The Dynkin diagram cohomology of finite Coxeter groups

TL;DR

This paper investigates the cohomology of the Dynkin complex associated with finite Coxeter groups, demonstrating rigidity of quasi-Coxeter algebra structures and providing explicit cohomology computations.

Contribution

It computes the Dynkin complex cohomology for finite Coxeter groups and establishes the rigidity of quasi-Coxeter algebra structures on their group algebras.

Findings

Cohomology computed explicitly for finite Coxeter groups

Proves rigidity of quasi-Coxeter algebra structures on group algebras

Provides partial results for affine Coxeter groups

Abstract

Let D be a connected graph. The Dynkin complex CD(A) of a D-algebra A was introduced by the second author in [TL2] to control the deformations of quasi-Coxeter algebra structures on A. In the present paper, we study the cohomology of this complex when A is the group algebra of a Coxeter group W and D is the Dynkin diagram of W. We compute this cohomology when W is finite and prove in particular the rigidity of quasi-Coxeter algebra structures on kW. For an arbitrary W, we compute the top cohomology group and obtain a number of additional partial results when W is affine. Our computations are carried out by filtering CD(A) by the number of vertices of subgraphs of D. The corresponding graded complex turns out to be dual to the sum of the Coxeter complexes of all standard, irreducible parabolic subgroups of W.