Positive cluster complexes and $\tau$-tilting simplicial complexes of cluster-tilted algebras of finite type

TL;DR

This paper investigates the structure of positive cluster complexes in finite type, providing formulas for face vector differences caused by mutations, explicit descriptions, and applications to $ au$-tilting theory of cluster-tilted algebras.

Contribution

It introduces explicit formulas and methods for analyzing positive cluster complexes and connects these to $ au$-tilting modules in finite type cluster-tilted algebras.

Findings

Formulas for face vector differences under mutation

Explicit descriptions of positive cluster complexes of finite type

Method to compute face vectors of positive cluster complexes

Abstract

In this study, we consider the positive cluster complex, a full subcomplex of a cluster complex the vertices of which are all non-initial cluster variables. In particular, we provide a formula for the difference in face vectors of positive cluster complexes caused by a mutation for finite type. Moreover, we explicitly describe specific positive cluster complexes of finite type and calculate their face vectors. We also provide a method to compute the face vector of an arbitrary positive cluster complex of finite type using these results. Furthermore, we apply our results to the $\tau$-tilting theory of cluster-tilted algebras of finite representation type using the correspondence between clusters and support $\tau$-tilting modules.