Generalized cluster complexes and Coxeter combinatorics

TL;DR

This paper introduces a family of simplicial complexes linked to finite root systems and Coxeter groups, generalizing known structures and providing recursive algorithms for classical invariants.

Contribution

It generalizes cluster complexes to arbitrary root systems and develops recursive methods to compute Coxeter invariants from Coxeter diagrams.

Findings

Computed face numbers and h-vectors of the complexes.

Established links to algebraic combinatorics and Coxeter invariants.

Provided recursive algorithms for classical Coxeter invariants.

Abstract

We introduce and study a family of simplicial complexes associated to an arbitrary finite root system and a nonnegative integer parameter m. For m=1, our construction specializes to the (simplicial) generalized associahedra or, equivalently, to the cluster complexes for the cluster algebras of finite type. Our computation of the face numbers and h-vectors of these complexes produces the enumerative invariants defined in other contexts by C.A.Athanasiadis, suggesting links to a host of well studied problems in algebraic combinatorics of finite Coxeter groups, root 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. In types A and B, we rediscover the constructions and results obtained by E.Tzanaki <math.CO/0501100>.