Soliton cellular automaton associated with $D_n^{(1)}$-crystal $B^{2,s}$

TL;DR

This paper investigates a soliton cellular automaton linked to the $D_n^{(1)}$-crystal $B^{2,s}$, calculating the combinatorial R matrix and revealing its scattering rule's relation to a tensor product of simpler crystals.

Contribution

It provides the first explicit calculation of the combinatorial R matrix for the $B^{2,s} imes B^{2,1}$ crystal and connects the automaton's scattering rule to a known crystal structure.

Findings

Calculated the combinatorial R matrix for all elements of $B^{2,s} imes B^{2,1}$.

Identified the automaton's scattering rule with the combinatorial R matrix for a specific crystal tensor product.

Established a link between the soliton scattering and the crystal structure of $U_q(A_1^{(1)}) imes U_q(D_{n-2}^{(1)})$.

Abstract

A solvable vertex model in ferromagnetic regime gives rise to a soliton cellular automaton which is a discrete dynamical system in which site variables take on values in a finite set. We study the scattering of a class of soliton cellular automata associated with the $U_q(D_n^{(1)})$-perfect crystal $B^{2,s}$. We calculate the combinatorial $R$ matrix for all elements of $B^{2,s} \otimes B^{2,1}$. In particular, we show that the scattering rule for our soliton cellular automaton can be identified with the combinatorial $R$ matrix for $U_q(A_1^{(1)}) \oplus U_q(D_{n-2}^{(1)})$-crystals.