Characterization of ${\cal B}(\infty)$ using marginally large tableaux and rigged configurations in the $A_n$ case via integer sequences

TL;DR

This paper introduces cascading sequences to characterize marginally large tableaux and establishes an explicit bijection with rigged configurations for the crystal ${\cal B}(\infty)$ in the $A_n$ case, advancing combinatorial understanding.

Contribution

It provides a new characterization of marginally large tableaux and rigged configurations via cascading sequences, and constructs an explicit bijection between them.

Findings

Cascading sequences effectively characterize marginally large tableaux.

An explicit bijection between marginally large tableaux and rigged configurations is established.

The approach enhances combinatorial understanding of crystal structures in type $A_n$.

Abstract

Rigged configurations are combinatorial objects prominent in the study of solvable lattice models. Marginally large tableaux are semi-standard Young tableaux of special form that give a realization of the crystals ${\cal B}(\infty)$. We introduce cascading sequences to characterize marginally large tableaux. Then we use cascading sequences and a non-explicit crystal isomorphism between marginally large tableaux and rigged configurations to give a characterization of the latter set, and to give an explicit bijection between the two sets.