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Cluster fans, stability conditions, and domains of semi-invariants


Author: Calin Chindris
Journal: Trans. Amer. Math. Soc. 363 (2011), 2171-2190
MSC (2000): Primary 16G20; Secondary 05E15
DOI: https://doi.org/10.1090/S0002-9947-2010-05184-0
Published electronically: November 9, 2010
MathSciNet review: 2746679
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that the cone of finite stability conditions of a quiver $ Q$ without oriented cycles has a fan covering given by (the dual of) the cluster fan of $ Q$. Along the way, we give new proofs of Schofield's results (1991) on perpendicular categories. From our results, we recover Igusa-Orr-Todorov-Weyman's theorem on cluster complexes and domains of semi-invariants for Dynkin quivers. For arbitrary quivers, we also give a description of the domains of semi-invariants labeled by real Schur roots in terms of quiver exceptional sets.


References [Enhancements On Off] (What's this?)

  • 1. A. B. Buan, R. Marsh, M. Reineke, I. Reiten, and G. Todorov.
    Tilting theory and cluster combinatorics.
    Adv. Math., 204(2):572-618, 2006. MR 2249625 (2007f:16033)
  • 2. W. Crawley-Boevey.
    Exceptional sequences of representations of quivers.
    In Representations of algebras (Ottawa, ON, 1992), volume 14 of CMS Conf. Proc., pages 117-124. Amer. Math. Soc., Providence, RI, 1993. MR 1265279
  • 3. W. Crawley-Boevey and Ch. Geiss.
    Horn's problem and semi-stability for quiver representations.
    In Representations of algebra. Vol. I, II, pages 40-48. Beijing Norm. Univ. Press, Beijing, 2002. MR 2067369 (2005e:16024)
  • 4. H. Derksen and J. Weyman. Semi-invariants of quivers and saturation for Littlewood- Richardson coefficients. J. Amer. Math. Soc., 13(3):467-479, 2000. MR 1758750 (2001g:16031)
  • 5. H. Derksen and J. Weyman.
    The combinatorics of quiver representations.
    Preprint, arXiv.math.RT/0608288, 2006.
  • 6. M. Domokos and A. N. Zubkov.
    Semi-invariants of quivers as determinants.
    Transform. Groups, 6(1):9-24, 2001. MR 1825166 (2002d:16015)
  • 7. K. Igusa, K. Orr, G. Todorov, and J. Weyman.
    Cluster complexes via semi-invariants.
    Preprint, ar Xiv:0708.0798v2, 2008.
  • 8. K. Igusa and K. E. Orr.
    Links, pictures and the homology of nilpotent groups.
    Topology, 40(6):1125-1166, 2001. MR 1867241 (2003a:57011)
  • 9. C. Ingalls and H. Thomas.
    Noncrossing partitions and representations of quivers.
    Compos. Math. 145(6):1533-1562, (2009). MR 2575093
  • 10. B. Keller.
    $ A$-infinity algebras in representation theory.
    In Representations of algebra. Vol. I, II, pages 74-86. Beijing Norm. Univ. Press, Beijing, 2002. MR 2067371 (2005b:16021)
  • 11. B. Keller.
    $ A$-infinity algebras, modules and functor categories.
    In Trends in representation theory of algebras and related topics, volume 406 of Contemp. Math., pages 67-93. Amer. Math. Soc., Providence, RI, 2006. MR 2258042 (2007g:18002)
  • 12. A.D. King.
    Moduli of representations of finite-dimensional algebras.
    Quart. J. Math. Oxford Ser.(2), 45(180):515-530, 1994. MR 1315461 (96a:16009)
  • 13. A. Knutson and T. Tao.
    The honeycomb model of $ \textrm{gl}_n(\mathbb{C})$ tensor products. $ \mathrm{I}$. Proof of the saturation conjecture.
    J. Amer. Math. Soc., 12(4):1055-1090, 1999. MR 1671451 (2000c:20066)
  • 14. D. Luna.
    Slices étales.
    In Sur les groupes algébriques, pages 81-105. Bull. Soc. Math. France, Paris, Mémoire 33. Soc. Math. France, Paris, 1973. MR 0318167 (47:6714)
  • 15. R. Marsh, M. Reineke, and A. Zelevinsky.
    Generalized associahedra via quiver representations.
    Trans. Amer. Math. Soc., 355(10):4171-4186, 2003. MR 1990581 (2004g:52014)
  • 16. C.M. Ringel.
    Representations of $ \textit{K}$-species and bimodules.
    J. Algebra, 41(2):269-302, 1976. MR 0422350 (54:10340)
  • 17. M. Sato and T. Kimura.
    A classification of irreducible prehomogeneous vector spaces and their relative invariants.
    Nagoya Math. J., 65:1-155, 1977. MR 0430336 (55:3341)
  • 18. A. Schofield.
    Semi-invariants of quivers.
    J. London Math. Soc. (2), 43(3):385-395, 1991. MR 1113382 (92g:16019)
  • 19. A. Schofield.
    General representations of quivers.
    Proc. London Math. Soc. (3), 65(1):46-64, 1992. MR 1162487 (93d:16014)
  • 20. A. Schofield and M. van den Bergh.
    Semi-invariants of quivers for arbitrary dimension vectors.
    Indag. Math. (N.S.), 12(1):125-138, 2001. MR 1908144 (2003e:16016)

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Additional Information

Calin Chindris
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Address at time of publication: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: calin-chindris@uiowa.edu, chindrisc@missouri.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-05184-0
Keywords: Clusters, domains of semi-invariants, exceptional sequences, stability
Received by editor(s): March 5, 2009
Received by editor(s) in revised form: August 17, 2009
Published electronically: November 9, 2010
Article copyright: © Copyright 2010 American Mathematical Society

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