A promotion operator on rigged configurations

Rigged configurations appear in the Bethe Ansatz study of exactly solvable lattice models as combinatorial objects to index the solutions of the Bethe equations [5, 6]. Based on work by Kerov, Kirillov and Reshetikhin [5, 6], it was shown in [7] that there is a statistic preserving bijection Φ between LittlewoodRichardson tableaux and rigged configurations. The description of the bijection Φ is based on a quite technical recursive algorithm. Littlewood-Richardson tableaux can be viewed as highest weight crystal elements in a tensor product of Kirillov–Reshetikhin (KR) crystals of type A n . KR crystals are affine finite-dimensional crystals corresponding to affine Kac–Moody algebras, in the setting of [7] of type A n . The highest weight condition is with respect to the finite subalgebra An. The bijection Φ can be generalized by dropping the highest weight requirement on the elements in the KR crystals [1], yielding the set of crystal paths P . On the corresponding set of unrestricted rigged configurations RC, the An crystal structure is known explicitly [12]. One of the remaining open questions is to define the full affine crystal structure on the level of rigged configurations. Given the affine crystal structure on both sides, the bijection Φ has a much more conceptual interpretation as an affine crystal isomorphism. In type A n , the affine crystal structure can be defined using the promotion operator pr, which corresponds to the Dynkin diagram automorphism mapping node i to i + 1 modulo n + 1. On crystals, the promotion operator is defined using jeu-de-taquin [13, 15]. In [12], Schilling proposed an algorithm pr on RC and conjectured [12, Conjecture 4.12] that pr corresponds to the promotion operator pr under the bijection Φ. Several necessary conditions of promotion operators were established and it was shown that in special cases pr is the correct promotion operator.