Abstract
Let W be a finite crystallographic reflection group. The generalized Catalan number of W coincides both with the number of clusters in the cluster algebra associated to W, and with the number of noncrossing partitions for W. Natural bijections between these two sets are known. For any positive integer m, both m-clusters and m-noncrossing partitions have been defined, and the cardinality of both these sets is the Fuss–Catalan number C m (W). We give a natural bijection between these two sets by first establishing a bijection between two particular sets of exceptional sequences in the bounded derived category D b(H) for any finite-dimensional hereditary algebra H.
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Armstrong, D.: Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Am. Math. Soc. 202(949) (2009)
Assem I., Souto Salorio M.J., Trepode S.: Ext-projectives in suspended subcategories. J. Pure Appl. Algebra 212, 423–434 (2008)
Athanasiadis, C., Brady, T., McCammond, J., Watt, C.: h-Vectors of generalized associahedra and noncrossing partitions. Int. Math. Res. Notes (2006) (article ID 69705)
Beligiannis, A., Reiten, I.: Homological and homotopical aspects of torsion theories. Mem. Am. Math. Soc. 188(883) (2007)
Bessis D.: The dual braid monoid. Ann. Sci. École Norm. Sup. 36, 647–683 (2003)
Bondal, A.I.: Representations of associative algebras and coherent sheaves. Izv. Akad. Nauk SSSR Ser. Mat. 53, 25–44 (1989); English transl. in Math USSR-Izv. 34(1), 23–44 (1990)
Bongartz, K.: Tilted algebras. Representations of Algebras (Puebla, 1980). Lecture Notes in Mathematics, vol. 903, pp. 26–38. Springer, Berlin (1981)
Bretscher, O., Läser, C., Riedtmann, C.: Self-injective and simply-connected algebras. Manuscr. Math. 36(3), 253–307 (1981/82)
Bridgeland T.: T-structures on some local Calabi–Yau varieties. J. Algebra 289(2), 453–483 (2005)
Buan A.B., Marsh R.J., Reineke M., Reiten I., Todorov G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)
Buan, A.B., Reiten, I., Thomas, H.: m-noncrossing partitions and m-clusters. In: 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), Discrete Mathematics and Theoretical Computer Science Proceedings, pp. 145–154 (2009)
Buan, A.B., Reiten, I., Thomas, H.: Three kinds of mutation. J. Algebra (to appear). arXiv:1005.0276
Coelho F., Happel D., Unger L.: Complements to partial tilting modules. J. Algebra 170(1), 184–205 (1994)
Crawley-Boevey, W.: Exceptional sequences of representations of quivers. Representations of Algebras (Ottawa, ON, 1992). CMS Conference Proceedings, vol. 14, pp. 117–124. American Mathematical Society, Providence (1993)
Fomin S., Reading N.: Generalized cluster complexes and Coxeter combinatorics. Int. Math. Res. Notes (44), 2709–2757 (2005)
Fomin S., Zelevinsky A.: Y-systems and generalized associahedra. Ann. Math. (2) 158(3), 977–1018 (2003)
Gorodentsev, A.L., Kuleshov, S.A.: Helix theory. Mosc. Math. J. 4(2), 377–440, 535 (2004)
Happel D.: Triangulated categories in the representation theory of finite-dimensional algebras. London Mathematical Society Lecture Note Series, vol. 119. Cambridge University Press, Cambridge (1988)
Happel D., Ringel C.M.: Tilted algebras. Trans. Am. Math. Soc. 274(12), 399–443 (1982)
Happel D., Unger L.: Almost complete tilting modules. Proc. Am. Math. Soc 107(3), 603–610 (1989)
Hubery, A.: The cluster complex of an hereditary Artin algebra. Preprint, arXiv:0812.1475
Igusa K., Schiffler R.: Exceptional sequences and clusters, with an appendix written by the authors and H. Thomas. J. Algebra 323(8), 2183–2202 (2010)
Ingalls C., Thomas H.: Noncrossing partitions and representations of quivers. Compos. Math. 145(6), 1533–1562 (2009)
Keller B.: On orbit triangulated categories. Doc. Math. 10, 551–581 (2005)
Keller B., Vossieck D.: Aisles in the derived category, Deuxième Contact Franco-Belge en Algèbre (Faulx-les-Tombes, 1987). Bull. Soc. Math. Belg. Ser. A 40(2), 239–253 (1988)
Reading N.: Coxeter sortable elements and noncrossing partitions. Trans. Am. Math. Soc. 359(12), 5931–5958 (2007)
Riedtmann, C.: Representation-finite selfinjective algebras of class A n . In: Representation theory II (Prof. Second International Conference, Carleton University, Ottawa, ON, 1979). Lecture Notes in Mathematics, vol. 832. pp. 449–520. Springer, Berlin (1980)
Riedtmann C.: Configurations of \({\mathbb {Z}D_n}\). J. Algebra 82(2), 309–327 (1983)
Riedtmann C., Schofield A.: On open orbits and their complements. J. Algebra 130(2), 388–411 (1990)
Riedtmann C., Schofield A.: On a simplicial complex associated with tilting modules. Commun. Math. Helv. 66(1), 70–78 (1991)
Ringel C.M.: Representations of K-species and bimodules. J. Algebra 41(2), 269–302 (1976)
Ringel, C.M.: The braid group action on the set of exceptional sequences of a hereditary Artin algebra. Abelian Group Theory and Related Topics (Oberwolfach, 1993). Contemporary Mathematics, vol. 171, pp. 339–352. American Mathematical Society, Providence (1994)
Simoes, R.C.: Hom-configurations and noncrossing partitions. Preprint, arXiv:1012.1276
Thomas H.: Defining an m-cluster category. J. Algebra 318(1), 37–46 (2007)
Unger L.: Schur modules over wild, finite-dimensional path algebras with three simple modules. J. Pure Appl. Algebra 64(2), 205–222 (1990)
Wraalsen A.: Rigid objects in higher cluster categories. J. Algebra 321(2), 532–547 (2009)
Zhou Y., Zhu B.: Cluster combinatorics of d-cluster categories. J. Algebra 321(10), 2898–2915 (2009)
Zhu B.: Generalized cluster complexes via quiver representations. J. Algebr. Combin. 27(1), 35–54 (2008)
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A. B. Buan, I. Reiten, and H. Thomas were supported by STOR-FORSK Grant 167130 from NFR. A.B.B. and I.R. were supported by Grant 196600 from NFR. H.T. was supported by an NSERC Discovery Grant.
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Buan, A.B., Reiten, I. & Thomas, H. From m-clusters to m-noncrossing partitions via exceptional sequences. Math. Z. 271, 1117–1139 (2012). https://doi.org/10.1007/s00209-011-0906-7
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DOI: https://doi.org/10.1007/s00209-011-0906-7