Sách Promotion operator on rigged configurations of type A

1 Introduction
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 bi-
jection Φ between Littlewood-Richardson 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 ten-
sor product of Kirillov–Reshetikhin (KR) crystals of type A
(1)
n . KR crystals are affine finite-
dimensional crystals corresponding to affine Kac–Moody algebras, in the setting of [7] of type
A
(1)
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 [14]. One of the remaining
open questions is to define the full affine crystal structure on the level of rigged configurations.