Quantum Weyl Reciprocity and Tilting Modules
نویسندگان
چکیده
Quantum Weyl reciprocity relates the representation theory of Hecke algebras of type A with that of q-Schur algebras. This paper establishes that Weyl reciprocity holds integrally (i. e., over the ringZ[q;q ] of Laurent polynomials) and that it behaves well under base-change. A key ingredient in our approach involves the theory of tilting modules for q-Schur algebras. New results obtained in that direction include an explicit determination of the Ringel dual algebra of a q-Schur algebra in all cases. In particular, in the most interesting situation, the Ringel dual identi es with a natural quotient algebra of the Hecke algebra. Weyl reciprocity refers to the connection between the representation theories of the general linear group GLn(k) and the symmetric group Sr . Let V be a vector space (over a eld k) of dimension n and form the tensor space V . The natural (left) action of GLn(k) on V r commutes with the (right) permutation action of Sr. Let A (resp., R) be the algebra generated by the image of GLn(k) (resp., Sr) in the algebra End(V ) of linear operators on V . Classically [We], when k = C , these algebras satisfy the double centralizer property (1) a) A = EndR(V ) and b) R = EndA(V ): Further, the set (n; r) of partitions of r into at most n nonzero parts indexes both the irreducibleA-modules L( ) and the irreducibleR-modules S . The L( ) are the irreducible polynomial representations of GLn(C ) of homogeneous degree r, while the S are Specht modules for Sr. Weyl reciprocity also entails the decomposition
منابع مشابه
Rigidity of Tilting Modules
Let Uq denote the quantum group associated with a finite dimensional semisimple Lie algebra. Assume that q is a complex root of unity of odd order and that Uq is obtained via Lusztig’s q-divided powers construction. We prove that all regular projective (tilting) modules for Uq are rigid, i.e., have identical radical and socle filtrations. Moreover, we obtain the same for a large class of Weyl m...
متن کاملTilting Modules for Cyclotomic Schur Algebras
This paper investigates the tilting modules of the cyclotomic q–Schur algebras, the Young modules of the Ariki–Koike algebras, and the interconnections between them. The main tools used to understand the tilting modules are contragredient duality, and the Specht filtrations and dual Specht filtrations of certain permutation modules. Surprisingly, Weyl filtrations — which are in general more pow...
متن کاملTensor Ideals in the Category of Tilting Modules
Let g be a complex finite dimensional simple Lie algebra with the root datum (Y,X, . . . ), see [Lu2]. Let Wf denote the Weyl group, R denote the root system, R+ denote the set of positive roots. Let X+ denote the set of dominant integral weights. Let h denote the Coxeter number of g. Let us fix l ∈ N, l > h. We assume that l is odd (and not divisible by 3, if g is of type G2). Let W denote the...
متن کاملQuantum Weyl Reciprocity for Cohomology
Classical Weyl reciprocity relates the representation theories of the general linear group GLnðkÞ and the symmetric group Sr by means of the ðGLnðkÞ;SrÞ-bimodule T :1⁄4 V , where V 1⁄4 k. For example, when k 1⁄4 C, the poset 0þðn; rÞ of partitions of r with at most n non-zero parts indexes the irreducible polynomial GLnðCÞ-modules Lð Þ, which are homogeneous of degree r, and a subset of the com...
متن کاملDimensions of Quantized Tilting Modules
We will follow the notations of [7]. Let (Y,X, . . . ) be a simply connected root datum of finite type. Let p be a prime number bigger than the Coxeter number h. Let ζ be a primitive p-th root of unity in C. Let U be the quantum group with divided powers associated to these data. Let T be the category of tilting modules over U , see e.g. [1]. Recall that any tilting module is a sum of indecompo...
متن کامل