Clusters and seeds in acyclic cluster algebras

Authors:
Aslak Bakke Buan, Robert J. Marsh, Idun Reiten and Gordana Todorov; \protect\break with an Appendix coauthored in addition by P. Caldero; B. Keller

Journal:
Proc. Amer. Math. Soc. **135** (2007), 3049-3060

MSC (2000):
Primary 16G20, 16G70

Published electronically:
June 19, 2007

MathSciNet review:
2322734

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Abstract | References | Similar Articles | Additional Information

Abstract: Cluster algebras are commutative algebras that were introduced by Fomin and Zelevinsky in order to model the dual canonical basis of a quantum group and total positivity in algebraic groups. Cluster categories were introduced as a representation-theoretic model for cluster algebras. In this article we use this representation-theoretic approach to prove a conjecture of Fomin and Zelevinsky, that for cluster algebras with no coefficients associated to quivers with no oriented cycles, a seed is determined by its cluster. We also obtain an interpretation of the monomial in the denominator of a non-polynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra.

**[BFZ]**Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky,*Cluster algebras. III. Upper bounds and double Bruhat cells*, Duke Math. J.**126**(2005), no. 1, 1–52. MR**2110627**, 10.1215/S0012-7094-04-12611-9**[BMR1]**Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten,*Cluster-tilted algebras*, Trans. Amer. Math. Soc.**359**(2007), no. 1, 323–332 (electronic). MR**2247893**, 10.1090/S0002-9947-06-03879-7**[BMR2]**Buan A., Marsh R., Reiten I.*Cluster mutation via quiver representations*, preprint arxiv:math.RT/0412077, to appear in Commentarii Mathematici Helvetici.**[BMRRT]**Buan A., Marsh R., Reineke M., Reiten I., Todorov G.*Tilting theory and cluster combinatorics*, Advances in Mathematics, 204 (2), (2006), 572-618.**[CC]**Caldero P., Chapoton F.*Cluster algebras as Hall algebras of quiver representations*, Commentarii Mathematici Helvetici, 81, (2006), 595-616.**[CCS1]**P. Caldero, F. Chapoton, and R. Schiffler,*Quivers with relations arising from clusters (𝐴_{𝑛} case)*, Trans. Amer. Math. Soc.**358**(2006), no. 3, 1347–1364. MR**2187656**, 10.1090/S0002-9947-05-03753-0**[CCS2]**Caldero P., Chapoton F., Schiffler R.,*Quivers with relations and cluster tilted algebras.*Algebras and Representation Theory, 9, (2006), no. 4, 359-376.**[CK1]**Caldero P., Keller B.*From triangulated categories to cluster algebras*, to appear in Inventiones Math.**[CK2]**Caldero P., Keller B.*From triangulated categories to cluster algebras II*, Annales Scientifiques de l'Ecole Normale Supérieure, 4eme serie, 39, (2006), 983-1009.**[FZ1]**Sergey Fomin and Andrei Zelevinsky,*Cluster algebras. I. Foundations*, J. Amer. Math. Soc.**15**(2002), no. 2, 497–529 (electronic). MR**1887642**, 10.1090/S0894-0347-01-00385-X**[FZ2]**Sergey Fomin and Andrei Zelevinsky,*Cluster algebras. II. Finite type classification*, Invent. Math.**154**(2003), no. 1, 63–121. MR**2004457**, 10.1007/s00222-003-0302-y**[Kel]**Bernhard Keller,*On triangulated orbit categories*, Doc. Math.**10**(2005), 551–581. MR**2184464****[Ker]**Otto Kerner,*Representations of wild quivers*, Representation theory of algebras and related topics (Mexico City, 1994), CMS Conf. Proc., vol. 19, Amer. Math. Soc., Providence, RI, 1996, pp. 65–107. MR**1388560****[MRZ]**Robert Marsh, Markus Reineke, and Andrei Zelevinsky,*Generalized associahedra via quiver representations*, Trans. Amer. Math. Soc.**355**(2003), no. 10, 4171–4186. MR**1990581**, 10.1090/S0002-9947-03-03320-8**[RT]**Reiten I., Todorov G.*unpublished*.

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Additional Information

**Aslak Bakke Buan**

Affiliation:
Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet, N-7491 Trondheim, Norway

Email:
aslakb@math.ntnu.no

**Robert J. Marsh**

Affiliation:
Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, England

Email:
rjm25@mcs.le.ac.uk

**Idun Reiten**

Affiliation:
Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet, N-7491 Trondheim, Norway

Email:
idunr@math.ntnu.no

**Gordana Todorov**

Affiliation:
Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, Massachusetts 02115

Email:
todorov@neu.edu

DOI:
https://doi.org/10.1090/S0002-9939-07-08801-6

Received by editor(s):
December 1, 2005

Received by editor(s) in revised form:
June 4, 2006

Published electronically:
June 19, 2007

Communicated by:
Martin Lorenz

Article copyright:
© Copyright 2007
by the authors