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Crystal Isomorphisms in Fock Spaces and Schensted Correspondence in Affine Type A

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Abstract

We are interested in the structure of the crystal graph of level l Fock spaces representations of \(\mathcal{U}_{q}^{\prime} (\widehat{\mathfrak{s}\mathfrak{l}_{e}}) \). Since the work of Shan (Ann. Sci. Éc Norm. Supér. 44:147–182, 2011), we know that this graph encodes the branching rule for a corresponding cyclotomic rational Cherednik algebra. Besides, it appears to be closely related to the Harish-Chandra branching graph for the appropriate finite unitary group, according to [8]. In this paper, we make explicit a particular isomorphism between connected components of the crystal graphs of Fock spaces. This so-called “canonical” crystal isomorphism turns out to be expressible only in terms of:

  • Schensted’s classic bumping procedure,

  • the cyclage isomorphism defined in Jacon and Lecouvey (Algebras and Representation Theory 13:467–489, 2010),

  • a new crystal isomorphism, easy to describe, acting on cylindric multipartitions.

We explain how this can be seen as an analogue of the bumping algorithm for affine type A. Moreover, it yields a combinatorial characterisation of the vertices of any connected component of the crystal of the Fock space.

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  1. Lehrstuhl D für Mathematik, RWTH., Pontdriesch 14-16, 52062, Aachen, Germany

    Thomas Gerber

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Gerber, T. Crystal Isomorphisms in Fock Spaces and Schensted Correspondence in Affine Type A . Algebr Represent Theor 18, 1009–1046 (2015). https://doi.org/10.1007/s10468-015-9530-2

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