Generalized Cluster Complexes and Coxeter Combinatorics

In the first part of this paper (Sections 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12), we introduce and study a simplicial complex ∆(Φ) associated to a finite root system Φ and a nonnegative integer parameter m. For m = 1, our construction specializes to the (simplicial) generalized associahedra ∆(Φ) introduced in [13] and identified in [14] as the underlying complexes for the cluster algebras of finite type. We enumerate the faces of the complexes ∆(Φ) and determine their Euler characteristics. For the classical types in the Cartan-Killing classification, we provide explicit combinatorial descriptions of these complexes in terms of dissections of a convex polygon into smaller polygons. In types An and Bn, we rediscover the constructions given by Tzanaki [29]. Face numbers of the complexes ∆(Φ) provide natural generalizations of the Fuss-Catalan, Kirkman-Cayley, and Przytycki-Sikora numbers to arbitrary types. Our computations of h-vectors of these complexes recover enumerative invariants defined in other contexts by Athanasiadis [3], suggesting links to a host of well-studied problems in algebraic combinatorics of finite Coxeter groups, root systems, and hyperplane arrangements. The second part of the paper (Sections 13 and 14) is devoted to combinatorial algorithms for determining Coxeter-theoretic invariants. Starting with a Coxeter diagram of a finite Coxeter group (or with the corresponding Dynkin diagram or Cartan matrix), we compute the Coxeter number, the exponents, and other related invariants by a procedure (in fact, by several alternative procedures) which is entirely combinatorial in