Affine crystal structure on rigged configurations of type $D_{n}^{(1)}$

@article{Okado2011AffineCS,
  title={Affine crystal structure on rigged configurations of type \$D\_\{n\}^\{(1)\}\$},
  author={Masato Okado and Reiho Sakamoto and Anne Schilling},
  journal={Journal of Algebraic Combinatorics},
  year={2011},
  volume={37},
  pages={571-599}
}
Extending the work in Schilling (Int. Math. Res. Not. 2006:97376, 2006), we introduce the affine crystal action on rigged configurations which is isomorphic to the Kirillov–Reshetikhin crystal Br,s of type $D_{n}^{(1)}$ for any r,s. We also introduce a representation of Br,s (r≠n−1,n) in terms of tableaux of rectangular shape r×s, which we coin Kirillov–Reshetikhin tableaux (using a nontrivial analogue of the type A column splitting procedure) to construct a bijection between elements of a… 
Rigged configurations of type $D_4^{(3)}$ and the filling map
We give a statistic preserving bijection from rigged configurations to a tensor product of Kirillov–Reshetikhin crystals $\otimes_{i=1}^{N}B^{1,s_i}$ in type $D_4^{(3)}$ by using virtualization into
Crystal Structure on Rigged Configurations and the Filling Map
TLDR
Under the bijection between rigged configurations and tensor products of Kirillov-Reshetikhin crystals specialized to a single tensor factor, a new tableaux model is obtained for Kirillovo-Reshevikin crystals.
Type $${{\varvec{D}}}_{{\varvec{n}}}^\mathbf{(1)}$$Dn(1) rigged configuration bijection
We establish a bijection between the set of rigged configurations and the set of tensor products of Kirillov–Reshetikhin crystals of type $$D^{(1)}_n$$Dn(1) in full generality. We prove the
An Explicit Algorithm of Rigged Configuration Bijection for the Adjoint Crystal of Type $G_{2}^{(1)}$
We construct an explicit algorithm of the static-preserving bijection between the rigged configurations and the highest weight paths of the form (B2,1)⊗L in the G (1) 2 adjoint crystals.
Uniform description of the rigged configuration bijection
We give a uniform description of the bijection $$\Phi $$ Φ from rigged configurations to tensor products of Kirillov–Reshetikhin crystals of the form $$\bigotimes _{i=1}^N B^{r_i,1}$$ ⨂ i = 1 N B r i
Rigged Configurations and Kashiwara Operators
For types A (1) and D (1) n we prove that the rigged configuration bijection in- tertwines the classical Kashiwara operators on tensor products of the arbitrary Kirillov{ Reshetikhin crystals and the
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(∞)$B(\infty )$ models given by rigged
Tableau models for semi-infinite Bruhat order and level-zero representations of quantum affine algebras.
We prove that semi-infinite Bruhat order on an affine Weyl group is completely determined from those on the quotients by affine Weyl subgroups associated with various maximal (standard) parabolic
...
...

References

SHOWING 1-10 OF 39 REFERENCES
A CRYSTAL TO RIGGED CONFIGURATION BIJECTION FOR NONEXCEPTIONAL AFFINE ALGEBRAS
Author(s): Okado, Masato; Schilling, Anne; Shimozono, Mark | Abstract: Kerov, Kirillov, and Reshetikhin defined a bijection between highest weight vectors in the crystal graph of a tensor power of
Combinatorial R-matrices for Kirillov--Reshetikhin crystals of type D^{(1)}_n, B^{(1)}_n, A^{(2)}_{2n-1}
We calculate the image of the combinatorial R-matrix for any classical highest weight element in the tensor product of Kirillov--Reshetikhin crystals $B^{r,k}\otimes B^{1,l}$ of type $D^{(1)}_n,
Crystal interpretation of Kerov–Kirillov–Reshetikhin bijection II. Proof for $\mathfrak{sl}_{n}$ case
Abstract In proving the Fermionic formulae, a combinatorial bijection called the Kerov–Kirillov–Reshetikhin (KKR) bijection plays the central role. It is a bijection between the set of highest paths
A bijection between Littlewood-Richardson tableaux and rigged configurations
Abstract. We define a bijection from Littlewood-Richardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasi-particle
KKR type bijection for the exceptional affine algebra E_6^{(1)}
For the exceptional affine type E_6^{(1)} we establish a statistic-preserving bijection between the highest weight paths consisting of the simplest Kirillov-Reshetikhin crystal and the rigged
X=M for symmetric powers
GENERALIZED ENERGIES AND INTEGRABLE $D^{(1)}_{n}$ CELLULAR UTOMATON
We introduce generalized energies for a class of U_q(D^{(1)}_n) crystals by using the piecewise linear functions that are building blocks of the combinatorial R. They include the conventional energy
Demazure Crystals, Kirillov-Reshetikhin Crystals, and the Energy Function
TLDR
A formula of the Demazure character is obtained in terms of the energy function, which has applications to Macdonald polynomials and q-deformed Whittaker functions.
...
...