Hom-configurations and noncrossing partitions
Raquel Coelho Simões
DOI: 10.1007/s10801-011-0305-5
Abstract
We study maximal Hom-free sets in the τ [2]-orbit category C( Q) of the bounded derived category for the path algebra associated to a Dynkin quiver Q, where τ denotes the Auslander-Reiten translation and [2] denotes the square of the shift functor. We prove that these sets are in bijection with periodic combinatorial configurations, as introduced by Riedtmann, certain Hom\thinspace \leq 0-configurations, studied by Buan, Reiten and Thomas, and noncrossing partitions of the Coxeter group associated to Q which are not contained in any proper standard parabolic subgroup. Note that Reading has proved that these noncrossing partitions are in bijection with positive clusters in the associated cluster algebra. Finally, we give a definition of mutation of maximal Hom-free sets in C( Q) \mathcal {C}(Q) and prove that the graph of these mutations is connected.
Pages: 313–343
Keywords: keywords cluster combinatorics; derived category; exceptional sequences; hom-configurations; mutations; noncrossing partitions; perpendicular categories; reflection functors; quiver representations
Full Text: PDF
References
1. Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Al- gebras, vol
1. Techniques of Representation Theory. London Mathematical Society Student Texts, vol.
65. Cambridge University Press, Cambridge (2006)
2. Bessis, D.: The dual braid monoid. Ann. Sci. Éc. Norm. Super. 36, 647-683 (2003)
3. Biane, P.: Some properties of crossings and partitions. Discrete Math. 175, 41-53 (1997)
4. Brady, T.: A partial order on the symmetric group and new K(π, 1)'s for the braid group. Adv. Math. 161(1), 20-40 (2001)
5. Brady, T., Watt, C.: Noncrossing partition lattices in finite real reflection groups. Trans. Am. Math. Soc. 360, 1983-2005 (2008)
6. Bretscher, O., Läser, C., Riedtmann, C.: Selfinjective and simply connected algebras. Manuscr. Math. 36(3), 253-307 (1982)
7. Brüning, K.: Thick subcategories of the derived category of a hereditary algebra. Homol. Homot. Appl. 9(2), 165-176 (2007)
8. Buan, A.B., Marsh, R.J., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204, 572-618 (2006)
9. Buan, A.B., Reiten, I., Thomas, H.: m-noncrossing partitions and m-clusters. DMTCS proc. (subm.), extended abstract
10. Buan, A., Reiten, I., Thomas, H.: From m-clusters to m-noncrossing partitions via exceptional sequences. Preprint [math.RT]
11. Crawley-Boevey, W.: Exceptional sequences of representations of quivers. In: Dlab, V., Lenzing, H. (eds.) Representations of Algebras, Proc. Ottawa
1992. Canadian Math. Soc. Conf. Proc, vol. 14, pp. 117-124. Amer. Math. Soc., Providence (1993)
12. Fomin, S., Zelevinsky, A.: Cluster algebras I: Foundations. J. Am. Math. Soc. 15(2), 497-529 (2002)
13. Fomin, S., Zelevinsky, A.: Cluster algebras II: Finite type classification. Invent. Math. 154(1), 63-121 (2003)
14. Fomin, S., Zelevinsky, A.: Y-systems and generalized associahedra. Ann. Math. (2) 158(3), 977-1018 (2003)
15. Gabriel, P.: Auslander-Reiten sequences and representation-finite algebras. In: Representation Theory, I, Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979, pp. 1-71. Springer, Berlin (1980)
16. 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)
17. Igusa, K., Schiffler, R.: Exceptional sequences and clusters. J. Algebra 323(8), 2183-2202 (2010)
18. Ingalls, C., Thomas, H.: Noncrossing partitions and representations of quivers. Compos. Math. 145(6), 1533-1562 (2009)
19. Keller, B.: On triangulated orbit categories. Doc. Math. 10, 551-581 (2005)
20. Kreweras, G.: Sur les partitions non croisées d'un cycle. Discrete Math. 1, 333-350 (1972)
21. Reading, N.: Clusters, Coxeter-sortable elements and noncrossing partitions. Trans. Am. Math. Soc. 359, 5931-5958 (2007)
22. Riedtmann, C.: Representation-finite selfinjective algebras of class An. In: Representation Theory II, Prof. Second Internat. Conf., Carleton Univ., Ottawa, Ont.
1979. Lecture Notes in Math., vol. 832, pp. 449-520. Springer, Berlin (1980)
23. Riedtmann, C.: Configurations of ZDn. J. Algebra 82(2), 309-327 (1983)
24. Ringel, C.: The braid group action on the set of exceptional sequences of a hereditary Artin algebra. In: Abelian Group Theory and Related Topics, Oberwolfach,
1993. Contemp. Math., vol. 171, pp.
1. Techniques of Representation Theory. London Mathematical Society Student Texts, vol.
65. Cambridge University Press, Cambridge (2006)
2. Bessis, D.: The dual braid monoid. Ann. Sci. Éc. Norm. Super. 36, 647-683 (2003)
3. Biane, P.: Some properties of crossings and partitions. Discrete Math. 175, 41-53 (1997)
4. Brady, T.: A partial order on the symmetric group and new K(π, 1)'s for the braid group. Adv. Math. 161(1), 20-40 (2001)
5. Brady, T., Watt, C.: Noncrossing partition lattices in finite real reflection groups. Trans. Am. Math. Soc. 360, 1983-2005 (2008)
6. Bretscher, O., Läser, C., Riedtmann, C.: Selfinjective and simply connected algebras. Manuscr. Math. 36(3), 253-307 (1982)
7. Brüning, K.: Thick subcategories of the derived category of a hereditary algebra. Homol. Homot. Appl. 9(2), 165-176 (2007)
8. Buan, A.B., Marsh, R.J., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204, 572-618 (2006)
9. Buan, A.B., Reiten, I., Thomas, H.: m-noncrossing partitions and m-clusters. DMTCS proc. (subm.), extended abstract
10. Buan, A., Reiten, I., Thomas, H.: From m-clusters to m-noncrossing partitions via exceptional sequences. Preprint [math.RT]
11. Crawley-Boevey, W.: Exceptional sequences of representations of quivers. In: Dlab, V., Lenzing, H. (eds.) Representations of Algebras, Proc. Ottawa
1992. Canadian Math. Soc. Conf. Proc, vol. 14, pp. 117-124. Amer. Math. Soc., Providence (1993)
12. Fomin, S., Zelevinsky, A.: Cluster algebras I: Foundations. J. Am. Math. Soc. 15(2), 497-529 (2002)
13. Fomin, S., Zelevinsky, A.: Cluster algebras II: Finite type classification. Invent. Math. 154(1), 63-121 (2003)
14. Fomin, S., Zelevinsky, A.: Y-systems and generalized associahedra. Ann. Math. (2) 158(3), 977-1018 (2003)
15. Gabriel, P.: Auslander-Reiten sequences and representation-finite algebras. In: Representation Theory, I, Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979, pp. 1-71. Springer, Berlin (1980)
16. 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)
17. Igusa, K., Schiffler, R.: Exceptional sequences and clusters. J. Algebra 323(8), 2183-2202 (2010)
18. Ingalls, C., Thomas, H.: Noncrossing partitions and representations of quivers. Compos. Math. 145(6), 1533-1562 (2009)
19. Keller, B.: On triangulated orbit categories. Doc. Math. 10, 551-581 (2005)
20. Kreweras, G.: Sur les partitions non croisées d'un cycle. Discrete Math. 1, 333-350 (1972)
21. Reading, N.: Clusters, Coxeter-sortable elements and noncrossing partitions. Trans. Am. Math. Soc. 359, 5931-5958 (2007)
22. Riedtmann, C.: Representation-finite selfinjective algebras of class An. In: Representation Theory II, Prof. Second Internat. Conf., Carleton Univ., Ottawa, Ont.
1979. Lecture Notes in Math., vol. 832, pp. 449-520. Springer, Berlin (1980)
23. Riedtmann, C.: Configurations of ZDn. J. Algebra 82(2), 309-327 (1983)
24. Ringel, C.: The braid group action on the set of exceptional sequences of a hereditary Artin algebra. In: Abelian Group Theory and Related Topics, Oberwolfach,
1993. Contemp. Math., vol. 171, pp.
© 1992–2009 Journal of Algebraic Combinatorics
©
2012 FIZ Karlsruhe /
Zentralblatt MATH for the EMIS Electronic Edition