Related papers: R\'eflexions dans un cristal

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and let $\lambda$ be an arbitrary integral weight. We denote by $\mathbb{B}(\lambda)$ the crystal of all Lakshmibai-Seshadri paths of shape $\lambda$. Let $V(\lambda)$ be the…

Let B(\infty) be the crystal corresponding to the nilpotent part of a quantized Kac-Moody algebra. We suggest a general way to represent B(\infty) as the set of integer solutions of a system of linear inequalities. As an application, we…

We present a simple combinatorial model for the characters of the irreducible integrable highest weight modules for complex symmetrizable Kac-Moody algebras. This model can be viewed as a discrete counterpart to the Littelmann path model.…

The Kashiwara $B(\infty)$ crystal pertains to a Verma module for a Kac- Moody Lie algebra. Ostensibly it provides only a parametrisation of the global/canonical basis for the latter. Yet it is much more having a rich combinatorial structure…

Given a quantized enveloping algebra $U_q(\mathfrak g)$ and a pair of dominant weights ($\lambda$, $\mu$), we extend a conjecture raised by Lusztig in \cite{Lusztig:1992}to a more general form and then prove this extended Lusztig's…

We provide an explicit combinatorial description of the embedding of the crystal of Kashiwara-Nakashima tableaux in types $B$ and $C$ into that of $\bi$-Lusztig data associated to a family of reduced expressions $\bi$ of the longest element…

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…

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank 2, and set $\lambda=\Lambda_{1} - \Lambda_{2}$, where $\Lambda_{1}$, $\Lambda_{2}$ are the fundamental weights. Denote by $V(\lambda)$ the extremal weight module of extremal…

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…

Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra g has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given g, there is a limit crystal, usually denoted by…

In this paper, we develop the theory of abstract crystals for quantum Borcherds-Bozec algebras. Our construction is different from the one given by Bozec. We further prove the crystal embedding theorem and provide a characterization of…

The polyhedral realizations for crystal bases of the integrable highest weight modules of $U_q(\mathfrak{g})$ have been introduced in ([T.Nakashima, J. Algebra, vol.219, no. 2, (1999)]), which describe the crystal bases as sets of lattice…

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…

The main result in this paper is the character formula for arbitrary irreducible highest weight modules of W algebras. The key ingredient is the functor provided by quantum Hamiltonian reduction, that constructs the W algebras from affine…

Let $U_q(\g)$ be a quantum generalized Kac-Moody algebra and let $V(\Lambda)$ be the integrable highest weight $U_q(\g)$-module with highest weight $\Lambda$. We prove that the cyclotomic Khovanov-Lauda-Rouquier algebra $R^\Lambda$ provides…

Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac-Moody algebra. The underlying set consists of the irreducible components of Lusztig's quiver varieties, which are varieties of nilpotent…

In this paper, we consider polyhedral realizations for crystal bases $B(\lambda)$ of irreducible integrable highest weight modules of a quantized enveloping algebra $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a classical affine Lie algebra…

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…

We describe the embedding from the crystal of Kashiwara-Nakashima tableaux in type $D$ of an arbitrary shape into that of $\mathbf{i}$-Lusztig data associated to a family of reduced expressions $\mathbf{i}$ which are compatible with the…

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$. We give a polyhedral realization of the crystal basis for the extremal weight module of extremal weight $\lambda$, where $\lambda$ is an integral weight whose Weyl group…