相关论文: Embedded factor patterns for Deodhar elements in K…

Let $(W,S)$ be a finite Coxeter group. Kazhdan and Lusztig introduced the concept of $W$-graphs and Gyoja proved that every irreducible representation of the Iwahori-Hecke algebra $H(W,S)$ can be realized as a $W$-graph. Gyoja defined an…

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part…

The Iwahori-Hecke algebra $\mathcal{H}$ of a Coxeter system $(W,S)$ has a "standard basis" indexed by the elements of $W$ and a "bar involution" given by a certain antilinear map. Together, these form an example of what Webster calls a…

We study a category of Whittaker modules over a complex semisimple Lie algebra by realizing it as a category of twisted D-modules on the associated flag variety using Beilinson-Bernstein localization. The main result of this paper is the…

Let $(W,S)$ be an arbitrary Coxeter system. We introduce a family of polynomials, $\{ \tilde{\mathcal{R}}_{u,\underline{v}}(t)\}$, indexed by pairs $(u,\underline{v})$ formed by an element $u\in W$ and a (non-necessarily reduced) word…

We study the representation theory of finite W-algebras. After introducing parabolic subalgebras to describe the structure of W-algebras, we define the Verma modules and give a conjecture for the Kac determinant. This allows us to find the…

We introduce deformations of Kazhdan-Lusztig elements and specialised nonsymmetric Macdonald polynomials, both of which form a distinguished basis of the polynomial representation of a maximal parabolic subalgebra of the Hecke algebra. We…

The super Weyl group of a basic classical Lie superalgebra was introduced and studied in \cite{PS}, which turns out to play an important role for the study of representations of the basic classical Lie superalgebras and algebraic…

In this paper we show that the leading coefficient $\mu(y,w)$ of some Kazhdan-Lusztig polynomials $P_{y,w}$ with $y,w$ in an affine Weyl group of type $\tilde A_n $ is $n+2$. This fact has some consequences on the dimension of first…

This paper is a sequel to work of Dynkin on subroot lattices of root lattices and to work of Carter on presentations of Weyl group elements as products of reflections. The quotients $L/L_1$ are calculated for all irreducible root lattices…

The lowest two-sided cell of the extended affine Weyl group $W_e$ is the set $\{w \in W_e: w = x \cdot w_0 \cdot z, \text{for some} x,z \in W_e\}$, denoted $W_{(\nu)}$. We prove that for any $w \in W_{(\nu)}$, the canonical basis element…

In this paper we show that the leading coefficient $\mu(y,w)$ of some Kazhdan-Lusztig polynomials $P_{y,w}$ with $y,w$ in an affine Weyl group of type $\tilde B_n$ (resp. $\tilde C_n$ or $\tilde D_n$) is $n$ (resp. $n+1$).

We give a concrete combinatorial interpretation of the coefficients of the Kazhdan-Lusztig polynomials of Dowling geometries, a family of matroids which generalizes braid matroids of types A and B. Furthermore, we interpret the coefficients…

For a classical group $G$ and a Coxeter element $c$ of the Weyl group, it is known that the coordinate ring $\mathbb{C}[G^{e,c^2}]$ of the double Bruhat cell $G^{e,c^2}:=B\cap B_-c^2B_-$ has a structure of cluster algebra of finite type,…

A Gelfand model for an algebra is a module given by a direct sum of irreducible submodules, with every isomorphism class of irreducible modules represented exactly once. We introduce the notion of a perfect model for a finite Coxeter group,…

We study a diagrammatic categorification (the "anti-spherical category") of the anti-spherical module for any Coxeter group. We deduce that Deodhar's (sign) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients, and that a…

We propose a combinatorial interpretation of the coefficient of $q$ in Kazhdan- Lusztig polynomials and we prove it for finite simply-laced Weyl groups.

In this paper, we let $\Hecke$ be the Hecke algebra associated with a finite Coxeter group $W$ and with one-parameter, over the ring of scalars $\Alg=\mathbb{Z}(q, q^{-1})$. With an elementary method, we introduce a cellular basis of…

Recently, Wang and the second author constructed a bar involution and canonical basis for a quasi-permutation module of the Hecke algebra associated to a type B Weyl group $W$, where the basis is parameterized by left cosets of a…

A Coxeter group is said to be \emph{$\mathbf{a}(2)$-finite} if it has finitely many elements of $\mathbf{a}$-value 2 in the sense of Lusztig. In this paper, we give explicit combinatorial descriptions of the left, right, and two-sided…