Related papers: On Combinatorial Models for Affine Crystals
We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but…
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 first describe how the Kashiwara involution on crystals of affine type $A$ is encoded by the combinatorics of aperiodic multisegments. This yields a simple relation between this involution and the Zelevinsky involution on the set of…
The Lascoux-Leclerc-Thibon-Ariki theory asserts that the K-group of the representations of the affine Hecke algebras of type A is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with a Lie…
Using the previous obtained universal $R$-matrix for the quantized nontwisted affine Lie algebras $U_q(A_1^{(1)})$ and $U_q(A_2^{(1)})$, we determine the explicitly spectral-dependent universal $R$-matrix for the corresponding quantum Lie…
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…
Kang et al. provided a path realization of the crystal graph of a highest weight module over a quantum affine algebra, as certain semi-infinite tensor products of a single perfect crystal. In this paper, this result is generalized to give a…
In this paper, we recall Lepowsky's and Wakimoto's product character formulas formulated in a new way by using arrays of specialized weighted crystals of negative roots for affine Lie algebras of type $C_l^{(1)}$, $D_{l+1}^{(2)}$ and…
We obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dimensional irreducible representations of simple Lie algebras of type A and D. The first realization we consider is a geometric construction in…
The classical Gindikin-Karpelevich formula appears in Langlands' calculation of the constant terms of Eisenstein series on reductive groups and in Macdonald's work on p-adic groups and affine Hecke algebras. The formula has been generalized…
We investigate the characters of some finite-dimensional representations of the quantum affine algebras $U_q(\hat{g})$ using the action of the copy of $U_q(g)$ embedded in it. First, we present an efficient algorithm for computing the…
We derive decomposition formulas for supercharacters of quantum affine ortho-symplectic superalgebras and twisted quantum affine superalgebras into supercharacters of their finite-type quantum sub-superalgebras, by employing Cauchy-type…
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.
Using combinatorics of Young tableaux, we give an explicit construction of irreducible graded modules over Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^{\lambda}$ of type $A_{n}$. Our construction is compatible…
Lusztig $q$-weight multiplicities extend the Kostka-Foulkes polynomials to a broader range of Lie types. In this work, we investigate these multiplicities through the framework of Kirillov-Reshetikhin crystals. Specifically, for type $C$…
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…
In this paper we continue the study of the higher-rank graphs associated to finite-dimensional complex semisimple Lie algebras, introduced by the author and R. Yuncken, whose construction relies on Kashiwara's theory of crystals. First we…
We study a certain family of finite-dimensional simple representations over quantum affine superalgebras associated to general linear Lie superalgebras, the so-called fundamental representations: the denominators of rational $R$-matrices…
We prove the Kirillov-Reshetikhin conjecture for all untwisted quantum affine algebras : we prove that the character of Kirillov-Reshetikhin modules solve the Q-system and we give an explicit formula for the character of their tensor…
We compute $t$--analogs of $q$--characters of all $l$--fundamental representations of the quantum affine algebras of type $E_6^{(1)}$, $E_7^{(1)}$, $E_8^{(1)}$ by a supercomputer. In particular, we prove the fermionic formula for…