Related papers: Crystal isomorphisms for irreducible highest weigh…

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 show that the different labelings of the crystal graph for irreducible highest weight $\mathcal{U}\_q (\hat{\mathfrak{sl}}\_e)$-modules lead to different labelings of the simple modules for non semisimple Ariki-Koike algebras by using…

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 characterization of the highest weight vertices in the crystal graph of the level l Fock spaces. This characterization is based on the notion of totally periodic symbols viewed as affine analogues of reverse lattice words…

This paper provides a combinatorial dictionary between three sets of objects: Bernstein-Zelevinsky multisegments, Kleshchev multipartitions, and the irreducible modules of the affine Hecke algebra $H_n$ (for generic $q$). In particular, we…

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…

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…

Using combinatorics of Young walls, we give a new realization of arbitrary level irreducible highest weight crystals $\mathcal{B}(\lambda)$ for quantum affine algebras of type $A_n^{(1)}$, $B_n^{(1)}$, $C_n^{(1)}$, $A_{2n-1}^{(2)}$,…

We use Khovanov-Lauda-Rouquier algebras to categorify a crystal isomorphism between a highest weight crystal and the tensor product of a perfect crystal and another highest weight crystal, all in level 1 type A affine. The nodes of the…

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 introduce the notion of dual perfect bases and dual perfect graphs. We show that every integrable highest weight module $V_q(\lambda)$ over a quantum generalized Kac-Moody algebra $U_{q}(\mathcal{g})$ has a dual perfect basis and its…

In this paper, we give a new realization of crystal bases for irreducible highest weight modules over $U_q(G_2)$ in terms of monomials. We also discuss the natural connection between the monomial realization and tableau realization.

We present a list of ``local'' axioms and an explicit combinatorial construction for the regular $B_2$-crystals (crystal graphs of highest weight integrable modules over $U_q(sp_4)$). Also a new combinatorial model for these crystals is…

We give a new combinatorial model for the crystals of integrable highest weight modules over the classical Lie algebras of type $B$ and $C$ in terms of classical Young tableux. We then obtain a new description of its Littlewood-Richardson…

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver…

For a weighted graph $E$, we construct representation graphs $F$, and consequently, $L_K(E)$-modules $V_F$, where $L_K(E)$ is the Leavitt path algebra associated to $E$, with coefficients in a field $K$. We characterise representation…

For a finite dimensional Lie algebra $\mathfrak{g}$, the Duflo map $S\mathfrak{g}\rightarrow U\mathfrak{g}$ defines an isomorphism of $\mathfrak{g}$-modules. On $\mathfrak{g}$-invariant elements it gives an isomorphism of algebras.…

We classify all irreducible highest-weight unitary modules over the non-compact real form $\mathfrak{u}(p,q|n)$ of the general linear Lie superalgebra $\mathfrak{gl}_{p+q|n}$. The classification is given by explicit necessary and sufficient…

This paper provides results on the modular representation theory of the supergroup $GL(m|n).$ Working over a field of arbitrary characteristic, we prove that the explicit combinatorics of certain crystal graphs describe the representation…

We give the crystal structure of the Grothendieck group $G_0(R)$ of irreducible modules over the quiver Hecke algebra $R$ constructed in \cite{TW2023}. This leads to the categorification of the crystal $B(\infty)$ of the quantum Borcherds…