Promotion operator on rigged configurations of type A

TL;DR

This paper proves a conjecture that the promotion operator on rigged configurations of type A aligns with the crystal structure, establishing an affine crystal isomorphism via a generalized bijection.

Contribution

It provides a proof that the promotion operator on rigged configurations corresponds to an affine crystal isomorphism, confirming a key conjecture in the field.

Findings

The promotion operator on rigged configurations is an affine crystal isomorphism.

The bijection between crystals and rigged configurations preserves crystal structure.

The proof confirms the conjecture for type A rigged configurations.

Abstract

Recently, the analogue of the promotion operator on crystals of type A under a generalization of the bijection of Kerov, Kirillov and Reshetikhin between crystals (or Littlewood--Richardson tableaux) and rigged configurations was proposed. In this paper, we give a proof of this conjecture. This shows in particular that the bijection between tensor products of type A_n^{(1)} crystals and (unrestricted) rigged configurations is an affine crystal isomorphism.