Cluster mutation-periodic quivers and associated Laurent sequences
Allan P. Fordy
and Robert J. Marsh
DOI: 10.1007/s10801-010-0262-4
Abstract
We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity.
The periodicity means that sequences given by recurrence relations arise in a natural way from the associated cluster algebras. We present a number of interesting new families of nonlinear recurrences, necessarily with the Laurent property, of both the real line and the plane, containing integrable maps as special cases. In particular, we show that some of these recurrences can be linearised and, with certain initial conditions, give integer sequences which contain all solutions of some particular Pell equations. We extend our construction to include recurrences with parameters, giving an explanation of some observations made by Gale.
Pages: 19–66
Keywords: keywords cluster algebra; quiver mutation; periodic quiver; somos sequence; integer sequences; pell's equation; Laurent phenomenon; integrable map; linearisation; Seiberg duality; supersymmetric quiver gauge theory
Full Text: PDF
References
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65. Cambridge University Press, Cambridge (2006)
2. Assem, I., Reutenauer, C., Smith, D.: Frises. Preprint (2009). [math.RA]
3. Bellon, M., Viallet, C.-M.: Algebraic entropy. Commun. Math. Phys. 204, 425-437 (1999)
4. Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations I: Mutations. Sel. Math., New Ser. 14, 59-119 (2008)
5. Feng, B., Hanany, A., He, Y.-H.: D-brane gauge theories from toric singularities and toric duality. Nucl. Phys. B 595, 165-200 (2001)
6. Feng, B., Hanany, A., He, Y.-H., Uranga, A.M.: Toric duality as Seiberg duality and brane diamonds. J. High Energy Phys. 12, 035 (2001)
7. Fomin, S., Zelevinsky, A.: Cluster algebras I: foundations. J. Am. Math. Soc. 15, 497-529 (2002)
8. Fomin, S., Zelevinsky, A.: The Laurent phenomenon. Adv. Appl. Math. 28, 119-144 (2002)
9. Fordy, A.P.: Mutation-periodic quivers, integrable maps and associated Poisson algebras. Preprint (2010). [nlin.SI]
10. Fordy, A.P., Hone, A.N.W.: Integrable maps and Poisson algebras derived from cluster algebras. In preparation
11. Franco, S., Hanany, A., Kennaway, K.D., Vegh, D., Wecht, B.: Brane dimers and quiver gauge theories. J. High Energy Phys. 01, 096 (2006) J Algebr Comb (2011) 34: 19-66
12. Fu, C., Keller, B.: On cluster algebras with coefficients and 2-Calabi-Yau categories. Trans. Am. Math. Soc. 362, 859-895 (2010)
13. Gale, D.: The strange and surprising saga of the Somos sequences. Math. Intell. 13, 40-42 (1991)
14. Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster algebras and Poisson geometry. Mosc. Math. J. 3, 899-934 (2003)
15. Grammaticos, B., Ramani, A., Papageorgiou, V.G.: Do integrable mappings have the Painlevé property? Phys. Rev. Lett. 67, 1825-1828 (1991)
16. Halburd, R.G.: Diophantine integrability. J. Phys. A 38, L263-L269 (2005)
17. Heideman, P., Hogan, E.: A new family of Somos-like recurrences. Electron. J. Comb. 15, R54 (2008)
18. Hoggatt, V.E., Bicknell-Johnson, M.: A primer for the Fibonacci numbers XVII: generalized Fi- bonacci numbers satisfying un+1un - 1 - u2n = \pm
1. Fibonacci Q. 2, 130-137 (1978)
19. Hone, A.N.W.: Laurent polynomials and superintegrable maps. SIGMA 3, 022 (2007), 18 pages
20. Kashiwara, M.: On crystal bases of the q-analogue of universal enveloping algebras. Duke Math. J. 63, 465-516 (1991)
21. Keller, B.: The periodicity conjecture for pairs of Dynkin diagrams. Preprint (2010). [math.RT]
22. Keller, B., Scherotzke, S.: Linear recurrence relations for cluster variables of affine quivers. Preprint (2010). [math.RT]
23. Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3, 447- 498 (1990)
24. Mukhopadhyay, S., Ray, K.: Seiberg duality as derived equivalence for some quiver gauge theories. J. High Energy Phys. 02, 070 (2004)
25. Nakanishi, T.: Periodic cluster algebras and dilogarithm identities. Preprint (2010). [math.QA]
26. Oota, T., Yasui, Y.: New example of infinite family of quiver gauge theories. Nucl. Phys. B 762, 377-391 (2007)
27. Quispel, G.R.W., Roberts, J.A.G., Thompson, C.J.: Integrable mappings and soliton equations. Phys. Lett. A 126, 419-421 (1988)
28. Sloane, N.J.A.: The on-line encyclopedia of integer sequences. (2009)
29. Stienstra, J.: Hypergeometric systems in two variables, quivers, dimers and dessins d'enfants. In: Yui, N., Verrill, H., Doran, C.F. (eds.) Modular Forms and String Duality. Fields Inst. Commun., vol. 54, pp. 125-161. Am. Math. Soc., Providence (2008)
30. Veselov, A.P.: Integrable maps. Russ. Math. Surv. 46(5), 1-51 (1991)
31. Vitoria, J.: Mutations vs. Seiberg duality. J. Algebra 321(3), 816-828 (2009) \surd
32. Zay, B.: An application of the continued fractions for D in solving some types of Pell's equations.
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