Jeu de taquin, uniqueness of rectification, and ultradiscrete KP

TL;DR

This paper explores the tropical and integrable system aspects of the rectification algorithm on skew Young tableaux, revealing new combinatorial insights and connections to ultradiscrete KP equations.

Contribution

It introduces a new combinatorial map equivalent to the rectification algorithm, simplifying the proof of its uniqueness and linking it to tropical integrable systems.

Findings

The rectification algorithm can be interpreted as a tropical integrable system evolution.

A new combinatorial map simplifies understanding of rectification properties.

The uniqueness of rectification is reduced to a straightforward combinatorial problem.

Abstract

In this paper, we study tropical-theoretic aspects of the ``rectification algorithm'' on skew Young tableaux. It is shown that the algorithm is interpreted as a time evolution of some tropical integrable system. By using this fact, we construct a new combinatorial map that is essentially equivalent to the rectification algorithm. Some of properties of the rectification can be seen more clearly via this map. For example, the uniqueness of a rectification boils down to an easy combinatorial problem. Our method is mainly based on the two previous researches: the theory of geometric tableaux by Noumi-Yamada, and the study on the relationship between jeu de taquin slides and the ultradiscrete KP equation by Mikami and Katayama-Kakei.