Related papers: Type $D_n^{(1)}$ rigged configuration bijection
The branching coefficients of the tensor product of finite-dimensional irreducible $U_{q}(\mathfrak{g})$-modules, where $\mathfrak{g}$ is $\mathfrak{so}(2n+1,\mathbb{C})$ ($B_{n}$-type), $\mathfrak{sp}(2n,\mathbb{C})$ ($C_{n}$-type), and…
We study the crystal of quantum nilpotent subalgebra of $U_q(D_n)$ associated to a maximal Levi subalgebra of type $A_{n-1}$. We show that it has an affine crystal structure of type $D_n^{(1)}$ isomorphic to a limit of perfect…
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…
The rigged configuration realization $RC(\infty)$ of the crystal $B(\infty)$ was originally presented as a certain connected component within a larger crystal. In this work, we make the realization more concrete by identifying the elements…
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…
It has previously been shown that, at least for non-exceptional Kac-Moody Lie algebras, there is a close connection between Demazure crystals and tensor products of Kirillov-Reshetikhin crystals. In particular, certain Demazure crystals are…
We introduce a probability distribution on the set of states in a generalized box-ball system associated with Kirillov-Reshetikhin (KR) crystals of type $A^{(1)}_n$. Their conserved quantities induce $n$-tuple of random Young diagrams in…
The affine Dynkin diagram of type $A_n^{(1)}$ has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type $A_n$…
In this note, inspired by the proof of the Kirillov-Reshetikhin conjecture, we consider tensor products of Kirillov-Reshetikhin modules of a fixed node and various level. We fix a positive integer and attach to each of its partitions such a…
In this paper we prove that every Kirillov-Reshetikhin module of type $G_2^{(1)}$ and $D_4^{(3)}$ has a crystal pseudobase (crystal base modulo signs), by applying the criterion for the existence of a crystal pseudobase due to Kang et al.
The behavior of the Kirillov-Schilling-Shimozono bijection is examined under the similarity map on Kirillov-Reshetikhin crystals. It enables us to define this bijection over $\mathbb{Q}$. Conjectures on the extension to $\mathbb{R}$ is also…
For an affine algebra of nonexceptional type in the large rank we show the fermionic formula depends only on the attachment of the node 0 of the Dynkin diagram to the rest, and the fermionic formula of not type A can be expressed as a sum…
For coherent families of crystals of affine Lie algebras of type B^{(1)}_n, D^{(1)}_n, A^{(2)}_{2n} and D^{(2)}_{n+1} we describe the combinatorial R matrix using column insertion algorithms for B,C,D Young tableaux.
We show that the Kirillov-Reshetikhin crystal B^{r,s} for nonexceptional affine types is simple and have the similarity property. As a corollary of the first fact we can derive that the tensor product of KR crystals is connected. Variations…
In this paper, we study a tensor product of perfect Kirillov-Reshetikhin crystals (KR crystals for short) whose levels are not necessarily equal. We show that, by tensoring with a certain highest weight element, such a crystal becomes…
The X=M conjecture of Hatayama et al. asserts the equality between the one-dimensional configuration sum X expressed as the generating function of crystal paths with energy statistics and the fermionic formula M for all affine Kac--Moody…
Let $U$ be a tensor product of highest weight modules of $GL_n(\mathbb C)$ corresponding to multiples of fundamental weights (i.e. rectangles). We consider three ways to stratify $U^{\otimes k}$ into components: using isotypic components of…
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…
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…
We study the polytope model for the affine type $A$ Kirillov-Reshetikhin crystals and prove that the action of the affine Kashiwara operators can be described in a remarkable simple way. Moreover, we investigate the combinatorial $R$-matrix…