Linearization of the box-ball system: an elementary approach

TL;DR

This paper presents an elementary approach to linearizing the box-ball system using rigged configurations, extending the method to systems with finite carriers and connecting to discrete integrable equations.

Contribution

It introduces a simple understanding of rigged configurations for rak{sl}_2-type and provides an elementary proof of the linearization property, applicable to finite carrier systems and related algebraic structures.

Findings

Elementary proof of linearization for rak{sl}_2-type box-ball systems

Extension of linearization to systems with finite carriers

New fermionic formula for finite carrier case

Abstract

Kuniba, Okado, Takagi and Yamada have found that the time-evolution of the Takahashi-Satsuma box-ball system can be linearized by considering rigged configurations associated with states of the box-ball system. We introduce a simple way to understand the rigged configuration of $\mathfrak{sl}_2$-type, and give an elementary proof of the linearization property. Our approach can be applied to a box-ball system with finite carrier, which is related to a discrete modified KdV equation, and also to the combinatorial $R$-matrix of $A_1^{(1)}$-type. We also discuss combinatorial statistics and related fermionic formulas associated with the states of the box-ball systems. A fermionic-type formula we obtain for the finite carrier case seems to be new.