Related papers: On the maximally clustered elements of Coxeter gro…

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,…

Studying geodesics in Cayley graphs of groups has been a very active area of research over the last decades. We introduce the notion of a uniquely labelled geodesic, abbreviated with u.l.g. These will be studied first in finite Coxeter…

We discuss the symmetries of the Lagrangian of the leptonic sector. We consider the case when this symmetry group is a Coxeter group. The number of elements of the PMNS matrix predicted by this group structure would depend on the number of…

In [5], Elnitsky constructed three elegant bijections between classes of reduced words for Type $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{D}$ families of Coxeter groups and certain tilings of polygons. This paper offers a particular…

We show that certain right-angled Coxeter groups have finite index subgroups that quotient to $\mathbb Z$ with finitely generated kernels. The proof uses Bestvina-Brady Morse theory facilitated by combinatorial arguments. We describe a…

We study classes of right-angled Coxeter groups with respect to the strong submodel relation of parabolic subgroup. We show that the class of all right-angled Coxeter group is not smooth, and establish some general combinatorial criteria…

A classical result of H. S. M. Coxeter asserts that a certain quotient $B(m,n)$ of the braid group $B(m)$ on $m$ strands is finite if and only if $(m,n)$ corresponds to the type of one of the five Platonic solids. If ${\bf k}$ is a knot or…

Let $\mathcal{W}$ be the set of strongly real elements of $W$, a Coxeter group. Then for $w \in \mathcal{W}$, $e(w)$, the excess of $w$, is defined by $e(w) = \min\{\ell(x) + \ell(y) - \ell(w) \; | \; w=xy, x^2 = y^2 = 1\}$. When $W$ is…

We study the connectivity of Morse boundaries of Coxeter groups. We define two conditions on the defining graph of a Coxeter group: wide-avoidant and wide-spherical-avoidant. We show that wide-spherical-avoidant, one-ended, affine-free…

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 the combinatorics of finite finite Coxeter groups, there is a simple formula giving the number of maximal chains of noncrossing partitions. It is a reinterpretation of a result by Deligne which is due to Chapoton, and the goal of this…

In this paper we introduce the galaxy of Coxeter groups -- an infinite dimensional, locally finite, ranked simplicial complex which captures isomorphisms between Coxeter systems. In doing so, we would like to suggest a new framework to…

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$ and $B_-$ be its two opposite Borel subgroups. For two elements $u$, $v$ of the Weyl group $W$, it is known that the coordinate ring ${\mathbb C}[G^{u,v}]$ of the…

Let $W$ be an extended affine Weyl group. We prove that minimal length elements $w_{\co}$ of any conjugacy class $\co$ of $W$ satisfy some special properties, generalizing results of Geck and Pfeiffer \cite{GP} on finite Weyl groups. We…

For the coinvariant rings of finite Coxeter groups of types other than H$_4$, we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and…

For every quiver (valued) of finite representation type we define a finitely presented group called a picture group. This group is very closely related to the cluster theory of the quiver. For example, positive expressions for the Coxeter…

We study two different types of (maximal) almost disjoint families: very mad families and (maximal) cofinitary groups. For the very mad families we prove the basic existence results. We prove that MA implies there exist many pairwise…

Kastermans proved that consistently $\bigoplus_{\aleph_1} \mathbb{Z}_2$ has a cofinitary representation. We present a short proof that $\bigoplus_{\mathfrak{c}} \mathbb{Z}_2$ always has an arithmetic cofinitary representation. Further, for…

We give explicit necessary and sufficient conditions for the abstract commensurability of certain families of 1-ended, hyperbolic groups, namely right-angled Coxeter groups defined by generalized theta-graphs and cycles of generalized…

The deletion order of a finitely generated Coxeter group W is a total order on the elements which, as is proved, is a refinement of the Bruhat order. This order is applied in [8] to construct Elnitsky tilings for any finite Coxeter group.…