Soliton cellular automaton associated with $G_2^{(1)}$ crystal base

TL;DR

This paper computes the combinatorial R matrix for a G_2^{(1)}-based crystal and explores the resulting soliton cellular automaton, revealing its scattering behavior and soliton identification with A_1^{(1)}-crystals.

Contribution

It introduces a new soliton cellular automaton associated with G_2^{(1)}-crystals and characterizes its scattering rules using combinatorial R matrices.

Findings

Solitons of length l correspond to A_1^{(1)}-crystals _{3l}

The scattering rule matches the combinatorial R matrix for A_1^{(1)}-crystals

Explicit calculation of the combinatorial R matrix for _l _1

Abstract

We calculate the combinatorial $R$ matrix for all elements of $\mathcal{B}_l\otimes \mathcal{B}_1$ where $\mathcal{B}_l$ denotes the $G_2^{(1)}$-perfect crystal of level $l$, and then study the soliton cellular automaton constructed from it. The solitons of length $l$ are identified with elements of the $A_1^{(1)}$-crystal $\tilde{\mathcal{B}}_{3l}$. The scattering rule for our soliton cellular automaton is identified with the combinatorial $R$ matrix for $A_1^{(1)}$-crystals.