zbMATH — the first resource for mathematics

On the quiver Grassmannian in the acyclic case. (English) Zbl 1153.14032
Let \(Q\) be a quiver without oriented cycles, let \(A\) be its path algebra and let \(M\) be a finite-dimensional \(A\)-module. The subject of this paper is the geometry of the so-called quiver Grassmannian \(Gr_{\underline e}(M)\) of \(A\)-submodules of dimension \(\underline e\), where \(\underline e\) is a dimension vector; in particular, the Euler characteristics \(\chi_c(Gr_{\underline e}(M))\) of these Grassmannians. This is related to cluster algebras as follows. If \(M\) is an exceptional module (that is, without self-extensions), then there is a suitably defined Laurent polynomial \(\chi_M\) defined in terms of Euler characteristics corresponding to various dimension vectors. It is known that the \(\chi_M\)’s, where \(M\) runs over the set of exceptional modules, form a set of generators of the cluster algebra, called cluster variables. In this paper, an inductive formula to compute \(\chi_c(Gr_{\underline e}(M))\) is given, when is \(M\) is preprojective. This is used to prove a conjecture of Fomin and Zelevinsky for acyclic cluster algebras, namely that \(\chi_c(Gr_{\underline e}(M))\) is positive when \(M\) is exceptional. It is also shown that the quiver Grassmannian is smooth when \(M\) is exceptional.

14L30 Group actions on varieties or schemes (quotients)
14M15 Grassmannians, Schubert varieties, flag manifolds
16G20 Representations of quivers and partially ordered sets
Full Text: DOI arXiv
[1] Auslander, M.; Reiten, I.; Smalø, S.O., Representation theory of Artin algebras, () · Zbl 0834.16001
[2] Bongartz, K.; Dudek, D., Decomposition classes for representations of time quivers, J. algebra, 240, 1, 268-288, (2001) · Zbl 1036.16011
[3] Buan, A.; Marsh, R.; Reiten, I.; Todorov, G., Cluster and seeds in acyclic cluster algebras, with an appendix by A. B. buan, R. J. marsh, P. caldero, B. Keller, I. Reiten, and G. Todorov, Proc. amer. math. soc., 135, 10, 3049-3060, (2007), (electronic) · Zbl 1190.16022
[4] Caldero, P.; Chapoton, F., Cluster algebras as Hall algebras of quiver representations, Comment. math. helv., 81, 595-616, (2006) · Zbl 1119.16013
[5] Caldero, P.; Keller, B., From triangulated categories to cluster algebras, Inv. math., 172, 1, 169-211, (2008), (43) · Zbl 1141.18012
[6] Caldero, P.; Keller, B., From triangulated categories to cluster algebras II, Ann. sc. ec. norm. sup. 4eme serie, 39, 983-1009, (2006) · Zbl 1115.18301
[7] Caldero, P.; Zelevinsky, A., Laurent expansions in cluster algebras via quiver representations, Mosc. math. J., 6, 411-429, (2006), (Special Issue in Honor of Alexander Alexandrovich Kirillov. on the occasion of his seventieth birthday) · Zbl 1133.16012
[8] Chriss, N.; Ginzburg, V., Representation theory and complex geometry, (1997), Birkhäuser Boston · Zbl 0879.22001
[9] W. Crawley-Boevey, On homomorphisms from a fixed representation to a general representation of a quiver, Trans. Amer. Math. Soc. 348 (5) · Zbl 0855.16014
[10] Dlab, V.; Ringel, C.M., Indecomposable representations of graphs and algebras, Mem. amer. math. soc., 6, 173, (1976) · Zbl 0332.16015
[11] Fomin, S.; Zelevinsky, A., Cluster algebras. I. foundations, J. amer. math. soc., 15, 2, 497-529, (2002) · Zbl 1021.16017
[12] Fomin, S.; Zelevinsky, A., Cluster algebras. II. finite type classification, Invent. math., 154, 1, 63-121, (2003) · Zbl 1054.17024
[13] Green, J.A., Hall algebras, hereditary algebras and quantum groups, Invent. math., 120, 2, 361-377, (1995) · Zbl 0836.16021
[14] Hartshorne, R., ()
[15] A. Hubery, Hall polynomials for affine quivers. math.RT/0703178 · Zbl 1241.16011
[16] Kac, V.G., Root systems, representations of quivers and invariant theory, () · Zbl 0497.17007
[17] Kronecker, L., Algebraische reduktion der scharen bilinearer formen, (1890), Sitzungsberichte Akad Berlin, 1225-1237 · JFM 22.0169.01
[18] Le Bruyn, L., Noncommutative geometry@n, (2005), neverendingbooks
[19] Lusztig, G., Introduction to quantum groups, () · Zbl 0304.58022
[20] Lusztig, G., Canonical bases and Hall algebras, (), 365-399 · Zbl 0934.17010
[21] Reineke, M., Counting rational points of quiver moduli, Int. math. res. not., (2006), Art ID 70456 · Zbl 1113.14018
[22] Ringel, C.M., Hall algebras and quantum groups, Invent. math., 101, 3, 583-591, (1990) · Zbl 0735.16009
[23] Schofield, A., General representations of quivers, Proc. London math. soc., 65, 3, 46-64, (1992) · Zbl 0795.16008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.