Particle Configurations and Coxeter Operads

TL;DR

This paper explores generalized moduli spaces derived from Coxeter complexes, constructing explicit models for Weyl groups, and introduces a Coxeter operad with detailed classifications and topological invariants.

Contribution

It provides explicit configuration space models for classical Weyl groups, introduces a Coxeter operad, and classifies the building sets of Coxeter complexes.

Findings

Explicit models for Weyl group configuration spaces

Definition of the Coxeter operad from compactified spaces

Classification of Coxeter complex building sets and Euler characteristic computations

Abstract

There exist natural generalizations of the real moduli space of Riemann spheres based on manipulations of Coxeter complexes. These novel spaces inherit a tiling by the graph-associahedra convex polytopes. We obtain explicit configuration space models for the classical infinite families of finite and affine Weyl groups using particles on lines and circles. A Fulton-MacPherson compactification of these spaces is described and this is used to define the Coxeter operad. A complete classification of the building sets of these complexes is also given, along with a computation of their Euler characteristics.