Related papers: A two-sided analogue of the Coxeter complex

We revisit the structure of the normalizer $N_W(P)$ of a parabolic subgroup $P$ in a finite Coxeter group $W$, originally described by Howlett. Building on Howlett's Lemma, which provides canonical complements for reflection subgroups, and…

For a Coxeter element $c$ of a finite Coxeter group, we consider a family of subword complexes parameterized by reduced expressions of the longest element. This family generalizes $c-$cluster complexes. We describe vertices of these…

We study the spectral radii of elements in the hyperbolic Coxeter group $W(E_{10})$ by introducing a filtration indexed by reflections conjugate to a distinguished simple reflection $s_0$. This filtration organizes $W(E_{10})$ into double…

Let $W$ be a Weyl group corresponding to the root system $A_{n-1}$ or $B_n$. We define a simplicial complex $ \Delta^m_W $ in terms of polygon dissections for such a group and any positive integer $m$. For $ m=1 $, $ \Delta^m_W$ is…

In [APS], the authors characterize the partitions of $n$ whose corresponding representations of $S_n$ have nontrivial determinant. The present paper extends this work to all irreducible finite Coxeter groups $W$. Namely, given a nontrivial…

In this paper we extend the computations in parts I and II of this series of papers and complete the proof of a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group W acting on the pth graded component of its…

We prove that among all right-angled Coxeter groups in hyperbolic 3-space, the group generated by reflections in the faces of a right-angled triangular bipyramid with three ideal and two finite vertices has the smallest covolume. The group…

Let W be an irreducible finitely generated Coxeter group. The geometric representation of W in GL(V) provides a discrete embedding in the orthogonal group of the Tits form (the associated bilinear form of the Coxeter group). If the Tits…

In this paper we construct the quasi regular polyhedra and their duals which are the generalizations of the Archimedean and Catalan solids respectively. This work is an extension of two previous papers of ours which were based on the…

Let $(W,S)$ be a Coxeter system and let $w \mapsto w^*$ be an involution of $W$ which preserves the set of simple generators $S$. Lusztig and Vogan have recently shown that the set of twisted involutions (i.e., elements $w \in W$ with…

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…

Here, for $W$ the Coxeter group $\mathrm{D}_n$ where $n > 4$, it is proved that the maximal rank of an abstract regular polytope for $W$ is $n - 1$ if $n$ is even and $n$ if $n$ is odd. Further it is shown that $W$ has abstract regular…

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 notion of a reduced expression for a double coset in a Coxeter group was introduced by Williamson, and recent work of Elias and Ko has made this theory more accessible and combinatorial. One result of Elias-Ko is that any coset admits a…

Our main theorems provide a single geometric setting in which polynomial representatives for Schubert classes in the integral cohomology ring of the flag manifold are determined uniquely, and have positive coefficients for geometric…

We study a family of posets and the associated chain and order polytopes. We identify the order polytope as a maximal Kogan face in a Gelfand-Tsetlin polytope of a multiple of a fundamental weight. We show that the character of such a Kogan…

This article studies a large, general class of orthogonal polytopes which we may call "generic orthotopes". These objects emerged from a desire to represent a Coxeter complex by an orthogonal polytope that is particularly nice with respect…

Let $ (W,S)$ be a Coxeter system. We investigate the equation $ w(\Phi_{x}) = \Phi_{y}$ where $ w,x,y\in W$ and $ \Phi_{x}$, $\Phi_{y}$ denote the left inversion sets of $ x$ and $ y$. We then define a commutative square diagram called a…

We construct several quantum coset W-algebras, e.g. sl(2,R)/U(1) and sl(2,R)+sl(2,R) / sl(2,R), and argue that they are finitely nonfreely generated. Furthermore, we discuss in detail their role as unifying W-algebras of Casimir W-algebras.…

Let C be a one- or two-sided Kazhdan--Lusztig cell in a Coxeter group (W,S), and let Reduced(C) denote the set of reduced expressions of all w in C, regarded as a language over the alphabet S. Casselman has conjectured that Reduced(C) is…