Geometric lifting of the integrable cellular automata with periodic boundary conditions

TL;DR

This paper introduces a new class of discrete integrable systems as a geometric lifting of periodic box-ball systems, using a novel Lax representation and proving the existence of unique solutions for periodic boundary conditions.

Contribution

It constructs a family of integrable systems based on geometric lifting, extending the periodic box-ball systems with guaranteed unique solutions and commuting time evolutions.

Findings

Existence of unique solutions for the carrier equation with periodic boundary conditions.

Construction of a commuting family of time evolutions for all states.

Extension of box-ball systems through geometric lifting and tropicalization.

Abstract

Inspired by G. Frieden's recent work on the geometric R-matrix for affine type A crystal associated with rectangular shaped Young tableaux, we propose a method to construct a novel family of discrete integrable systems which can be regarded as a geometric lifting of the generalized periodic box-ball systems. By converting the conventional usage of the matrices for defining the Lax representation of the discrete periodic Toda chain, together with a clever use of the Perron-Frobenious theorem, we give a definition of our systems. It is carried out on the space of real positive dependent variables, without regarding them to be written by subtraction-free rational functions of independent variables but nevertheless with the conserved quantities which can be tropicalized. We prove that, in this setup an equation of an analogue of the `carrier' of the box-ball system for assuring its periodic boundary condition always has a unique solution. As a result, any states in our systems admit a commuting family of time evolutions associated with any rectangular shaped tableaux, in contrast to the case of corresponding generalized periodic box-ball systems where some states did not admit some of such time evolutions.