Kirillov--Schilling--Shimozono bijection as energy functions of crystals

TL;DR

This paper reformulates the Kirillov--Schilling--Shimozono bijection algebraically using crystal energy functions, providing an intrinsic characterization within tensor products of crystals.

Contribution

It introduces an algebraic reformulation of the KSS bijection via combinatorial R and energy functions, enhancing understanding of its intrinsic crystal structure.

Findings

Provides an algebraic description of the KSS bijection.

Characterizes the KSS map as an intrinsic property of crystal tensor products.

Connects the bijection to energy functions in crystal theory.

Abstract

The Kirillov--Schilling--Shimozono (KSS) bijection appearing in theory of the Fermionic formula gives an one to one correspondence between the set of elements of tensor products of the Kirillov--Reshetikhin crystals (called paths) and the set of rigged configurations. It is a generalization of Kerov--Kirillov--Reshetikhin bijection and plays inverse scattering formalism for the box-ball systems. In this paper, we give an algebraic reformulation of the KSS map from the paths to rigged configurations, using the combinatorial R and energy functions of crystals. It gives a characterization of the KSS bijection as an intrinsic property of tensor products of crystals.