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Communications in Mathematical Physics

, Volume 182, Issue 2, pp 469–483 | Cite as

Schur duality in the toroidal setting

  • M. Varagnolo
  • E. Vasserot
Article

Abstract

The classical Frobenius-Schur duality gives a correspondence between finite dimensional representations of the symmetric and the linear groups. The goal of the present paper is to extend this construction to the quantum toroidal setup with only elementary (algebraic) methods. This work can be seen as a continuation of [J, D1 and C2] (see also [C-P and G-R-V]) where the cases of the quantum groups U q (sl(n)), Y(sl(n)) (the Yangian) and U q (sl(n)) are given. In the toroidal setting the two algebras involved are deformations of Cherednik's double affine Hecke algebra introduced in [C1] and of the quantum toroidal group as given in [G-K-V]. Indeed, one should keep in mind the geometrical construction in [G-R-V] and [G-K-V] in terms of equivariant K-theory of some flag manifolds. A similar K-theoretic construction of Cherednik's algebra has motivated the present work. At last, we would like to lay emphasis on the fact that, contrary to [J, D1 and C2], the representations involved in our duality are infinite dimensional. Of course, in the classical case, i.e.,q=1, a similar duality holds between the toroidal Lie algebra and the toroidal version of the symmetric group.

The authors would like to thank V. Ginzburg for a useful remark on a preceding version of this paper.

Keywords

Quantum Group Braid Group Finite Dimensional Representation Flag Manifold Universal Central Extension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • M. Varagnolo
    • 1
  • E. Vasserot
    • 2
  1. 1.Dipartimento di MatematicaRomaItaly
  2. 2.Université de Cergy-PontoiseCergy-PontoiseFrance

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