Schur duality in the toroidal setting |
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Authors: | M Varagnolo E Vasserot |
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Institution: | (1) Dipartimento di Matematica, via della Ricerca Scientifica, 00133 Roma, Italy;(2) Université de Cergy-Pontoise, 2 avenue A. Chauvin, Pontoise, 95302 Cergy-Pontoise, France |
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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.
Communicated by M. Jimbo |
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Keywords: | |
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