these formulas can be transported to many other weight spaces by using appropriate reflections of the affine Weyl group (see [LM]). Equation (5) was also found independently by Chuang and Tan [CT] for Ò-regular. In the case Ò ¾, one recovers by putting Õ ½ some decomposition numbers for Hecke algebras previously calculated by James and Mathas [JM]. It is easy to deduce from Theorem 0 the following interesting symmetry Corollary Let and be two partitions satisfying (). Then Õµ ¼ ¼ Õµ Õµ ¼ ¼ Õµ 6

17 Comparison with the Macdonald polynomials. As mentioned in Section, the classical Fock space can be identified in a natural way to the ring of symmetric functions. Section 9 suggests yet another way of seing symmetric functions as vectors of Õ, namely by mapping the Schur function to the canonical base vector Òµ. This provides a linear embedding of ËÝÑ in Õ such that ËÝѵ is the space of highest weight vectors under the action of Í Õ. We have µ Î µ, µ Ë µ and Ô µ µ, where ½ Ö and ¾ µ is the Õ-boson operator of [KMS]. The natural scalar product on Õ is given by µ µ Æ are almost or- is orthonormal This is the scalar product with respect to which the canonical bases and thonormal in the sense of [Lu],.., that is, is orthonormal at Õ ¼ and at Õ ½. Proposition We have µ Ô µ Ô µ Æ Þ ½ ½ Õ ¾Ò ½ Õ ¾ where Þ É Ñ µ Ñ µ. Proof It is proved in [KMS] that the Õ-bosons operators satisfy the commutation rule Ð Æ Ð ½ Õ ¾Ò ½ Õ ¾ It is also known that and are adjoint to each other with respect to. It follows that where Ô µ Ô µ µ µ µ µ ½ Ö. Now, since µ ¼ for ¼, we have µ µ ½ Õ ¾Ò µ ½ Õ ¾ It follows by induction on Ö that Ö ¾Ò ½ Õ ½ µ Ö Ö ½ Õ ¾ µ and therefore ½ Õ Ö ¾Ò Ö Ö µ ÖÖ ½ Õ ¾ µ The result follows because and Ð commute when Ð. ¾ 7

18 . Put Ø Õ ¾ and Ô Õ ¾Ò Ø Ò. Then by Proposition, for any symmetric functions and we have µ µ ÔØ (7) where ÔØ is Macdonald s scalar product ([Mcd], p. 06). In other words Macdonald s scalar product for this choice of parameters can be seen as the restriction of the natural scalar product of the Fock space representation of Í Õ ÐÒ µ to its subspace of highest weight vectors. This implies, using (7) and (), that the Macdonald scalar product of two Schur functions can be expressed in terms of parabolic Kazhdan-Lusztig polynomials, or Õ-Littlewood-Richardson coefficients: Corollary ÔØ Ò Õ ½ µ Ò Õ ½ µ ¼µ Ò ½µ Õ ½ µ ¼µ Ò ½µ Õ ½ µ Let be the subring of à consisting of the rational functions without pole at Ú ½, and denote by Ä the free -module with basis µ. Let È È Ô Øµ denote the Macdonald symmetric function corresponding to our choice of parameters. Proposition 5 The images È µ of the Macdonald symmetric functions satisfy the two conditions È µ È µ È µ Òµ mod Õ ½ Ä Proof The basis È can be obtained via the Gram-Schmidt orthogonalization process applied to the basis Ñ, or equivalently to the basis. This gives È where det «ÔØ «and is obtained from by replacing by in the column. Let Ê be the subfield of à consisting of the rational functions invariant under Õ Õ ½. We have Ô Ô ÔØ ¾ Õ Ò ½µ Ê. Since is a É-linear combination of Ô s, we also have ÔØ ¾ Õ Ò ½µ Ê for any partitions and of. Hence ¾ Ê, and since µ Òµ is bar invariant, we get that È µ is bar invariant. Finally, since is almost orthonormal at Õ ½, we have ÔØ Æ mod Õ ½ Õ ½ Therefore ½ ¼ mod Õ ½ Õ ½ hence È µ Òµ mod Õ ½ ¾ as required. Note that È µ satisfies conditions very similar to the defining properties of Òµ. The only difference is that Òµ is required to belong to Ä while È µ is allowed to belong to the larger lattice Ä. On the other hand, the vectors È µ are pairwise orthogonal with respect to whereas the vectors Òµ are only orthogonal at Õ ½. The results of this section are very similar to the main results of [BFJ], but not obviously the same. Here we have used a Heisenberg algebra of intertwining operators of the Í Õ -module Õ, whereas [BFJ] uses a Heisenberg subalgebra of Í Õ coming from Drinfeld s new realization. Also, 8

19 we work in the so-called principal picture while [BFJ] considers the homogeneous picture. Finally, we obtain Macdonald s scalar product Ø Ò Ø by working with ÐÒ at level ½, while in [BFJ] the scalar product Ø ¾ Ø is expected to be associated with о at level (in [BFJ], only the case ½ is treated). This is probably another instance of the classical level-rank duality due to Frenkel [F, F]. The higher level Õ-deformed Fock spaces of Takemura and Uglov [U, TU] provide the natural setting for the quantized version of the level-rank duality, and we should probably obtain some deeper understanding by trying to interpret the Macdonald polynomials in this broader context. References [A] S. ARIKI, On the decomposition numbers of the Hecke algebra of Ò ½ ѵ, J. Math. Kyoto Univ. 6 (996), [BFJ] J. BECK, I. B. FRENKEL, N. JING, Canonical basis and Macdonald polynomials, Adv. Math. 0 (998), [CL] C. CARRÉ, B. LECLERC, Splitting the square of a Schur function into its symmetric and antisymmetric parts, J. Algebraic Combinatorics, (995), 0. [CK] J. CHUANG, R. KESSAR, Symmetric groups, wreath products, and Broué s abelian defect group conjecture, (000), Preprint. [CT] J. CHUANG, K. M. TAN, Canonical basis of the basic Í Õ Ð Òµ-module, Preprint 00. [D] V. V. DEODHAR, On some geometric aspects of Bruhat orderings II. The parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra, (987), [Dr] V. G. DRINFELD, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. (985), [E] K. ERDMANN, Decomposition numbers for symmetric groups and composition factors of Weyl modules, J. Algebra 80 (996), 6 0. [F] I. B. FRENKEL, Representations of affine Lie algebras, Hecke modular forms and Korteweg-de-Vries type equation, Lect. Notes Math., 9 (98), 7 0. [F] I. B. FRENKEL, Representations of affine Kac-Moody algebras and dual resonance models, Lectures in Appl. Math., (985), 5 5. [GW] [H] [HM] F. GOODMAN, H. WENZL, Crystal bases of quantum affine algebras and affine Kazhdan-Lusztig polynomials, Internat. Math. Res. Notices, 5 (999), T. HAYASHI, Õ-analogues of Clifford and Weyl algebras spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys. 7 (990), 9. A. HIDA, H. MIYACHI, Some blocks of finite general linear groups in non defining characteristic, (000), Preprint. [J] G. JAMES, The decomposition matrices of Ä Ò Õµ for Ò ½¼, Proc. London Math. Soc., 60 (990), [JK] G.D. JAMES, A. KERBER, The representation theory of the symmetric group, Encyclopedia of Mathematics 6, Addison- Wesley, 98. [JM] G. JAMES, A. MATHAS, Hecke algebras of type at Õ ½, J. Algebra 8 (996), [Ji] M. JIMBO, A Õ-difference analogue of Í µ and the Yang-Baxter equation, Lett. Math. Phys. 0 (985), [JiMi] M. JIMBO, T. MIWA, Solitons and infinite dimensional Lie algebras, Publ. RIMS, Kyoto Univ. 9 (98), [Ka] V. G. KAC, Infinite dimensional Lie algebras, rd Ed. Cambridge University Press, 990. [K] M. KASHIWARA, On level zero representations of quantized affine algebras, 000, math.qa/0009. [KMS] M. KASHIWARA, T. MIWA, E. STERN, Decomposition of Õ-deformed Fock spaces, Selecta Math. (995), [KMPY] M. KASHIWARA, T. MIWA, J.-U. PETERSEN, C. M. YUNG, Perfect crystals and Õ-deformed Fock spaces, Selecta Math. (996),

20 [KT] [LLT] [LLT] M. KASHIWARA, T. TANISAKI, Parabolic Kazhdan-Lusztig polynomials and Schubert varieties, 999, math.rt/ A. LASCOUX, B. LECLERC, J.-Y. THIBON, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Commun. Math. Phys. 8 (996), A. LASCOUX, B. LECLERC, J.-Y. THIBON, Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras, and unipotent varieties, J. Math. Phys. 8 (997), [L] B. LECLERC, Decomposition numbers and canonical bases, Algebras and representation theory, (000), [LM] B. LECLERC, H. MIYACHI, Some closed formulas for canonical bases of Fock spaces, 00, math.qa/0007. [LT] B. LECLERC, J.-Y. THIBON, Canonical bases of Õ-deformed Fock spaces, Int. Math. Res. Notices, 9 (996), [LT] B. LECLERC, J.-Y. THIBON, Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials, in Combinatorial methods in representation theory, Ed. M. Kashiwara et al., Adv. Stud. Pure Math. 8 (000), [Li] D. E. LITTLEWOOD, Modular representations of symmetric groups, Proc. Roy. Soc. 09 (95), 5. [Lu] G. LUSZTIG, Green polynomials and singularities of unipotent classes, Advances in Math. (98), [Lu] G. LUSZTIG, Singularities, character formulas, and a Õ-analog of weight multiplicities, Analyse et topologie sur les espaces singuliers (II-III), Astérisque 0-0 (98), [Lu] G. LUSZTIG, Introduction to quantum groups, Progress in Math. 0, Birkhauser 99. [MM] K.C. MISRA, T. MIWA, Crystal base of the basic representation of Í Õ Ð Òµ, Commun. Math. Phys. (990), [Mcd] I.G. MACDONALD, Symmetric functions and Hall polynomials, Oxford U. Press, 995. [Sc] O. SCHIFFMANN, The Hall algebra of the cyclic quiver and canonical bases of Fock spaces, Internat. Math. Res. Notices 8 (000), 0. [So] W. SOERGEL, Kazhdan-Lusztig-Polynome und eine Kombinatorik für Kipp-Moduln, Represent. Theory (997), 7 68 (english 8 ). [TU] [U] K. TAKEMURA, D. UGLOV, Representations of the quantum toroidal algebra on highest weight modules of the quantum affine algebra of type Ð Æ, Publ. RIMS, Kyoto Univ. 5 (999), D. UGLOV, Canonical bases of higher-level Õ-deformed Fock spaces and Kazhdan-Lusztig polynomials, in Physical Combinatorics Ed. M. Kashiwara, T. Miwa, Progress in Math. 9, Birkhauser 000. [vl] M. A. A. VAN LEEUWEN, Some bijective correspondences involving domino tableaux, Electron. J. Comb. 7 (000), R5, 5 p. [VV] M. VARAGNOLO, E. VASSEROT, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 00 (999), B. LECLERC : Département de Mathématiques, Université de Caen, Campus II, Bld Maréchal Juin, BP 586, 0 Caen cedex, France leclerc@math.unicaen.fr 0