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Quantum cluster characters for valued quivers. (English) Zbl 1371.16014
Let \(F\) be a finite field, and let \((Q,d)\) be a valued quiver without oriented cycles. In a previous paper [Int. Math. Res. Not. 2011, No. 14, 3207–3236 (2011; Zbl 1237.16013)], the author gave a quantum analogue of the cluster character for valued quiver representations [P. Caldero and F. Chapoton, Comment. Math. Helv. 81, No. 3, 595–616 (2006; Zbl 1119.16013)], replacing the Euler-PoincarĂ© characteristics by the cardinalities of valued quiver Grassmannians. It was conjectured that the quantum cluster character provides a bijection between exceptional representations \(V\) of \((Q,d)\) and non-initial quantum cluster variables of the corresponding quantum cluster algebra. Using a recent approach of A. Hubery [“Acyclic cluster algebras via Ringel-Hall algebras”, Preprint, http://www.maths.leeds.ac.uk/\( \sim \)ahubery/Cluster.pdf], the conjecture is proved. As a corollary, it follows that counting polynomials exist for the Grassmannians \(\mathrm{Gr}^V_e\) of subrepresentations in \(V\) of any type \(e\).

MSC:
16G20 Representations of quivers and partially ordered sets
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[1] Berenstein, Arkady; Zelevinsky, Andrei, Quantum cluster algebras, Adv. Math., 195, 2, 405-455, (2005) · Zbl 1124.20028
[2] Buan, Aslak Bakke; Marsh, Robert; Reineke, Markus; Reiten, Idun; Todorov, Gordana, Tilting theory and cluster combinatorics, Adv. Math., 204, 2, 572-618, (2006) · Zbl 1127.16011
[3] Caldero, Philippe; Chapoton, Fr\'ed\'eric, Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv., 81, 3, 595-616, (2006) · Zbl 1119.16013
[4] Caldero, Philippe; Keller, Bernhard, From triangulated categories to cluster algebras. II, Ann. Sci. \'Ecole Norm. Sup. (4), 39, 6, 983-1009, (2006) · Zbl 1115.18301
[5] [dingsheng] M. Ding and J. Sheng, Multiplicative properties of a quantum Caldero-Chapoton map associated to valued quivers, preprint: math/1109.5342v1, 2011.
[6] Ding, Ming; Xu, Fan, A quantum analogue of generic bases for affine cluster algebras, Sci. China Math., 55, 10, 2045-2066, (2012) · Zbl 1271.16017
[7] [ef] A. Efimov, Quantum cluster variables via vanishing cycles, preprint: math.AG/1112.3601v1, 2011.
[8] Fei, Jiarui, Counting using Hall algebras I. Quivers, J. Algebra, 372, 542-559, (2012) · Zbl 1283.16014
[9] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras. I. Foundations, J. Amer. Math. Soc., 15, 2, 497-529 (electronic), (2002) · Zbl 1021.16017
[10] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras. II. Finite type classification, Invent. Math., 154, 1, 63-121, (2003) · Zbl 1054.17024
[11] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras. IV. Coefficients, Compos. Math., 143, 1, 112-164, (2007) · Zbl 1127.16023
[12] Gabriel, Peter, Unzerlegbare Darstellungen. I, Manuscripta Math., 6, 71-103; correction, ibid. 6 (1972), 309, (1972) · Zbl 0232.08001
[13] Happel, Dieter; Ringel, Claus Michael, Tilted algebras, Trans. Amer. Math. Soc., 274, 2, 399-443, (1982) · Zbl 0503.16024
[14] Happel, Dieter; Unger, Luise, Almost complete tilting modules, Proc. Amer. Math. Soc., 107, 3, 603-610, (1989) · Zbl 0675.16012
[15] [hub1] A. Hubery, Acyclic cluster algebras via Ringel-Hall algebras, preprint: www.maths.leeds.ac.uk/\(~\)ahubery/Cluster.pdf. · Zbl 1253.16014
[16] Hubery, Andrew, Quiver representations respecting a quiver automorphism: a generalisation of a theorem of Kac, J. London Math. Soc. (2), 69, 1, 79-96, (2004) · Zbl 1062.16021
[17] Kac, V. G., Infinite root systems, representations of graphs and invariant theory, Invent. Math., 56, 1, 57-92, (1980) · Zbl 0427.17001
[18] Qin, Fan, Quantum cluster variables via Serre polynomials, J. Reine Angew. Math., 668, 149-190, (2012) · Zbl 1252.13013
[19] Ringel, Claus Michael, Representations of \(K\)-species and bimodules, J. Algebra, 41, 2, 269-302, (1976) · Zbl 0338.16011
[20] Ringel, Claus Michael, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics 1099, xiii+376 pp., (1984), Springer-Verlag: Berlin:Springer-Verlag · Zbl 0546.16013
[21] Rupel, Dylan, On a quantum analog of the Caldero-Chapoton formula, Int. Math. Res. Not. IMRN, 14, 3207-3236, (2011) · Zbl 1237.16013
[22] Rupel, Dylan, Proof of the Kontsevich non-commutative cluster positivity conjecture, C. R. Math. Acad. Sci. Paris, 350, 21-22, 929-932, (2012) · Zbl 1266.16028
[23] [schiffmann] O. Schiffmann, Lectures on Hall algebras, preprint: math/0611617v1, 2009.
[24] [zel] A. Zelevinsky, Quantum Cluster Algebras: Oberwolfach talk, February 2005, Unpublished lecture notes: math.QA/0502260, 2005.
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