Positive cluster complexes and τ-tilting simplicial complexes of cluster-tilted algebras of finite type

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.

KEYWORD:

• Cluster algebras
• cluster complexes
• face vectors
• f-vectors
• mutations
• representation theory of algebras
• support tau-tilting modules

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13F60
• 16G20

Acknowledgments

The author thanks Haruhisa Enomoto for the discussion on positive complexes. The author also thanks Osamu Iyama, Sota Asai, and Aaron Chan for their helpful comments at the Tokyo-Nagoya Algebra Seminar. Iyama also checked the paper and commented on its structure. The author would also like to thank Tomoki Nakanishi.

Additional information

Funding

This work was supported by JSPS KAKENHI (Grant numbers JP20J12675, JP22J00523).