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Generic bases for cluster algebras and the Chamber Ansatz


Authors: Christof Geiß, Bernard Leclerc and Jan Schröer
Journal: J. Amer. Math. Soc. 25 (2012), 21-76
MSC (2010): Primary 13F60, 14M15, 14M99, 16G20, 20G44
DOI: https://doi.org/10.1090/S0894-0347-2011-00715-7
Published electronically: August 10, 2011
MathSciNet review: 2833478
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Abstract: Let $ Q$ be a finite quiver without oriented cycles, and let $ \Lambda$ be the corresponding preprojective algebra. Let $ \mathfrak{g}$ be the Kac-Moody Lie algebra with Cartan datum given by $ Q$, and let $ W$ be its Weyl group. With $ w \in W$, there is associated a unipotent cell $ N^w$ of the Kac-Moody group with Lie algebra $ \mathfrak{g}$. In previous work we proved that the coordinate ring $ \mathbb{C}[N^w]$ of $ N^w$ is a cluster algebra in a natural way. A central role is played by generating functions $ \varphi_X$ of Euler characteristics of certain varieties of partial composition series of $ X$, where $ X$ runs through all modules in a Frobenius subcategory $ \mathcal{C}_w$ of the category of nilpotent $ \Lambda$-modules. The first aim of this article is to compare the function $ \varphi_X$ with the so-called cluster character of $ X$, which is defined in terms of the Euler characteristics of quiver Grassmannians. We show that for every $ X$ in $ \mathcal{C}_w$, $ \varphi_X$ coincides, after an appropriate change of variables, with the cluster character of Fu and Keller associated with $ X$ using any cluster-tilting object $ T$ of $ \mathcal{C}_w$. A crucial ingredient of the proof is the construction of an isomorphism between varieties of partial composition series of $ X$ and certain quiver Grassmannians. This isomorphism is obtained in a very general setup and should be of interest in itself. Another important tool of the proof is a representation-theoretic version of the Chamber Ansatz of Berenstein, Fomin and Zelevinsky, adapted to Kac-Moody groups. As an application, we get a new description of a generic basis of the cluster algebra $ \mathcal{A}(\underline{\Gamma}_T)$ obtained from $ \mathcal{C}[N^w]$ via specialization of coefficients to 1. Here generic refers to the representation varieties of a quiver potential arising from the cluster-tilting module $ T$. For the special case of coefficient-free acyclic cluster algebras this proves a conjecture by Dupont.


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  • [A] C. Amiot, Cluster categories for algebras of global dimension $ 2$ and quivers with potential. Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2525-2590. MR 2640929 (2011c:16026)
  • [ART] C. Amiot, I. Reiten, G. Todorov, The ubiquity of generalized cluster categories. Adv. Math. 226 (2011), no. 4, 3813-3849. arXiv:0911.4819v1 [math.RT]. MR 2764906
  • [BFZ] A. Berenstein, S. Fomin, A. Zelevinsky, Parametrizations of canonical bases and totally positive matrices, Adv. Math. 122 (1996), 49-149. MR 1405449 (98j:17008)
  • [BZ] A. Berenstein, A. Zelevinsky, Total positivity in Schubert varieties. Comment. Math. Helv. 72 (1997), no. 1, 128-166. MR 1456321 (99g:14064)
  • [Bo] K. Bongartz, Minimal singularities for representations of Dynkin quivers. Comment. Math. Helv. 69 (1994), no. 4, 575-611. MR 1303228 (96f:16016)
  • [BIRS] A. Buan, O. Iyama, I. Reiten, J. Scott, Cluster structures for $ 2$-Calabi-Yau categories and unipotent groups. Compos. Math. 145 (2009), 1035-1079. MR 2521253 (2010h:18021)
  • [BIRSm] A. Buan, O. Iyama, I. Reiten, D. Smith, Mutation of cluster-tilting objects and potentials. Amer. J. Math. (to appear), 41pp., Preprint (2008), arXiv:0804.3813v4 [math.RT].
  • [CC] P. Caldero, F. Chapoton, Cluster algebras as Hall algebras of quiver representations. Comment. Math. Helv. 81 (2006), 595-616. MR 2250855 (2008b:16015)
  • [CK] P. Caldero, B. Keller, From triangulated categories to cluster algebras. Invent. Math. 172 (2008), 169-211. MR 2385670 (2009f:16027)
  • [CB] W. Crawley-Boevey, On the exceptional fibres of Kleinian singularities. Amer. J. Math. 122 (2000), 1027-1037. MR 1781930 (2001f:14009)
  • [CBS] W. Crawley-Boevey, J. Schröer, Irreducible components of varieties of modules. J. Reine Angew. Math. 553 (2002), 201-220. MR 1944812 (2004a:16020)
  • [DWZ1] H. Derksen, J. Weyman, A. Zelevinsky, Quivers with potentials and their representations. I. Mutations. Selecta Math. (N.S.) 14 (2008), no. 1, 59-119. MR 2480710 (2010b:16021)
  • [DWZ2] H. Derksen, J. Weyman, A. Zelevinsky, Quivers with potentials and their representations II: applications to cluster algebras. J. Amer. Math. Soc. 23 (2010), no. 3, 749-790. MR 2629987
  • [DK] Y. Drozd, V. Kirichenko, Finite-dimensional algebras. Translated from the 1980 Russian original and with an appendix by Vlastimil Dlab. Springer-Verlag, Berlin, 1994. xiv+249pp. MR 1284468 (95i:16001)
  • [D] G. Dupont, Generic variables in acyclic cluster algebras. 63pp, Preprint (2008), arXiv:0811.2909v1 [math.RT].
  • [FZ1] S. Fomin, A. Zelevinsky, Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15 (2002), no. 2, 497-529. MR 1887642 (2003f:16050)
  • [FZ2] S. Fomin, A. Zelevinsky, Cluster algebras. IV. Coefficients. Compositio Math. 143 (2007), no. 1, 112-164. MR 2295199 (2008d:16049)
  • [FK] Changjian Fu, B. Keller, On cluster algebras with coefficients and $ 2$-Calabi-Yau categories. Trans. Amer. Math. Soc. 362 (2010), no. 2, 859-895. MR 2551509 (2011b:13076)
  • [G] P. Gabriel, Finite representation type is open. Proceedings of the International Conference on Representations of Algebras (Carleton Univ., Ottawa, Ont., 1974), Paper No. 10, 23pp. Carleton Math. Lecture Notes, No. 9, Carleton Univ., Ottawa, Ont., 1974. MR 0376769 (51:12944)
  • [GLS1] C. Geiß, B. Leclerc, J. Schröer, Semicanonical bases and preprojective algebras. Annales Scient. l'École Normale Supérieure 38 (2005), 193-253. MR 2144987 (2007h:17018)
  • [GLS2] C. Geiß, B. Leclerc, J. Schröer, Rigid modules over preprojective algebras. Invent. Math. 165 (2006), no. 3, 589-632. MR 2242628 (2007g:16023)
  • [GLS3] C. Geiß, B. Leclerc, J. Schröer, Partial flag varieties and preprojective algebras. Ann. Institut Fourier, 58 (2008), 825-876. MR 2427512 (2009f:14104)
  • [GLS4] C. Geiß, B. Leclerc, J. Schröer, Cluster algebra structures and semicanonical bases for unipotent groups. 121pp., Preprint (2007), arXiv:math/0703039.
  • [GLS5] C. Geiß, B. Leclerc, J. Schröer, Kac-Moody groups and cluster algebras. Adv. Math. 228 (2011), no. 1, 329-433. arXiv:1001.3545v2 [math.RT].
  • [H] D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras. London Mathematical Society Lecture Note Series, 119. Cambridge University Press, Cambridge, 1988. x+208pp. MR 935124 (89e:16035)
  • [KR] B. Keller, I. Reiten, Cluster tilted algebras are Gorenstein and stably Calabi-Yau. Adv. Math. 211 (2007), 123-151. MR 2313531 (2008b:18018)
  • [KY] B. Keller, D. Yang, Derived equivalences from mutations of quivers with potential. Adv. Math. 226 (2011), no. 3, 2118-2168. arXiv:0906.0761v3 [math.RT]. MR 2739775
  • [Ku] S. Kumar, Kac-Moody groups, their flag varieties and representation theory. Progress in Mathematics, 204. Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1923198 (2003k:22022)
  • [L1] G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras. J. Amer. Math. Soc. 4 (1991), no. 2, 365-421. MR 1088333 (91m:17018)
  • [L2] G. Lusztig, Total positivity in reductive groups, in: Lie Theory and Geometry, Progress in Math. 123 (1994), 531-568. MR 1327548 (96m:20071)
  • [N] K. Nagao, Donaldson-Thomas theory and cluster algebras. 33pp., Preprint (2010), arXiv:1002.4884 [math.AG].
  • [P] Y. Palu, Cluster characters for triangulated $ 2$-Calabi-Yau categories. Ann. Inst. Fourier (Grenoble) 58 (2008), no. 6, 2221-2248. MR 2473635 (2009k:18013)
  • [Pl] P.-G. Plamondon, Cluster algebras via cluster categories with infinite-dimensional morphism spaces. Compositio Math. (to appear), 32pp, Preprint (2010), arXiv:1004.0830 [math.RT].

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Additional Information

Christof Geiß
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, 04510 México D.F., México
Email: christof@math.unam.mx

Bernard Leclerc
Affiliation: LMNO, Université de Caen, CNRS, UMR 6139, F-14032 Caen Cedex, France
Email: leclerc@math.unicaen.fr

Jan Schröer
Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: schroer@math.uni-bonn.de

DOI: https://doi.org/10.1090/S0894-0347-2011-00715-7
Received by editor(s): May 13, 2010
Received by editor(s) in revised form: May 13, 2011
Published electronically: August 10, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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