Essays on Coxeter groups Coxeter elements in finite Coxeter groups

On the Maximally Clustered Elements of Coxeter Groups

We continue the study of the maximally clustered elements for simply laced Coxeter groups which were recently introduced by Losonczy. Such elements include as a special case the freely braided elements of Losonczy and the author, which in turn constitute a superset of the iji-avoiding elements of Fan. Our main result is to classify the MC-finite Coxeter groups, namely those Coxeter groups havin...

Characterization of cyclically fully commutative elements in finite and affine Coxeter groups

An element of a Coxeter group is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. An element of a Coxeter group is cyclically fully commutative if any of its cyclic shifts remains fully commutative. These elements were studied by Boothby et al. In particular the authors precisely identified the Coxeter groups ...

Complete Growth Series of Coxeter Groups with more than Three Generators

The Enumeration of Coxeter Elements

Let (W, S, 0) be a Coxeter system: a Coxeter group W with S its distinguished generator set and 0 its Coxeter graph. In the present paper, we always assume that the cardinality l = |S| of S is finite. A Coxeter element of W is by definition a product of all generators s ∈ S in any fixed order. We use the notation C(W ) to denote the set of all the Coxeter elements in W . These elements play an ...

On the Isomorphism Problem for Finitely Generated Coxeter Groups. I Basic Matching

The isomorphism problem for finitely generated Coxeter groups is the problem of deciding if two finite Coxeter matrices define isomorphic Coxeter groups. Coxeter [3] solved this problem for finite irreducible Coxeter groups. Recently there has been considerable interest and activity on the isomorphism problem for arbitrary finitely generated Coxeter groups. In this paper, we determine some stro...