Kashiwara and Zelevinsky involutions in affine type A

Nicolas Jacon and Cédric Lecouvey

Vol. 243 (2009), No. 2, 287–311

DOI: 10.2140/pjm.2009.243.287

Abstract

We describe how the Kashiwara involution ∗ on crystals of affine type A is encoded by the combinatorics of aperiodic multisegments. This affords an elementary proof that ∗ coincides with the Zelevinsky involution τ on the set of simple modules for the affine Hecke algebras. We then give efficient procedures for computing ∗ and τ. Remarkably, these procedures do not use the underlying crystal structure. They also allow one to explicitly match to each other the Ginzburg and Ariki parametrizations of simple modules associated to affine and cyclotomic Hecke algebras, respectively.

Keywords

affine Hecke algebra, Zelevinsky involution, crystals, Kashiwara involution

Mathematical Subject Classification 2000

Primary: 20C08, 20C20, 81R50

Milestones

Received: 1 December 2008

Revised: 22 April 2009

Accepted: 24 April 2009

Published: 1 December 2009