The main goal of the paper is to show that the fully commutative elements in the affine Coxeter group e Cn form a union of two-sided cells. Then we completely answer the question of when the fully commutative elements of W form or do not form a union of two-sided cells in the case where W is either a finite or an affine Coxeter group. Let W be a Coxeter group with S the distinguished generator ...

This paper presents a technique for constructing new chiral or regular polyhedra (or maps) from self-dual abstract chiral polytopes of rank 4. From improperly self-dual chiral polytopes we derive “Petrie-Coxeter-type” polyhedra (abstract chiral analogues of the classical Petrie-Coxeter polyhedra) and investigate their groups of automorphisms.

The modular symmetries of string loop threshold corrections for gauge coupling constants are studied in the presence of discrete Wilson lines for all examples of abelian orbifolds, where the point group is realised by the action of Coxeter elements or generalised Coxeter elements on the root lattices of the Lie groups.

We introduce and study deformations of finite-dimensional modules over rational Cherednik algebras. Our main tool is a generalization of usual harmonic polynomials for each Coxeter groups — the so-called quasiharmonic polynomials. A surprising application of this approach is the construction of canonical elementary symmetric polynomials and their deformations for all Coxeter groups.

We conjecture that every planar graph of odd-girth at least 11 admits a homomorphism to the Coxeter graph. Supporting this conjecture, we prove that every planar graph of odd-girth at least 17 admits a homomorphism to the Coxeter graph.

We construct for every graph product of finitely generated abelian groups a CAT(0) cubical complex on which it acts properly and cocompactly. The complex generalizes (up to subdivision) the Salvetti complex of a right-angled Artin group and the Coxeter complex of a right-angled Coxeter group.

We introduce a notion of essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter polytopes. We determine a potentially large combinatorial class of polytopes containing, in particular, all the compact hyperbolic Coxeter polytopes of dimens...

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 operations (in the sense of Lusztig). We show that the Kazhdan–Lusztig bases of these groups have a nice projection property to the Temperley–Lieb type quotient, an...

We combine the theory of Coxeter groups, the covering theory of graphs introduced by Malnic, Nedela and Skoviera and the theory of reflections of graphs in order to obtain the following characterization of a Coxeter group: Let π : Γ → (v,D, ι,−1) be a 1-covering of a monopole admitting semiedges only. The graph Γ is the Cayley graph of a Coxeter group if and only if π is regular and any deck tr...

When W is a finite Coxeter group of classical type (A, B, or D), noncrossing partitions associated to W and compatibility of almost positive roots in the associated root system are known to be modeled by certain planar diagrams. We show how the classical-type constructions of planar diagrams arise uniformly from projections of smallW -orbits to the Coxeter plane. When the construction is applie...