Some cells in the weighted Coxeter group (C˜n,ℓ˜2n+1)

The sorting order on a Coxeter group

Let (W,S) be an arbitrary Coxeter system. For each sequence ω = (ω1, ω2, . . .) ∈ S∗ in the generators we define a partial order—called the ω-sorting order—on the set of group elements Wω ⊆ W that occur as finite subwords of ω. We show that the ω-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and strong Bruhat orders on the group. Moreover, t...

What Is a Coxeter Group?

Coxeter groups arise in a wide variety of areas, so every mathematician should know some basic facts about them, including their connection to “Dynkin diagrams.” Proofs about these “groups generated by reflections” mainly use group theory, geometry, and combinatorics. This talk will briefly explain: what it means to say that G is a “group generated by reflections” (or, equivalently, that G is a...

On the Center of a Coxeter Group

In this paper, we show that the center of every Coxeter group is finite and isomorphic to (Z2) n for some n ≥ 0. Moreover, for a Coxeter system (W, S), we prove that Z(W ) = Z(W S\S̃) and Z(W S̃ ) = 1, where Z(W ) is the center of the Coxeter group W and S̃ is the subset of S such that the parabolic subgroup W S̃ is the essential parabolic subgroup of (W, S) (i.e. W S̃ is the minimum parabolic subgr...

Weighted L–cohomology of Coxeter groups

Given a Coxeter system .W;S/ and a positive real multiparameter q , we study the “weighted L–cohomology groups,” of a certain simplicial complex † associated to .W;S/ . These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to .W;S/ and the multiparameter q . They have a “von Neumann dimension” with respect to the associated “Hecke–von Neumann algebra” ...

The Cohomology of a Coxeter Group with Group Ring Coefficients