Crystal Structure on Rigged Configurations and the Filling Map

Abstract

In this paper, we extend work of the first author on a crystal structure on rigged configurations of simply-laced type to all non-exceptional affine types using the technology of virtual rigged configurations and crystals. Under the bijection between rigged configurations and tensor products of Kirillov-Reshetikhin crystals specialized to a single tensor factor, we obtain a new tableaux model for Kirillov-Reshetikhin crystals. This is related to the model in terms of Kashiwara-Nakashima tableaux via a filling map, generalizing the recently discovered filling map in type D (1) n .Again let B 1 and B 2 be abstract U q (g)-crystals. A crystal morphism ψ : B 1 −→ B 2 is a map

On higher level Kirillov–Reshetikhin crystals, Demazure crystals, and related uniform models

“…We expect the proof to be completely similar, based on the generalizations of Theorems 3.2 and 3.8 that were conjectured in Section 3. We have verified this conjecture for B r,s for s = 2, 3, 4 in types D One particular approach to Problem 4.3 could be through the use of the so-called Kirillov-Reshetikhin (KR) tableaux of [OSS13,SS15]. These arise from the bijection Φ with rigged configurations, which use column splitting to construct classical crystal embeddings as a core part of the bijection Φ.…”

“…(1) n [SS15]. Also in the nonexceptional case, a type-specific construction, based on virtual crystals and tableaux, is found in [FOS09].…”

“…A partially uniform model for B r,1 using Nakajima monomials was given by Hernandez and Nakajima [HN06]. A uniform model for the classical crystal structure of B r,s was given in terms of rigged configurations in [Kle98,OSS03,Sch06,SS15]. However, the affine crystal structure on rigged configurations has currently only been explicitly constructed for type A (1) n [SW10], D…”

“…How is the isomorphism in Corollary 4.1 expressed concretely when the corresponding tensor products of KR crystals are realized based on the tableau model[FOS09] and the rigged configuration model[SS15]?…”

Virtual Crystals and Nakajima Monomials

“…In terms of Kashiwara-Nakashima tableaux [32], which only is valid for classical types, 1 a virtualization map has been explicitly described in [1,71]. However, it is a little complicated with specific rules depending on the type.…”