Related papers: Partition models for the crystal of the basic $U_q…
In this paper, we develop the crystal basis theory for the quantum queer superalgebra $U_q(\mathfrak q(n))$. We define the notion of crystal bases and prove the tensor product rule for $U_q(\mathfrak q(n))$-modules in the category…
We develop a crystal base theory for the general linear Lie superalgebra $gl(m,n)$. We prove that any irreducible $U_q(gl(m,n))$-module in some category has a crystal base, and prove that its associated crystal base is parameterized by…
Let $V(\lambda)$ be the irreducible lowest weight $U_q(D(N,1))$-module with lowest weight $\lambda$. Assume $\lambda = n_0\omega_0-\sum_{i=0}^{N}n_i\omega_i$, where $\omega_0$ is the fundamental weight corresponding to the unique odd coroot…
We define three combinatorial models for \hat{sl(n)} crystals, parametrized by partitions, configurations of beads on an `abacus', and cylindric plane partitions, respectively. These are reducible, but we can identify an irreducible…
We give explicit resolutions of all finite dimensional, simple $U_q(\mathfrak{sl_3})$-modules. We use these resolutions to categorify the colored $\mathfrak{sl}_3$-invariant of framed links via a complex of complexes of graded…
In this paper, we develop the crystal basis theory for quantum generalized Kac-Moody algebras. For a quantum generalized Kac-Moody algebra $U_q(\mathfrak g)$, we first introduce the category $\mathcal O_{int}$ of $U_q(\mathfrak g)$-modules…
Regular $A_n$-crystals are certain edge-colored directed graphs which are related to representations of the quantized universal enveloping algebra $U_q(\mathfrak{sl}_{n+1})$. For such a crystal $K$ with colors $1,2,...,n$, we consider its…
We construct new families of U_q(gl_n)-modules by continuation from finite dimensional representations. Each such module is associated with a combinatorial object - admissible set of relations defined in \cite{FRZ}. More precisely, we prove…
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…
We consider the affine Lie algebra $\widehat{\mathfrak{sl}_2}$ and the Kostant-Kumar submodules of tensor products of its level 1 highest weight integrable representations. We construct crystals for these submodules in terms of the charged…
A regular $A_n$-crystal is an edge-colored directed graph, with $n$ colors, related to an irreducible highest weight integrable module over $U_q(sl_{n+1})$. Based on Stembridge's local axioms for regular simply-laced crystals and a…
Given a partition $\lambda$ corresponding to a dominant integral weight of $\mathfrak{sl}_n$, we define the structure of crystal on the set of 5-vertex ice models satisfying certain boundary conditions associated to $\lambda$. We then show…
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…
We give a simple description of the natural bijection between the set of FLOTW bipartitions and the set of Uglov bipartitions (which generalizes the set of Kleshchev bipartitions). These bipartitions, which label the crystal graphs of…
We give an expression of the $q$-analogues of the multiplicities of weights in irreducible $\sl_{n+1}$-modules in terms of the geometry of the crystal graph attached to the corresponding $U_q(\sl_{n+1})$-modules. As an application, we…
We introduce a semisimple tensor category $\mc{O}^{int}_q(m|n)$ of modules over an quantum ortho-symplectic superalgebra. It is a natural counterpart of the category of finitely dominated integrable modules over the quantum classical…
We study the crystal graphs of irreducible $U_{v}(\hat{sl}}_{e})$-modules of higher level l. Generalizing works of the first author, we obtain a simple description of the bijections between the classes of multipartitions which naturally…
Recently we defined imaginary crystal bases for $U_q(\widehat{\mathfrak{sl}(2)})$- modules in category $\mathcal O^q_{\text{red,im}}$ and showed the existence of such bases for reduced quantized imaginary Verma modules for…
Let ${\mathbf U}_q^-$ be the negative half of a quantum group of finite type. We construct the canonical basis of ${\mathbf U}_q^-$ by applying the folding theory of quantum groups, and piecewise linear parametrization of canonical basis.…
In this paper we classify all the cyclic finite dimensional indecomposable\\ modules of the perfect Lie algebras $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$, given by the semidirect sum of the simple Lie algebra $A_n$ with its standard…