相关论文: A Demazure Character Formula for the Product Monom…

Let g be a finite-dimensional complex simple Lie algebra. Fix a non-negative integer l, we consider the set of dominant weights {\lambda} of g such that l{\Lambda}_0+{\lambda} is a dominant weight for the corresponding untwisted affine…

We give an explicit characterization of the standard monomials for positroid varieties with respect to the Hodge degeneration and give a Gr\"obner basis. Furthermore, we show that promotion and evacuation biject standard monomials of a…

Type A Demazure atoms are pieces of Schur functions, or sets of tableaux whose weights sum to such functions. Inspired by colored vertex models of Borodin and Wheeler, we will construct solvable lattice models whose partition functions are…

We use Khovanov-Lauda-Rouquier (KLR) algebras to categorify a crystal isomorphism between a fundamental crystal and the tensor product of a Kirillov-Reshetikhin crystal and another fundamental crystal, all in affine type. The nodes of the…

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…

In this paper, we introduce a family of indecomposable finite--dimensional graded modules for the current algebra associated to a simple Lie algebra. These modules are indexed by a tuple of partitions one for each positive root of the…

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…

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 study certain faces of the normal polytope introduced by Feigin, Littelmann and the author whose lattice points parametrize a monomial basis of the PBW-degenerated of simple modules for $\mathfrak{sl}_{n+1}$. We show that lattice points…

In this paper, we give a characterization of the crystal bases $\mathcal{B}_{x}^{+}(\lambda)$, $x \in W_{\mathrm{af}}$, of Demazure submodules $V_{x}^{+}(\lambda)$, $x \in W_{\mathrm{af}}$, of a level-zero extremal weight module…

In the prequel to this paper, we showed how results of Mason involving a new combinatorial formula for polynomials that are now known as Demazure atoms (characters of quotients of Demazure modules, called standard bases by Lascoux and…

We present a combinatorial model, called \emph{perforated tableaux}, to study $A_{n-1}$ crystals, unifying several previously studied combinatorial models. We identify nodes in the $k$-fold tensor product of the standard crystal with length…

Following an idea of A. Berenstein, we define a commutor for the category of crystals of a finite dimensional complex reductive Lie algebra. We show that this endows the category of crystals with the structure of a coboundary category.…

We give a parametrization for crystal bases of Demazure modules as a set of lattice points in some convex polytope and we also describe explicitly the extremal vectors as solutions of some system of linear equations.

The alcove model of the first author and A. Postnikov uniformly describes highest weight crystals of semisimple Lie algebras. We construct a generalization, called the quantum alcove model. In joint work of the first author with S. Naito,…

We show that a tensor product of nonexceptional type Kirillov--Reshetikhin (KR) crystals is isomorphic to a direct sum of Demazure crystals; we do this in the mixed level case and without the perfectness assumption, thus generalizing a…

We establish the equality of the specialization $E_{w\lambda}(x;q,0)$ of the nonsymmetric Macdonald polynomial $E_{w\lambda}(x;q,t)$ at $t=0$ with the graded character $\mathop{\rm gch} U_{w}^{+}(\lambda)$ of a certain Demazure-type…

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra with highest root $\theta$ and let $\mathfrak{g}[t]$ be the corresponding current algebra. In this paper, we consider the $\mathfrak{g}[t]$-stable Demazure modules…

Naoi showed that tensor products of perfect Kirillov-Reshetikhin crystals are isomorphic to certain generalized Demazure crystals. We extend Naoi's results to address distinguished subsets of these tensor products. In type A, these are…

The classical Pieri formula gives a multiplicity free expansion of an irreducible module with a fundamental one for the complex general linear group. In this article we replace the tensor product by the fusion product and prove an analogue…