Related papers: On the tensor product of two basic representations…

For an untwisted affine Lie algebra we prove an embedding of any higher level Demazure module into a tensor product of lower level Demazure modules (e.g. level one in type A) which becomes in the limit (for anti-dominant weights) the…

We give a criterion for the Demazure crystal $B_w(\lambda)$ defined by Kashiwara to have a tensor product structure. We study the $\sln$ symmetric tensor case, and see some Demazure characters are expressed using Kostka-Foulkes polynomials.

Let B be the crystal basis of the minus part of the quantized enveloping algebra of a semi-simple Lie algebra. Kashiwara has shown that B has a combinatorial description in terms of an embedding of B into the tensor product of B and k…

Fix a simply-laced semisimple Lie algebra. We study the crystal $ B(n\lambda)$, were $\lambda$ is a dominant minuscule weight and $n$ is a natural number. On one hand, $B(n\lambda)$ can be realized combinatorially by height $n$ reverse…

Using new combinatorics of Young walls, we give a new construction of the arbitrary level highest weight crystal $B(\lambda)$ for the quantum affine algebras of types $A^{(2)}_{2n}$, $D^{(2)}_{n+1}$, $A^{(2)}_{2n-1}$, $D^{(1)}_n$,…

In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety $\Zl$ in a quiver variety, and show the following results: (1) The…

Let g = n^- + h + n^+ be a symmetrizable Kac-Moody algebra. Let B(\infty) be the Kashiwara crystal of U_q(n^-), let \lambda be a dominant integral weight, let T_\lambda = {t_\lambda} be the crystal with one element of weight \lambda, and…

We consider a tensor product $V(b)= \otimes_{i=1}^n\C^N(b_i)$ of the Yangian $Y(gl_N)$ evaluation vector representations. We consider the action of the commutative Bethe subalgebra $B^q \subset Y(gl_N)$ on a $gl_N$-weight subspace…

We study the structure of the finite-dimensional representations of $\mathfrak{sl}_2[t]$, the current Lie algebra type of $A_1$, which are obtained by taking tensor products of special Demazure modules. We show that these representations…

We show that every Weyl module for a current algebra has a filtration whose successive quotients are isomorphic to Demazure modules, and that the path model for a tensor product of level zero fundamental representations is isomorphic to a…

By utilizing the combinatorial properties of various tableau models, we establish an explicit correspondence between the polyhedral realizations of the crystal bases $\mathcal B(\lambda)$ (resp. $\mathcal B(\infty)$) of type $A_n$ and the…

We consider simple modules for a Hecke algebra with a parameter of quantum characteristic $e$. Equivalently, we consider simple modules $D^{\lambda}$, labelled by $e$-restricted partitions $\lambda$ of $n$, for a cyclotomic KLR algebra…

We prove that if $W$ and $W'$ are two $B$-pairs whose tensor product is crystalline (or semi-stable or de Rham or Hodge-Tate), then there exists a character $\mu$ such that $W(\mu^{-1})$ and $W'(\mu)$ are crystalline (or semi-stable or de…

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 give a combinatorial description of the composition factors of the induction product of two evaluation modules of the affine Iwahori-Hecke algebra of type GL(m). Using quantum affine Schur-Weyl duality, this yields a combinatorial…

A previous work gave a combinatorial description of the crystal $B(\infty)$, in terms of certain simple Young tableaux referred to as the marginally large tableaux, for finite dimensional simple Lie algebras. Using this result, we present…

We establish a connection between the representation theory of certain noncommutative singular varieties and two-dimensional lattice models. Specifically, we consider noncommutative biparametric deformations of the fiber product of two…

In this paper we describe a theory of (branched) crystals which is adapted to the study of representations in the BGG category $\cal O$ and which generalizes the theory of normal crystals of Kashiwara. In the case of $sl_2$ we show that one…

We explore combinatorics associated with the degenerate Hecke algebra at $q=0$, obtaining a formula for a system of orthogonal idempotents, and also exploring various pattern avoidance results. Generalizing constructions for the 0-Hecke…

In this paper, we give an explicit combinatorial realization of the crystal B(\lambda) for an irreducible highest weight U_q(q(n))-module V(\lambda) in terms of semistandard decomposition tableaux. We present an insertion scheme for…