Related papers: Geometric Crystal and Tropical R for D^(1)_n

A perfect crystal of any level is constructed for the Kirillov-Reshetikhin module of $U_q(D_4^{(3)})$ corresponding to the middle vertex of the Dynkin diagram. The actions of Kashiwara operators are given explicitly. It is also shown that…

Generally speaking, this thesis focuses on the interplay between the representations of Lie groups and probability theory. It subdivides into essentially three parts. In a first rather algebraic part, we construct a path model for geometric…

We realize the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety. As a by-product, we give a counterexample…

Yang-Baxterising a braid group representation associated with multideformed version of $GL_{q}(N)$ quantum group and taking the corresponding $q\rightarrow 1$ limit, we obtain a rational $R$-matrix which depends on $\left ( 1+ {N(N-1) \over…

The symmetric Grothendieck polynomials representing Schubert classes in the $K$-theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type $A_n$ crystal structure on these tableaux. This…

Let G be a Lie group with Lie algebra $ \Cal G: = T_\epsilon G$ and $T^*G = \Cal G^* \rtimes G$ its cotangent bundle considered as a Lie group, where G acts on $\Cal G^*$ via the coadjoint action. We show that there is a 1-1 correspondance…

Extending the work arXiv:math/0508107, we introduce the affine crystal action on rigged configurations which is isomorphic to the Kirillov-Reshetikhin crystal B^{r,s} of type D_n^(1) for any r,s. We also introduce a representation of…

This mostly expository article explores recent developments in the relations between the three objects in the title from an algebro-combinatorial perspective. We prove a formula for Whittaker functions of a real semisimple group as an…

We study, in the path realization, crystals for Demazure modules of affine Lie algebras of types $A^{(1)}_n,B^{(1)}_n,C^{(1)}_n,D^{(1)}_n, A^{(2)}_{2n-1},A^{(2)}_{2n}, and D^{(2)}_{n+1}$. We find a special sequence of affine Weyl group…

Crystal structures are characterized by atomic bases within a primitive unit cell that repeats along a regular lattice throughout 3D space. The periodic and infinite nature of crystals poses unique challenges for geometric graph…

For each reductive algebraic group G we introduce and study unipotent bicrystals which serve as a regular version of birational geometric and unipotent crystals introduced earlier by the authors. The framework of unipotent bicrystals…

We describe the upper seminormal crystal structure for the $\mu$-supported $\delta$-vectors for any quiver with potential with reachable frozen vertices, or equivalently for the tropical points of the corresponding cluster $\mc{X}$-variety.…

There are two parts to this work, which are largely independent. The first consists of a series of results concerning the crystal commutor of Henriques and Kamnitzer. We first describe the relationship between the crystal commutor and…

We give a realization of crystal graphs for basic representations of the quantum affine algebra $U_q(C_2^{(1)})$ in terms of new combinatorial objects called the Young walls.

A three-parametric $R$-matrix satisfying a graded Yang-Baxter equation is introduced.This $R$-matrix allows us to construct new quantum supergroups which are deformations of the supergroup $GL(1/1)$ and the universal enveloping algebra…

We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type $A$. Using geometric objects called Rhombic tilings we derive a "crossing formula"…

Let g be an affine Lie algebra with associated Yangian Y_hg. We prove the existence of two meromorphic R-matrices associated to any pair of representations of Y_hg in the category O. They are related by a unitary constraint and constructed…

We describe a geometric construction of all nondegenerate trigonometric solutions of the associative and classical Yang-Baxter equations. In the associative case the solutions come from symmetric spherical orders over the irreducible nodal…

A trick to obtain a systematic solution to the set-theoretical reflection equation is presented from a known one to the Yang-Baxter equation. Examples are given from crystals and geometric crystals associated to the quantum affine algebra…

We provide the explicit combinatorial structure of the Kirillov-Reshetikhin crystals B^{r,s} of type D_n(1), B_n(1), and A_{2n-1}(2). This is achieved by constructing the crystal analogue sigma of the automorphism of the D_n(1) (resp.…