Related papers: Geometric Crystal and Tropical R for D^(1)_n
We shall realize certain affine geometric crystal of type $D_4^{(3)}$ associated with the fundamental representation $W(\pi_1)$ explicitly . By its explicit form, we see that it has a positive structure.
We give a crystal structure on the set of Gelfand-Tsetlin patterns which parametrize bases for finite-dimensional irreducible representations of the general linear Lie algebra. The crystal data are given in closed form, expressed using…
Hatayama et al. conjectured fermionic formulas associated with tensor products of U'_q(g)-crystals B^{r,s}. The crystals B^{r,s} correspond to the Kirillov--Reshetikhin modules which are certain finite dimensional U'_q(g)-modules. In this…
We obtain the affirmative answer to the conjecture in [15]. More precisely, let X be the affine geometric crystal of type G^(1)_2 in [15] and UD(X,T,\theta) a ultra-discretization of X with respect to a certain positive structure \theta.…
Consider Kashiwara's crystal associated to a highest weight representation of a symmetric Kac-Moody algebra. There is a geometric realization of this object using Nakajima's quiver varieties, but in many particular cases it can also be…
We give a new combinatorial model of the Kirillov-Reshetikhin crystals of type $A_n^{(1)}$ in terms of non-negative integral matrices based on the classical RSK algorithm, which has a simple description of the affine crystal structure…
Let $g$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$ and $g^L$ be its Langlands dual. It is conjectured that for each Dynkin node $i \in I \setminus \{0\}$ the affine Lie algebra $g$ has a positive geometric crystal…
We establish a bijection between the set of rigged configurations and the set of tensor products of Kirillov--Reshetikhin crystals of type $D^{(1)}_n$ in full generality. We prove the invariance of rigged configurations under the action of…
We shall show that for type $A_n$ the realization of crystal bases obtained from the decorated geometric crystals intorduced by Berenstein and Kazhdan coincides with our polyhedral realizations of crystal bases. We also observe certain…
This is a continuation of [15, 16]. We shall show that for type D_n the realization of crystal bases obtained from the decorated geometric crystals in [2] coincides with the polyhedral realizations of crystal bases.
We biject two combinatorial models for tensor products of (single-column) Kirillov-Reshetikhin crystals of any classical type $A-D$: the quantum alcove model and the tableau model. This allows us to translate calculations in the former…
The main purpose of this paper is to give a combinatorial realization of Kirillov-Reshetikhin (KR simply) crystals $B^{r, s}$ for type $\text{E}_n^{(1)}$ with a minuscule node $r$ and $s \ge 1$. To do this, we describe explicitly the…
This contribution discusses the geometry of $k$D crystal cells given by $(k+1)$ points in a projective space $\R^{n+1}$. We show how the concepts of barycentric and fractional (crystallographic) coordinates, reciprocal vectors and dual…
Regular $A_n$-, $B_n$- and $C_n$-crystals are edge-colored directed graphs, with ordered colors $1,2,...,n$, which are related to representations of quantized algebras $U_q(\mathfrak{sl}_{n+1})$, $U_q(\mathfrak{sp}_{2n})$ and…
Kirillov-Reshetikhin crystals are colored directed graphs encoding the structure of certain finite-dimensional representations of affine Lie algebras. A tensor products of column shape Kirillov-Reshetikhin crystals has recently been…
Berenstein and Kazhdan's theory of geometric crystals gives rise to two commuting families of geometric crystal operators acting on the space of complex $m \times n$ matrices. These are birational actions, which we view as a…
For any simple complex algebraic group, we define upper/lower half-decorated geometric crystals and show that their tropicalization will be upper/lower normal Kashiwara's crystals. In particular, we show that the tropicalization of the…
We introduce the special set-theoretic Yang-Baxter algebra and show that it is a Hopf algebra subject to certain conditions. The associated universal R-matrix is also obtained via an admissible Drinfel'd twist. The structure of braces…
Let $\mathfrak g$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$. It is conjectured in \cite{KNO} that for each Dynkin node $k \in I \setminus \{0\}$ the affine Lie algebra $\mathfrak g$ has a positive geometric…
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 $D^{(1)}_n, B^{(1)}_n, A^{(2)}_{2n-1}$. The notion of…