Uniform description of the rigged configuration bijection

We give a uniform description of the bijection $\Phi$ from rigged configurations to tensor products of Kirillov--Reshetikhin crystals of the form $\bigotimes_{i=1}^N B^{r_i,1}$ in dual untwisted types: simply-laced types and types $A_{2n-1}^{(2)}$, $D_{n+1}^{(2)}$, $E_6^{(2)}$, and $D_4^{(3)}$. We give a uniform proof that $\Phi$ is a bijection and preserves statistics. We describe $\Phi$ uniformly using virtual crystals for all remaining types, but our proofs are type-specific. We also give a uniform proof that $\Phi$ is a bijection for $\bigotimes_{i=1}^N B^{r_i,s_i}$ when $r_i$, for all $i$, map to $0$ under an automorphism of the Dynkin diagram. Furthermore, we give a description of the Kirillov--Reshetikhin crystals $B^{r,1}$ using tableaux of a fixed height $k_r$ depending on $r$ in all affine types. Additionally, we are able to describe crystals $B^{r,s}$ using $k_r \times s$ shaped tableaux that are conjecturally the crystal basis for Kirillov--Reshetikhin modules for various nodes $r$.Comment: 60 pages, 5 figures, 3 tables; v2 incorporated changes from refere

A crystal to rigged configuration bijection and the filling map for type D4(3)D_4^{(3)}

We give a bijection $\Phi$ from rigged configurations to a tensor product of Kirillov--Reshetikhin crystals of the form $B^{r,1}$ and $B^{1,s}$ in type $D_4^{(3)}$. We show that the cocharge statistic is sent to the energy statistic for tensor products $\bigotimes_{i=1}^N B^{r_i,1}$ and $\bigotimes_{i=1}^N B^{1,s_i}$. We extend this bijection to a single $B^{r,s}$, show that it preserves statistics, and obtain the so-called Kirillov--Reshetikhin tableaux model for $B^{r,s}$. Additionally, we show that $\Phi$ commutes with the virtualization map and that $B^{1,s}$ is naturally a virtual crystal in type $D_4^{(1)}$, thus defining an affine crystal structure on rigged configurations corresponding to $B^{1,s}$.Comment: 40 pages, 6 figures; various revisions from referee comments and fixed minor typo

Existence of Kirillov-Reshetikhin crystals for near adjoint nodes in exceptional types

We prove that, in types $E_{6,7,8}^{(1)}$, $F_4^{(1)}$ and $E_6^{(2)}$, every Kirillov--Reshetikhin module associated with the node adjacent to the adjoint one (near adjoint node) has a crystal pseudobase, by applying the criterion introduced by Kang et.al. In order to apply the criterion, we need to prove some statements concerning values of a bilinear form. We achieve this by using the global bases of extremal weight modules.Comment: 39 pages. In version 1 the main theorem is proved in type $E_6^{(1)}$ only, but in version 2 and 3 we generalize the result in types $E_{6,7,8}^{(1)}$, $F_4^{(1)}$ and $E_6^{(2)}$. In version 3, some minor corrections are mad

Kirillov-Reshetikhin crystals B1,sB^{1,s} for sl^n\widehat{\mathfrak{sl}}_n using Nakajima monomials

We give a realization of the Kirillov--Reshetikhin crystal $B^{1,s}$ using Nakajima monomials for $\widehat{\mathfrak{sl}}_n$ using the crystal structure given by Kashiwara. We describe the tensor product $\bigotimes_{i=1}^N B^{1,s_i}$ in terms of a shift of indices, allowing us to recover the Kyoto path model. Additionally, we give a model for the KR crystals $B^{r,1}$ using Nakajima monomials.Comment: 24 pages, 6 figures; v2 improved introduction, added more figures, and other misc improvements; v3 changes from referee report

Alcove path model for B(∞)B(\infty)

We construct a model for $B(\infty)$ 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 $B(\infty)$ 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 $B(\infty)$ models given by rigged configurations and marginally large tableaux.Comment: 22 pages, 3 figure

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 $0$-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

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 $D_n^{(1)}$.Comment: 45 page

K-theoretic crystals for set-valued tableaux of rectangular shapes

In earlier work with C. Monical (2018), we introduced the notion of a K-crystal, with applications to K-theoretic Schubert calculus and the study of Lascoux polynomials. We conjectured that such a K-crystal structure existed on the set of semistandard set-valued tableaux of any fixed rectangular shape. Here, we establish this conjecture by explicitly constructing the K-crystal operators. As a consequence, we establish the first combinatorial formula for Lascoux polynomials $L_{w\lambda}$ when $\lambda$ is a multiple of a fundamental weight as the sum over flagged set-valued tableaux. Using this result, we then prove corresponding cases of conjectures of Ross-Yong (2015) and Monical (2016) by constructing bijections with the respective combinatorial objects.Comment: 20 pages, 2 figures; changed the statement of Conjecture 6.