Kashiwara and Zelevinsky involutions in affine type A

Details Kashiwara and Zelevinsky involutions in affine type A Kashiwara and Zelevinsky involutions in affine type A

2009-01-05

arxiv.org/abs/0901.0443v3

Mathematics

Representation Theory

24 pages, to appear in Pacific J. Math

Scientific paper

We first describe how the Kashiwara involution on crystals of affine type $A$ is encoded by the combinatorics of aperiodic multisegments. This yields a simple relation between this involution and the Zelevinsky involution on the set of simple modules for the affine Hecke algebras. We then give efficient procedures for computing these involutions. Remarkably, these procedures do not use the underlying crystal structure. They also permit to match explicitly the Ginzburg and Ariki parametrizations of the simple modules associated to affine and cyclotomic Hecke algebras, respectively .

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