Related papers: Cubulation of Bruhat graphs

Let $P_{\text{MAX}}(d,s)$ denote the maximum arithmetic genus of a locally Cohen-Macaulay curve of degree $d$ in $\mathbb{P}^3$ that is not contained in a surface of degree $<s$. A bound $P(d, s)$ for $P_{\text{MAX}}(d,s)$ has been proven…

We investigate the class of Kazhdan-Lusztig varieties, and its subclass of matrix Schubert varieties, endowed with a naturally defined torus action. Writing a matrix Schubert variety $\overline{X_w}$ as $\overline{X_w}=Y_w\times…

Kazhdan-Lusztig-Stanley polynomials are a combinatorial generalization of Kazhdan-Lusztig polynomials of for Coxeter groups that include g-polynomials of polytopes and Kazhdan-Lusztig polynomials of matroids. In the cases of Weyl groups,…

A Coxeter system is an ordered pair (W,S) where S is the generating set in a particular type of presentation for the Coxeter group W. A subgroup of W is called special if it is generated by a subset of S. Amalgamated product decompositions…

Let $X$ be a convex curve in the plane (say, the unit circle), and let $\mathcal S$ be a family of planar convex bodies, such that every two of them meet at a point of $X$. Then $\mathcal S$ has a transversal $N\subset\mathbb R^2$ of size…

We define ``star reducible'' Coxeter groups to be those Coxeter groups for which every fully commutative element (in the sense of Stembridge) is equivalent to a product of commuting generators by a sequence of length-decreasing star…

In this paper, we investigate various properties of strong and weak twisted Bruhat orders on a Coxeter group. In particular, we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of…

For $t,g>0$, a vertex-weighted graph of total weight $W$ is $(t,g)$-trimmable if it contains a vertex-induced subgraph of total weight at least $(1-1/t)W$ and with no simple path of more than $g$ edges. A family of graphs is trimmable if…

We attach to every Coxeter system (W,S) an extension C_W of the corresponding Iwahori-Hecke algebra. We construct a 1-parameter family of (generically surjective) morphisms from the group algebra of the corresponding Artin group onto C_W.…

We study the interplay between the discrete geometry of Bruhat poset intervals and subword complexes of finite Coxeter systems. We establish connections between the cones generated by cover labels for Bruhat intervals and of root…

We show that Lusztig's $a$-function of a Coxeter group is bounded if the Coxeter group has a complete graph (i.e. any two vertices are joined) and the cardinalities of finite parabolic subgroups of the Coxeter group have a common upper…

We consider the set $\Irr(W)$ of (complex) irreducible characters of a finite Coxeter group $W$. The Kazhdan--Lusztig theory of cells gives rise to a partition of $\Irr(W)$ into "families" and to a natural partial order $\leq_{\cLR}$ on…

In this paper we consider the Hecke algebra $\mathcal {H}$ associated to an extended affine Weyl group of type $\widetilde{B_2}$. We give some interesting formulas on $C_{rt}S_{\lambda}$, which imply some relations between the…

We classify and explicitly construct the irreducible graded representations of anti-spherical Hecke categories which are concentrated in one degree. Each of these homogeneous representations is one-dimensional and can be cohomologically…

Let $W$ be a Coxeter group and $r\in W$ a reflection. If the group of order 2 generated by $r$ is the intersection of all the maximal finite subgroups of $W$ that contain it, then any isomorphism from $W$ to a Coxeter group $W'$ must take…

We study the appearance of notable interval structures -- lattices, modular lattices, distributive lattices, and boolean lattices -- in both the Bruhat and weak orders of Coxeter groups. We collect and expand upon known results for…

Let (W,S) be a Coxeter system of finite rank (ie |S| is finite) and let A be the associated Coxeter (or Davis) complex. We study chains of pairwise parallel walls in A using Tits' bilinear form associated to the standard root system of…

Let $(W,S)$ be a Coxeter system and $\ast$ be an automorphism of $W$ with order $\leq 2$ such that $s^{\ast}\in S$ for any $s\in S$. Let $I_{\ast}$ be the set of twisted involutions relative to $\ast$ in $W$. In this paper we consider the…

Introduced by Kodama and Williams, Bruhat interval polytopes are generalized permutohedra closely connected to the study of torus orbit closures and total positivity in Schubert varieties. We show that the 1-skeleton posets of these…

The higher Bruhat orders $\mathcal{B}(n,k)$ were introduced by Manin-Schechtman to study discriminantal hyperplane arrangements and subsequently studied by Ziegler, who connected $\mathcal{B}(n,k)$ to oriented matroids. In this paper, we…