Coxeter Cochain Complexes

TL;DR

This paper introduces the Coxeter cochain complex for Coxeter groups, explores its properties, and connects it to simplicial cohomology, configuration space homology, and local Artin ring homology through representative computations.

Contribution

It defines a new cochain complex for Coxeter groups and links it to various topological and algebraic structures, providing initial computational insights.

Findings

Coxeter cohomology computed for specific groups

Connection established between Coxeter cohomology and configuration space homology

Relation identified between Coxeter cohomology and local Artin ring homology

Abstract

We define the Coxeter cochain complex of a Coxeter group (G,S) with coefficients in a Z[G]-module A. This is closely related to the complex of simplicial cochains on the abstract simplicial complex I(S) of the commuting subsets of S. We give some representative computations of Coxeter cohomology and explain the connection between the Coxeter cohomology for groups of type A, the (singular) homology of certain configuration spaces, and the (Tor) homology of certain local Artin rings.