Related papers: Leading coefficients of Kazhdan--Lusztig polynomia…

Macdonald defined two-parameter Kostka functions K_{\lambda\mu}(q,t) where \lambda, \mu are partitions. The main purpose of this paper is to extend his definition to include all compositions as indices. Following Macdonald, we conjecture…

For $w$ in the symmetric group $S_n$, let $\widetilde C_w$ be the corresponding modified, signless Kazhdan--Lusztig basis element of the type-$A$ Hecke algebra $H_n(q)$. An extension [Ann. Comb. 25, no. 3 (2021) pp. 757--787] of a result of…

We prove that the combinatorial concept of a special matching can be used to compute the parabolic Kazhdan-Lusztig polynomials of doubly laced Coxeter groups and of dihedral Coxeter groups. In particular, for this class of groups which…

We introduce the Z-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial. We then exploit a symmetry of the Z-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients. We solve this recursion,…

In this paper we prove the Kazhdan-Lusztig type character formula for irreducible highest weight modules with positive rational highest weights over symmetrizable Kac-Moody Lie algebras.

For arbitrary Coxeter systems, we prove that inverse Kazhdan-Lusztig polynomials satisfy a monotonicity property. This follows from the validity of Soergel's conjecture and the existence of injective morphisms between Rouquier complexes in…

In 1980, Lusztig introduced the periodic Kazhdan-Lusztig polynomials, which are conjectured to have important information about the characters of irreducible modules of a reductive group over a field of positive characteristic, and also…

We give an easy diagrammatical description of the parabolic Kazhdan-Lusztig polynomials for the Weyl group $W_n$ of type $D_n$ with parabolic subgroup of type $A_n$ and consequently an explicit counting formula for the dimension of the…

Let $W$ be a finite Coxeter group and $X$ a subset of $W$. The length polynomial $L_{W,X}(t)$ is defined by $L_{W,X}(t) = \sum_{x \in X} t^{\ell(x)}$, where $\ell$ is the length function on $W$. In this article we derive expressions for the…

The equivariant Kazhdan-Lusztig polynomial of a matroid was introduced by Gedeon, Proudfoot, and Young. Gedeon conjectured an explicit formula for the equivariant Kazhdan-Lusztig polynomials of thagomizer matroids with an action of…

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…

Let $(W, S)$ be a Coxeter system equipped with a fixed automorphism $\ast$ of order $\leq 2$ which preserves $S$. Lusztig (and with Vogan in some special cases) have shown that the space spanned by set of "twisted" involutions was naturally…

Let $(W,S)$ be a Coxeter system, and write $S=\{s_i:i\in I\}$, where $I$ is a finite index set. Fix a nonempty convex subset $\mathscr{L}$ of $W$. If $W$ is of type $A$, then $\mathscr{L}$ is the set of linear extensions of a poset, and…

Kazhdan-Lusztig polynomials are important and mysterious objects in representation theory. Here we present a new formula for their computation for symmetric groups based on the Bruhat graph. Our approach suggests a solution to the…

Fully commutative elements in types $B$ and $D$ are completely characterized and counted by Stembridge. Recently, Feinberg-Kim-Lee-Oh have extended the study of fully commutative elements from Coxeter groups to the complex setting, giving…

From a combinatorial perspective, we establish three inequalities on coefficients of $R$- and Kazhdan-Lusztig polynomials for crystallographic Coxeter groups: (1) Nonnegativity of $(q-1)$-coefficients of $R$-polynomials, (2) a new criterion…

We introduce the notions of boundary vertex, linear equivalence and effective boundary vertex in the context of Viennot's heaps of pieces. We prove that in the heap of a fully commutative element in a star reducible Coxeter group, every…

We define the equivariant Kazhdan-Lusztig polynomial of a matroid equipped with a group of symmetries, generalizing the nonequivariant case. We compute this invariant for arbitrary uniform matroids and for braid matroids of small rank.

We introduce the notion of 321-avoiding permutations in the affine Weyl group $W$ of type $A_{n-1}$ by considering the group as a George group (in the sense of Eriksson and Eriksson). This enables us to generalize a result of Billey,…

We give a new characterization of the peak subalgebra of the algebra of quasisymmetric functions and use this to construct a new basis for this subalgebra. As an application of these results we obtain a combinatorial formula for the…