Generalized cluster complexes and Coxeter combinatorics

We introduce and study a simplicial complex associated to an arbitrary finite root system and a nonnegative integer parameter m. For m=1, our construction specializes to the (simplicial) generalized associahedra, also known as cluster complexes of finite type.

Our computation of the face numbers and h-vectors of these complexes recovers the enumerative invariants (generalized Narayana numbers) defined by C.A.Athanasiadis. This suggests links to a host of well studied problems in algebraic combinatorics of finite Coxeter groups, roots systems, and hyperplane arrangements.

Recurrences satisfied by the face numbers of our complexes lead to combinatorial algorithms for determining Coxeter-theoretic invariants. That is, starting with a Coxeter diagram of a finite Coxeter group, one can compute the Coxeter number, the exponents, and other classical invariants by a recursive procedure that only uses most basic graph-theoretic concepts applied to the input diagram.

This is joint work with Nathan Reading. In types A and B, we rediscover the constructions and results obtained by E.Tzanaki .