Subword complexes, cluster complexes, and generalized multi-associahedra

TL;DR

This paper introduces multi-cluster complexes using subword complexes, unifying finite type cluster complexes and multi-associahedra, and demonstrates their universality and connections to known simplicial complexes.

Contribution

It defines multi-cluster complexes for any finite Coxeter group and integer k, linking them to existing complexes and establishing their universal realization property.

Findings

Multi-cluster complexes are spherical and generalize known complexes.

For k=1, they are isomorphic to classical cluster complexes.

They encompass multi-triangulations and symmetric multi-triangulations.

Abstract

In this paper, we use subword complexes to provide a uniform approach to finite type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. For k=1, we show that this subword complex is isomorphic to the cluster complex of the given type. We show that multi-cluster complexes of types A and B coincide with known simplicial complexes, namely with the simplicial complexes of multi-triangulations and centrally symmetric multi-triangulations respectively. Furthermore, we show that the multi-cluster complex is universal in the sense that every spherical subword complex can be realized as a link of a face of the multi-cluster complex.