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 -tilting theory of cluster-tilted algebras of finite representation type using the correspondence between clusters and support
-tilting modules.
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.