Particle Configurations And

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. Over the past decade, there has been an increased interest in the configuration space of n labeled particles on the projective line. The focus is on a quotient of this space by PGl 2 (C), the affine automorphisms on CP 1. The resulting variety is the moduli space of Riemann spheres with n punctures M 0,n = C n (CP 1)/PGl 2 (C). There is a compactification M 0,n of this space, a smooth variety of complex dimension n−3, coming from Geometric Invariant Theory [24]. The space M 0,n plays a crucial role as a fundamental building block in the theory of Gromov-Witten invariants, also appearing in symplectic geometry and quantum cohomology [20]. Our work is motivated by the real points M 0,n (R) of this space, the set of points fixed under complex conjugation. These real moduli spaces have importance in their own right, appearing in areas such as ζ-motives of Goncharov and Manin [17] and Lagrangian Floer theory of Fukaya [15]. Indeed, M 0,n (R) has even emerged in phylogenetic trees [2] and networks [21]. It was Kapranov [19] who first noticed a relationship between M 0,n (R) and the braid arrangement of hyperplanes, associated to the Coxeter group of type A: Blow-ups of certain cells of the A n Coxeter complex yield a space homeomorphic to a double cover of M 0,n+2 (R). This creates a natural tiling of M 0,n (R) by associahedra, the combinatorics of which is discussed in [12]. Davis et. al have generalized this construction to all Coxeter groups, along with studying the fundamental groups of these blown-up spaces [8]. Carr and Devadoss [6] looked at the inherent tiling of these spaces by the convex polytopes graph-associahedra. 1.2. We begin with elementary results and notation: Section 2 provides the background of Coxeter groups and their associated Coxeter complexes and Section 3 constructs the …