We establish the existence of an IC basis for the generalized Temperley–Lieb algebra associated to a Coxeter system of arbitrary type. We determine this basis explicitly in the case where the Coxeter system is simply laced and the algebra is finite dimensional.

These are the notes of a mini-course given at the conference “Arrangements in Pyrénées”, held in Pau (France) from 11th to 15th of June, 2012. Definition. Let S be a finite set. A Coxeter matrix on S is a square matrix M = (ms,t)s,t∈S indexed by the elements of S and satisfying: (a) ms,s = 1 for all s ∈ S; (b) ms,t = mt,s ∈ {2, 3, 4, . . . } ∪ {∞} for all s, t ∈ S, s 6= t. A Coxeter matrix is u...

The purpose of this note is to use covering space theory to construct new examples of buildings. On the level of Coxeter systems, this contsruction was described previously in Davis [4]. It goes as follows. Start with a Coxeter system (W ′, S). Change some m(s, t)’s in its Coxeter matrix from integers ≥ 2 to the symbol ∞ . This defines a new Coxeter system (W, S). Suppose C′ is a building of ty...

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 ...

We provide a basic formal framework for the theory of chamber complexes and Coxeter systems, and for buildings as thick chamber complexes endowed with a system of apartments. Along the way, we develop some of the general theory of abstract simplicial complexes and of groups (relying on the group add class for the basics), including free groups and group presentations, and their universal proper...

Associated to any Coxeter system (W,S), there is a labeled simplicial complex L and a contractible CW-complex ΣL (the Davis complex) on which W acts properly and cocompactly. ΣL admits a cellulation under which the nerve of each vertex is L. It follows that if L is a triangulation of S, then ΣL is a contractible n-manifold. In this case, the orbit space, KL := ΣL/W , is a Coxeter orbifold. We p...

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. For a recent survey, see Mühlherr [10...

Let (G, S) be a finitely generated Coxeter group, such that the Coxeter system is indecomposable and the canonical bilinear form is indefinite but non-degenerate. We show that the reduced C∗-algebra of G is simple with unique normalised trace. For an arbitrary finitely generated Coxeter group we prove the validity of a Haagerup inequality: There exist constants C > 0 and Λ ∈ N such that for a f...

In this paper, we study the weak order of Coxeter systems and the combinatorial properties of descent sets. There are three main results: (1) Given a Coxeter system (W,S), some word v ∈W , some subset A ⊆ S disjoint fromDR(v), and some w ∈WA, we proved that DR(vw) ⊆ DR(v) ∪ A. (2) We obtained an explicit map for A ∪ B to dominate B in the case when A,B are commuting disjoint sets, with B finite...