Mutation of torsion pairs in cluster categories of Dynkin type D

Gratz, S. (2016) Mutation of torsion pairs in cluster categories of Dynkin type D. Applied Categorical Structures, 24(1), pp. 79-104. (doi: 10.1007/s10485-014-9387-2)

Abstract

In cluster categories, mutation of torsion pairs provides a generalisation for the mutation of cluster tilting subcategories, which models the combinatorial structure of cluster algebras. In this paper we present a geometric model for mutation of torsion pairs in the cluster category CDnCDn of Dynkin type Dn. Using a combinatorial model introduced by Fomin and Zelevinsky in [7], subcategories in CDnCDn correspond to rotationally invariant collections of arcs in a regular 2n-gon, called diagrams of Dynkin type Dn. Torsion pairs in CDnCDn have been classified by Holm, Jørgensen and Rubey in [10] and correspond to diagrams of Dynkin type Dn satisfying a certain combinatorial condition, called Ptolemy diagrams of Dynkin type Dn. We define mutation of a diagram XX of Dynkin type Dn with respect to a compatible diagram DD of Dynkin type Dn consisting of pairwise non-crossing arcs. Such a diagram DD partitions the regular 2n-gon into cells and mutation of XX with respect to DD can be thought of as a rotation of each of these cells. We show that mutation of Ptolemy diagrams of Dynkin type Dn corresponds to mutation of the corresponding torsion pairs in the cluster category of Dynkin type Dn.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Gratz, Dr Sira
• Authors: Gratz, S.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Applied Categorical Structures
• Publisher: Springer
• ISSN: 0927-2852
• ISSN (Online): 1572-9095
• Published Online: 07 January 2015