17 research outputs found
Crystal structure on rigged configurations and the filling map
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 .Comment: 45 page
On higher level Kirillov--Reshetikhin crystals, Demazure crystals, and related uniform models
We show that a tensor product of nonexceptional type Kirillov--Reshetikhin
(KR) crystals is isomorphic to a direct sum of Demazure crystals; we do this in
the mixed level case and without the perfectness assumption, thus generalizing
a result of Naoi. We use this result to show that, given two tensor products of
such KR crystals with the same maximal weight, after removing certain
-arrows, the two connected components containing the minimal/maximal
elements are isomorphic. Based on the latter fact, we reduce a tensor product
of higher level perfect KR crystals to one of single-column KR crystals, which
allows us to use the uniform models available in the literature in the latter
case. We also use our results to give a combinatorial interpretation of the
Q-system relations. Our results are conjectured to extend to the exceptional
types.Comment: 15 pages, 1 figure; v2, incorporated changes from refere
Alcove path model for
We construct a model for using the alcove path model of Lenart
and Postnikov. We show that the continuous limit of our model recovers a dual
version of the Littelmann path model for given by Li and Zhang.
Furthermore, we consider the dual version of the alcove path model and obtain
analogous results for the dual model, where the continuous limit gives the Li
and Zhang model.Comment: 19 pages, 7 figures; improvements from comments, added more figure
Connecting marginally large tableaux and rigged configurations via crystals
We show that the bijection from rigged configurations to tensor products of
Kirillov-Reshetikhin crystals extends to a crystal isomorphism between the
models given by rigged configurations and marginally large
tableaux.Comment: 22 pages, 3 figure
Rigged configurations of type and the filling map
International audienceWe give a statistic preserving bijection from rigged configurations to a tensor product of Kirillov–Reshetikhin crystals in type by using virtualization into type . We consider a special case of this bijection with , and we obtain the so-called Kirillov–Reshetikhin tableaux model for the Kirillov–Reshetikhin crystal.Nous donnons une bijection prservant les statistiques entre les configurations gréées et les produits tensoriels de cristaux de Kirillov–Reshetikhin de type , via une virtualisation en type . Nous considérons un cas particulier de cette bijection pour et obtenons ainsi les modèles de tableaux appelés Kirillov–Reshetikhin pour le cristal Kirillov–Reshetikhin
A rigged configuration model for
We describe a combinatorial realization of the crystals and
using rigged configurations in all symmetrizable Kac-Moody types
up to certain conditions. This includes all simply-laced types and all
non-simply-laced finite and affine types
Rigged configuration bijection and proof of the conjecture for nonexceptional affine types
We establish a bijection between rigged configurations and highest weight
elements of a tensor product of Kirillov-Reshetikhin crystals for all
nonexceptional types. A key idea for the proof is to embed both objects into
bigger sets for simply-laced types or , whose bijections
have already been established. As a consequence we settle the conjecture
in full generality for nonexceptional types. Furthermore, the bijection extends
to a classical crystal isomorphism and sends the combinatorial -matrix to
the identity map on rigged configurations.Comment: 30 pages, 2 figures; v2 Referenced Naoi's work in the introduction,
clarified some notation; v3 Various additions for more self-containment
(e.g., the signature rule) and typos fixe
Rigged configurations and the -involution for generalized Kac--Moody algebras
We construct a uniform model for highest weight crystals and for
generalized Kac--Moody algebras using rigged configurations. We also show an
explicit description of the -involution on rigged configurations for
: that the -involution interchanges the rigging and the
corigging. We do this by giving a recognition theorem for using the
-involution. As a consequence, we also characterize as a
subcrystal of using the -involution. We show that the
category of highest weight crystals for generalized Kac--Moody algebras is a
coboundary category by extending the definition of the crystal commutor using
the -involution due to Kamnitzer and Tingley.Comment: 23 pages, 1 figur