Kashiwara and Zelevinsky involutions in affine type A

Nicolas Jacon (1) , Cédric Lecouvey (2)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

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 .

Domaines: Théorie des représentations [math.RT]; Combinatoire [math.CO]; Algèbres quantiques [math.QA]

Nicolas Jacon, Cédric Lecouvey. Kashiwara and Zelevinsky involutions in affine type A. Pacific Journal of Mathematics, 2009, 243 (No.2), pp.287-311. ⟨hal-00349853v3⟩