Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type A n
Thorsten Holm
, Peter Jørgensen
and Martin Rubey
DOI: 10.1007/s10801-011-0280-x
Abstract
We give a complete classification of torsion pairs in the cluster category of Dynkin type A n . Along the way we give a new combinatorial description of Ptolemy diagrams, an infinite version of which was introduced by Ng ( 1005.4364v1 [math.RT], 2010). This allows us to count the number of torsion pairs in the cluster category of type A n . We also count torsion pairs up to Auslander-Reiten translation.
Pages: 507–523
Keywords: keywords clique; cluster algebra; cluster tilting object; generating function; recursively defined set; species; triangulated category
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3. Beligiannis, A., Reiten, I.: Homological and homotopical aspects of torsion theories. Mem. Amer. Math. Soc. 188(883) (2007)
4. Bergeron, F., Labelle, G., Leroux, P.: Combinatorial Species and Tree-Like Structures. Encyclopedia Math. Appl., vol.
67. Cambridge University Press, Cambridge (1998)
5. Bondarko, M.V.: Weight structures vs. t -structures; weight filtrations, spectral sequences, and complexes (for motives and in general). K-Theory 6, 387-504 (2010)
6. Buan, A.B., Marsh, R.J., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204, 572-618 (2006)
7. Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (An case). Trans. Am. Math. Soc. 358, 1347-1364 (2006)
8. Dickson, S.E.: A torsion theory for abelian categories. Trans. Am. Math. Soc. 121, 223-235 (1966)
9. Holm, T., Jørgensen, P.: On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon. Math. Z. (2011, to appear). doi:, [math.RT]
10. Iyama, O.: Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories. Adv. Math. 210, 22-50 (2007)
11. Iyama, O., Yoshino, Y.: Mutation in triangulated categories and rigid Cohen-Macaulay modules. In- vent. Math. 172, 117-168 (2008)
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16. Pauksztello, D.: Compact corigid objects in triangulated categories and co-t -structures. Cent. Eur. J.
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