Rigged configurations as tropicalizations of loop Schur functions

TL;DR

This paper proposes a conjectural explicit formula connecting rigged configurations with tropicalized loop Schur functions, providing insights into their combinatorial structure and dynamics in integrable systems.

Contribution

It introduces a conjectured piecewise-linear formula for the Kirillov-Schilling-Shimozono bijection involving tropicalizations of cylindric and loop Schur functions, with partial proofs.

Findings

The formula correctly accounts for rigging changes during box-ball system evolution.

It is verified for column splitting cases and for crystals B^{1, s}.

The approach links combinatorics with tropical geometry in integrable models.

Abstract

We conjecture an explicit formula for the image of a tensor product of Kirillov-Reshetikhin crystals $\bigotimes_{i=1}^m B^{1, s_i}$ under the Kirillov-Schilling-Shimozono bijection. Our conjectured formula is piecewise-linear, where the shapes are given by the tropicalization of cylindric loop Schur functions and the riggings are given by the tropicalization of loop Schur functions. We prove that our formula changes the riggings by the correct amount based upon the time evolution of the corresponding box-ball system. We show that our formula is correct under the column splitting portion of the Kirillov-Schilling-Shimozono bijection and for $B^{1, s}$.