Rigged Configurations and Cylindric Loop Schur Functions

TL;DR

This paper introduces cylindric loop Schur functions and provides a piecewise-linear formula, derived via tropicalization, to determine the shapes of rigged configurations, which are key to understanding soliton dynamics in box-ball systems.

Contribution

It proposes a new explicit piecewise-linear formula for rigged configurations using cylindric loop Schur functions, advancing the combinatorial understanding of box-ball systems.

Findings

Proved the conjecture for the first shape of a rigged configuration.

Derived a tropicalization-based formula for soliton lengths.

Identified cylindric loop Schur functions as invariants of the geometric R-matrix.

Abstract

Rigged configurations are known to provide action-angle variables for remarkable discrete dynamical systems known as box-ball systems. We conjecture an explicit piecewise-linear formula to obtain the shapes of a rigged configuration from a tensor product of one-row crystals. We introduce cylindric loop Schur functions and show that they are invariants of the geometric R-matrix. Our piecewise-linear formula is obtained as the tropicalization of ratios of cylindric loop Schur functions. We prove our conjecture for the first shape of a rigged configuration, thus giving a piecewise-linear formula for the lengths of the solitons of a box-ball system.