RSK tableaux and box-ball systems

TL;DR

This paper explores the relationship between RSK tableaux and the dynamics of box-ball systems, establishing conditions under which their soliton decompositions align with Robinson-Schensted insertion tableaux, using combinatorial and algebraic tools.

Contribution

It proves that under certain conditions, the soliton decomposition of a permutation matches its RS insertion tableau, linking dynamical systems with combinatorial tableau theory.

Findings

When the soliton decomposition is a standard tableau, it equals the RS insertion tableau.

If the soliton shape matches the RS partition, the decompositions coincide.

The study employs row reading words, Knuth moves, and Greene's theorem to analyze properties.

Abstract

A box-ball system is a discrete dynamical system whose dynamics come from the balls jumping according to certain rules. A permutation on n objects gives a box-ball system state by assigning its one-line notation to n consecutive boxes. After a finite number of steps, a box-ball system will reach a steady state. From any steady state, we can construct a tableau called the soliton decomposition of the box-ball system. We prove that if the soliton decomposition of a permutation w is a standard tableau or if its shape coincides with the Robinson-Schensted (RS) partition of w, then the soliton decomposition of w and the RS insertion tableau of w are equal. We also use row reading words, Knuth moves, RS recording tableaux, and a localized version of Greene's theorem (proven recently by Lewis, Lyu, Pylyavskyy, and Sen) to study various properties of a box-ball system.