String of columns rewriting and confluence of the jeu de taquin

TL;DR

This paper explores the jeu de taquin algorithm on tableaux, introducing a string of columns rewriting system to analyze transformations, and relates it to insertion algorithms and algebraic properties.

Contribution

It introduces a novel string of columns rewriting system to model jeu de taquin transformations and connects it with insertion algorithms and algebraic structures.

Findings

Rewrites jeu de taquin as paths in a string rewriting system

Establishes algebraic properties of the plastic congruence

Links jeu de taquin to insertion algorithms on tableaux

Abstract

Sch\"utzenberger's jeu de taquin is an algorithm on the structure of tableaux, which transforms a skew tableau into a Young one by local transformation rules on the columns of the tableaux. This algorithm defines an equivalence relation on tableaux compatible with the plactic congruence, and gives a proof of the Littlewood-Richardson rule on Schur polynomials. In this article, we introduce the notion of string of columns rewriting system as mechanism of transformations of glued sequences of columns. We describe the execution of the jeu de taquin algorithm as rewriting paths of a string of columns rewriting. We deduce algebraic properties on the plastic congruence and we relate the jeu de taquin to insertion algorithms on tableaux.