Linearization of the box-ball system: an elementary approach

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.

30 pages, 16 figures; v2: Section 2 reorganized

References in corpus (3)

• Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection
• Bethe ansatz and inverse scattering transform in a periodic box-ball system
• Rigged Configurations and Catalan, Stretched Parabolic Kostka Numbers and Polynomials: Polynomiality, Unimodality and Log-concavity