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Applied Categorical Structures

, Volume 24, Issue 1, pp 79–104 | Cite as

Mutation of Torsion Pairs in Cluster Categories of Dynkin Type D

  • Sira GratzEmail author
Article

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 \(\mathcal {C}_{D_{n}}\) of Dynkin type D n . Using a combinatorial model introduced by Fomin and Zelevinsky in [7], subcategories in \(\mathcal {C}_{D_{n}}\) correspond to rotationally invariant collections of arcs in a regular 2n-gon, called diagrams of Dynkin type D n . Torsion pairs in \(\mathcal {C}_{D_{n}}\) have been classified by Holm, Jørgensen and Rubey in [10] and correspond to diagrams of Dynkin type D n satisfying a certain combinatorial condition, called Ptolemy diagrams of Dynkin type D n . We define mutation of a diagram \(\mathcal {X}\) of Dynkin type D n with respect to a compatible diagram \(\mathcal {D}\) of Dynkin type D n consisting of pairwise non-crossing arcs. Such a diagram \(\mathcal {D}\) partitions the regular 2n-gon into cells and mutation of \(\mathcal {X}\) with respect to \(\mathcal {D}\) can be thought of as a rotation of each of these cells. We show that mutation of Ptolemy diagrams of Dynkin type D n corresponds to mutation of the corresponding torsion pairs in the cluster category of Dynkin type D n .

Keywords

Cluster category Dynkin type D Triangulated category Torsion pairs 

Mathematics Subject Classification (2010)

13F60 16G20 16G70 18E30 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Institut für Algebra, Zahlentheorie und Diskrete MathematikFakultät für Mathematik und Physik, Leibniz Universität HannoverHannoverGermany

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