Super jeu de taquin and combinatorics of super tableaux of type A

TL;DR

This paper develops a combinatorial framework for super tableaux of type A, introducing super jeu de taquin and evacuation procedures, and linking them to super Littlewood-Richardson rule via growth diagrams.

Contribution

It introduces super jeu de taquin and evacuation procedures compatible with super plactic congruence, extending classical combinatorial algorithms to super tableaux of type A.

Findings

Super jeu de taquin transforms super skew tableaux into super Young tableaux.

Super evacuation is an involution compatible with super plactic congruence.

Super jeu de taquin is described using Fomin's growth diagrams for super Littlewood-Richardson rule.

Abstract

This paper presents a combinatorial study of the super plactic monoid of type A, which is related to the representations of the general linear Lie superalgebra. We introduce the analogue of the Sch\"{u}tzenberger's jeu de taquin on the structure of super tableaux over a signed alphabet. We show that this procedure which transforms super skew tableaux into super Young tableaux is compatible with the super plactic congruence and it is confluent. We deduce properties relating the super jeu de taquin to insertion algorithms on super tableaux. Moreover, we introduce the super evacuation procedure as an involution on super tableaux and we show its compatibility with the super plactic congruence. Finally, we describe the super jeu de taquin in terms of Fomin's growth diagrams in order to give a combinatorial version of the super Littlewood--Richardson rule.