Quantized Chebyshev polynomials and cluster characters with coefficients
Grégoire Dupont
Institut Camille Jordan, Université Lyon 1, 43, Bd du 11 novembre 1918, 69622 Villeurbanne Cedex, France
DOI: 10.1007/s10801-009-0198-8
Abstract
We introduce quantized Chebyshev polynomials as deformations of generalized Chebyshev polynomials previously introduced by the author in the context of acyclic coefficient-free cluster algebras. We prove that these quantized polynomials arise in cluster algebras with principal coefficients associated to acyclic quivers of infinite representation types and equioriented Dynkin quivers of type \mathbb A \mathbb{A} . We also study their interactions with bases and especially canonically positive bases in affine cluster algebras.
Pages: 501–532
Keywords: keywords cluster algebras; quantized Chebyshev polynomials; principal coefficients; regular components; orthogonal polynomials
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References
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2. Buan, A., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Advances in Mathematics 204(2), 572-618 (2006) MR2249625 (2007f:16033)
3. Caldero, P., Chapoton, F.: Cluster algebras as Hall algebras of quiver representations. Commentarii Mathematici Helvetici 81, 596-616 (2006) MR2250855 (2008b:16015)
4. Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters. Transactions of the AMS 358, 1347-1354 (2006) MR2187656 (2007a:16025) (1)
5. Cerulli Irelli, G.: Canonically positive basis of cluster algebras of type A . 2 [math.RT] (2009)
6. Caldero, P., Keller, B.: From triangulated categories to cluster algebras II. Annales Scientifiques de l'Ecole Normale Supérieure 39(4), 83-100 (2006) MR2316979 (2008m:16031)
7. Caldero, P., Keller, B.: From triangulated categories to cluster algebras. Inventiones Mathematicae 172, 169-211 (2008) MR2385670
8. Caldero, P., Zelevinsky, A.: Laurent expansions in cluster algebras via quiver representations. Moscow Mathematical Journal 6, 411-429 (2006) MR2274858 (2008j:16045)
9. Dupont, G.: Algèbres amassées affines. PhD thesis, Université Claude Bernard Lyon 1 November 2008
10. Dupont, G.: Cluster multiplication in stable tubes via generalized Chebyshev polynomials. [math.RT] (2008)
11. Dupont, G.: Generic variables in acyclic cluster algebras. [math.RT] (2008)
12. Ding, M., Xiao, J., Xu, F.: Integral bases of cluster algebras and representations of tame quivers. [math.RT] (2009)
13. Fu, C., Keller, B.: On cluster algebras with coefficients and 2-Calabi-Yau categories. Trans. AMS (2009, to appear)
14. Fomin, S., Zelevinksy, A.: Cluster algebras I: Foundations. J. Amer. Math. Soc. 15, 497-529 (2002) MR1887642 (2003f:16050)
15. Fomin, S., Zelevinksy, A.: Cluster algebras IV: Coefficients. Composition Mathematica 143(1), 112- 164 (2007) MR2295199 (2008d:16049)
16. Gabriel, P.: Auslander-Reiten sequences and representation-finite algebras. In: Representation theory, I, Proc. Workshop, Carleton Univ., Ottawa, ON,
1979. Lecture Notes in Math., vol. 831, pp. 1-71. Springer, Berlin (1980)
17. Keller, B.: On triangulated orbit categories. Documenta Mathematica 10, 551-581 (2005) MR2184464 (2007c:18006)
18. Nakajima, H.: Varieties associated with quivers. In: Representation theory of algebras and related topics, Mexico City,
1994. CMS Conf. Proc., vol. 19, pp. 139-157. Amer. Math. Soc., Providence (1996)
19. Palu, Y.: Cluster characters for 2-Calabi-Yau triangulated categories. Ann. Inst. Fourier (Grenoble) 58(6), 2221-2248 (2008)
20. Propp, J.: The combinatorics of frieze patterns and Markoff numbers. [math.CO] (2008)
21. Ringel, C.M.: Finite dimensional hereditary algebras of wild representation type. Math. Z. 161(3), 235-255 (1978)
22. Ringel, C.M.: Tame algebras and integral quadratic forms. Lecture Notes in Mathematics, vol. 1099, pp. 1-376. Springer, Berlin (1984) MR0774589 (87f:16027)
23. Sherman, P., Zelevinsky, A.: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types. Mosc. Math. J. 4, 947-974 (2004) MR2124174 (2006c:16052)
24. Yang, S.W., Zelevinsky, A.: Cluster algebras of finite type via Coxeter elements and principal minors.
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