Related papers: Coxeter embeddings are injective

We prove Lusztig's conjectures P1-P15 for Coxeter groups with complete graph, using deceasing induction on $ \mathbf{a} $-values and a kind of decomposition formula of Kazhdan-Lusztig basis elements. As a byproduct, we give a description of…

Let $W$ be a finite Coxeter group. We classify the reflection subgroups of $W$ up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup $R$ of $W$ the conjugacy class of its Coxeter…

We present a method to compute integral cohomology of posets. This toolbox is applicable as soon as the sub-posets under each object possess certain structure. This is the case for simplicial complexes and simplex-like posets. The method is…

We study the modular representations of finite groups of Lie type arising in the cohomology of certain quotients of Deligne-Lusztig varieties associated with Coxeter elements. These quotients are related to Gelfand-Graev representations and…

In a recent paper we claimed that both the group algebra of a finite Coxeter group $W$ as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each…

In this paper we prove, without the finite rank assumption, that any irreducible Coxeter group of infinite order is directly indecomposable as an abstract group. The key ingredient of the proof is that we can determine, for an irreducible…

We study faithful realisations of Coxeter groups over fusion rings and study Vinberg systems associated to them. We show that they induce embeddings of hyperplane complements, which provide geometrical realisations of certain types of…

We show that the standard generating set of a Coxeter group is of minimal cardinality provided that the non-diagonal entries of the Coxeter matrix are sufficiently large.

The analysis of manifold valued data using embedding based methods is linked to the problem of finding suitable embeddings. In this paper we are interested in embeddings of quotient manifolds $\mathrm{SO}(3)/\mathcal{S}$ of the rotation…

It is well known that a compact two dimensional surface is homeomorphic to a polygon with the edges identified in pairs. This paper not only presents a new proof of this statement but also generalizes it to any connected $n$-dimensional…

Let $A$ be an Artin group. A partition $\mathcal{P}$ of the set of standard generators of $A$ is called admissible if, for all $X,Y \in \mathcal{P}$, $X \neq Y$, there is at most one pair $(s,t) \in X \times Y$ which has a relation. An…

We use probabilistic methods to prove that many Coxeter groups are incoherent. In particular, this holds for Coxeter groups of uniform exponent > 2 with sufficiently many generators.

We consider Lusztig's $\mathbf{a}$-function on Coxeter groups (in the equal parameter case) and classify all Coxeter groups with finitely many elements of $\mathbf{a}$-value 2 in terms of Coxeter diagrams.

We introduce a class of affine Deligne--Lusztig varieties that we call of positive Coxeter type. We show that the affine Deligne--Lusztig varieties of positive Coxeter type have a very simple and explicitly described geometric structure.…

In a recent paper, Barot and Marsh presented an explicit construction of presentation of a finite Weyl group by any seed of corresponding cluster algebra, i.e. by any diagram mutation-equivalent to an orientation of a Dynkin diagram with…

Tverberg's theorem states that any set of $t(r,d)=(r-1)(d+1)+1$ points in $\mathbb{R}^d$ can be partitioned into $r$ subsets whose convex hulls have non-empty $r$-fold intersection. Moreover, generic collections of fewer points cannot be so…

We consider the class of those Coxeter groups for which removing from the Cayley graph any tubular neighbourhood of any wall leaves exactly two connected components. We call these Coxeter groups bipolar. They include both the virtually…

We demonstrate a novel route of spatially organizing the colloid arrangements on the polymer-grafted substrate by use of self-consistent field and density functional theories. We find that grafting of polymers onto a substrate can…

It is a theorem due to F. Haglund and D. Wise that reflection groups (aka Coxeter groups) virtually embed into right-angled reflection groups (aka right-angled Coxeter groups). In this article, we generalise this observation to rotation…

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…