Hostname: page-component-cb9f654ff-hn9fh Total loading time: 0 Render date: 2025-08-08T18:38:49.889Z Has data issue: false hasContentIssue false
  • English
  • Français

Noncrossing partitions and representations of quivers

Part of: Algebraic combinatorics Representation theory of rings and algebras

Published online by Cambridge University Press:  03 December 2009

Colin Ingalls  and
Hugh Thomas
Colin Ingalls
Affiliation:
Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick E3B 5A3, Canada (email: cingalls@unb.ca)
Hugh Thomas
Affiliation:
Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick E3B 5A3, Canada (email: hthomas@unb.ca)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We situate the noncrossing partitions associated with a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated with a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver Q without oriented cycles are in natural bijection with the cluster tilting objects in the associated cluster category. We also show that these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition.

Keywords

noncrossing partitionsDynkin quiversreflections groupscluster categoryquiver representationstorsion classwide subcategoriesCambrian latticesemistable subcategories

MSC classification

Secondary: 16G20: Representations of quivers and partially ordered sets 05E15: Combinatorial aspects of groups and algebras

Information

Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 6 , November 2009 , pp. 1533 - 1562
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Assem, I., Simson, D. and Skowroński, A., Elements of the representation theory of associative algebras, 1: Techniques of representation theory, London Mathematical Society Student Texts, vol. 65 (Cambridge University Press, Cambridge, 2006).Google Scholar
[2]Auslander, M. and Smalø, S., Preprojective modules over Artin algebras, J. Algebra 66(1) (1980), 61–122.CrossRefGoogle Scholar
[3]Athanasiadis, C., Brady, T., McCammond, J. and Watt, C., h-vectors of generalized associahedra and non-crossing partitions, Int. Math. Res. Not. 2006, 28pp., Art. ID 69705.Google Scholar
[4]Bessis, D., The dual braid monoid, Ann. Sci. École Norm. Sup. (4) 36 (2003), 647–683.Google Scholar
[5]Blass, A. and Sagan, B., Möbius functions of lattices, Adv. Math. 127 (1997), 94–123.CrossRefGoogle Scholar
[6]Bourbaki, N., Groupes et algèbres de Lie (Hermann, Paris, 1968).Google Scholar
[7]Brady, T. and Watt, C., K(π,1)’s for Artin groups of finite type, Geom. Dedicata 94 (2002), 225–250.Google Scholar
[8]Brady, T. and Watt, C., Non-crossing partition lattices in finite real reflection groups, Trans. Amer. Math. Soc. 360 (2008), 1983–2005.CrossRefGoogle Scholar
[9]Buan, A., Marsh, R., Reineke, M., Reiten, I. and Todorov, G., Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572–618.Google Scholar
[10]Buan, A., Marsh, R. and Reiten, I., Cluster mutation via quiver representations, Comment. Math. Helv. 83 (2008), 143–177.CrossRefGoogle Scholar
[11]Buan, A., Marsh, R., Reiten, I. and Todorov, G., Clusters and seeds in acyclic cluster algebras, Proc. Amer. Math. Soc. 135(10) (2007), 3049–3060 (with appendix by the above authors, P. Caldero, and B. Keller).Google Scholar
[12]Caldero, P. and Keller, B., From triangulated categories to cluster algebras II, Ann. Sci. École Norm. Sup. (4) 39 (2006), 983–1009.Google Scholar
[13]Caldero, P. and Keller, B., From triangulated categories to cluster algebras, Invent. Math. 172(1) (2008), 169–211.Google Scholar
[14]Crawley-Boevey, W., Exceptional sequences of representations of quivers, in Proceedings of the sixth international conference on representations of algebras (Ottawa, ON, 1992), Carleton–Ottawa Mathematical Lecture Note Series, vol. 14 (Carleton University, Ottawa, ON, 1992), p. 7.Google Scholar
[15]Digne, F., Présentations duales des groupes de tresses de type affine , Comment. Math. Helv. 81 (2006), 23–47.Google Scholar
[16]Dyer, M., On minimal lengths of expressions of Coxeter group elements as products of reflections, Proc. Amer. Math. Soc. 129 (2001), 2591–2595.CrossRefGoogle Scholar
[17]Fomin, S. and Zelevinsky, A., Cluster algebras I: foundations, J. Amer. Math. Soc. 15 (2002), 497–529.CrossRefGoogle Scholar
[18]Fomin, S. and Zelevinsky, A., Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), 977–1018.Google Scholar
[19]Gabriel, P. and Roiter, A., Representations of finite dimensional algebras (Springer, Berlin, 1992).Google Scholar
[20]Happel, D. and Unger, L., On the quiver of tilting modules, J. Algebra 284 (2005), 847–868.CrossRefGoogle Scholar
[21]Humphreys, J., Reflection groups and Coxeter groups (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[22]Igusa, K. and Schiffler, R., Exceptional sequences and clusters, with an appendix by the authors and H. Thomas, Preprint (2009), arXiv:0901.2590.Google Scholar
[23]Keller, B., On triangulated orbit categories, Doc. Math. 10 (2005), 551–581.Google Scholar
[24]King, A., Moduli of representations of finite-dimensional algebras, Q. J. Math. (2) 45(180) (1994), 515–530.CrossRefGoogle Scholar
[25]Kreweras, G., Sur les partitions non croisées d’un cycle, Discrete Math. 1 (1972), 333–350.CrossRefGoogle Scholar
[26]Markowsky, G., Primes, irreducibles, and extremal lattices, Order 9 (1992), 265–290.Google Scholar
[27]Marsh, R., Reineke, M. and Zelevinsky, A., Generalized associahedra via quiver representations, Trans. Amer. Math. Soc. 355 (2003), 4171–4186.Google Scholar
[28]McConnell, J. and Robson, J., Noncommutative noetherian rings (John Wiley & Sons, Chichester, 1987).Google Scholar
[29]McNamara, P. and Thomas, H., Poset edge-labelling and left modularity, European J. Combin. 27 (2006), 101–113.CrossRefGoogle Scholar
[30]Pilkington, A., Convex geometries on root systems, Comm. Alg. 34 (2006), 3183–3202.Google Scholar
[31]Reading, N., Cambrian lattices, Adv. Math. 205 (2006), 313–353.CrossRefGoogle Scholar
[32]Reading, N., Clusters, Coxeter-sortable elements and noncrossing partitions, Trans. Amer. Math. Soc. 359 (2007), 5931–5958.CrossRefGoogle Scholar
[33]Reading, N., Sortable elements and Cambrian lattices, Algebra Universalis 56 (2007), 411–437.Google Scholar
[34]Reading, N. and Speyer, D., Cambrian fans, J. Eur. Math. Soc. (JEMS) 11 (2009), 407–447.Google Scholar
[35]Reiner, V., Noncrossing partitions for classical reflection groups, Discrete Math. 177 (1997), 195–222.CrossRefGoogle Scholar
[36]Riedtmann, C. and Schofield, A., On a simplicial complex associated with tilting modules, Comment. Math. Helv. 66 (1991), 70–78.Google Scholar
[37]Ringel, C., The regular components of the Auslander-Reiten quiver of a tilted algebra, Chinese Ann. Math. Ser. B 9 (1988), 1–18.Google Scholar
[38]Rudakov, A., et al. Helices and vector bundles: seminaire Rudakov, London Mathematical Society Lecture Note Series, vol. 148 (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[39]Schofield, A., Semi-invariants of quivers, J. London Math. Soc. (2) 43 (1991), 385–395.Google Scholar
[40]Thomas, H., An analogue of distributivity for ungraded lattices, Order 23 (2006), 249–269.Google Scholar