Separation of colour degree of freedom from dynamics in a soliton cellular automaton

TL;DR

This paper introduces an algorithm that isolates the colour degrees of freedom in a coloured box-ball cellular automaton, simplifying it into a monochrome system while preserving key conserved quantities.

Contribution

The novel algorithm leverages crystal basis theory to extract and encode colour information as a conserved word, advancing understanding of integrable cellular automata.

Findings

The algorithm successfully reduces the coloured system to a monochrome one.

It identifies a conserved word representing colour degrees of freedom.

The method is based on tensor products of sl_n crystal bases.

Abstract

We present an algorithm to reduce the coloured box-ball system, a one dimensional integrable cellular automaton described by motions of several colour (kind) of balls, into a simpler monochrome system. This algorithm extracts the colour degree of freedom of the automaton as a word which turns out to be a conserved quantity of this dynamical system. It is based on the theory of crystal basis and in particular on the tensor products of sl_n crystals of symmetric and anti-symmetric tensor representations.