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Journal of Algebraic Combinatorics

, Volume 43, Issue 3, pp 589–633 | Cite as

Laurent phenomenon sequences

Article

Abstract

In this paper, we undertake a systematic study of sequences generated by recurrences \(x_{m+n}x_m = P(x_{m+1}, \ldots , x_{m+n-1})\) which exhibit the Laurent phenomenon. Some of the most famous among these are the Somos and the Gale-Robinson sequences. Our approach is based on finding period 1 seeds of Laurent phenomenon algebras of Lam–Pylyavskyy. We completely classify polynomials P that generate period 1 seeds in the cases of \(n=2,3\) and of mutual binomial seeds. We also find several other interesting families of polynomials P whose generated sequences exhibit the Laurent phenomenon. Our classification for binomial seeds is a direct generalization of a result by Fordy and Marsh, that employs a new combinatorial gadget we call a double quiver.

Keywords

Laurent phenomenon Cluster algebra LP algebra 

Notes

Acknowledgments

This research was conducted at the 2013 summer REU (Research Experience for Undergraduates) program at the University of Minnesota, Twin Cities, and was supported by NSF grants DMS-1067183 and DMS-1148634. We would like to thank Professors Dennis Stanton, Gregg Musiker, Joel Lewis, and especially our mentor Pavlo Pylyavskyy, who directed the program, for their advice and support throughout this project. We would also like to thank Al Garver, Andrew Hone and other REU participants for their help in editing this paper and the anonymous reviewers for careful reading and valuable comments.

References

  1. 1.
    Fomin, S., Zelevinsky, A.: The Laurent phenomenon. Adv. Appl. Math. 28(2), 119–144 (2002). ISSN: 0196-8858Google Scholar
  2. 2.
    Fordy, A.P., Marsh, R.J.: Cluster mutation-periodic quivers and associated Laurent sequences. J. Algebr. Comb. 34(1), 19–66 (2011). INNS: 0925-9899Google Scholar
  3. 3.
    Bousquet-Mélou, M., Propp, J., West, J.: Perfect matchings for the three-term Gale-Robinson sequences. Electron. J. Comb. 16(1), Research Paper 125, 37. ISSN: 1077-8926 http://www.combinatorics.org/Volume_16/Abstracts/v16i1r125.html (2009)
  4. 4.
    Lam, T., Pylyavskyy, P.: Laurent phenomenon algebras. arxiv.org/pdf/1206.2611v2.pdf (2012)
  5. 5.
    Musiker, G.: Undergraduate Thesis: cluster algebras, Somos sequences and exchange graphs. www.math.umn.edu/~musiker/Research.html (2002)
  6. 6.
    Stein, W.A., et al.: Sage Mathematics Software (Version 5.9). The Sage Development Team (2013)Google Scholar
  7. 7.
    Russell, M.C.: Noncommutative recursions and the Laurent phenomenon. Adv. Appl. Math. 64, 21–30 (2015). ISSN:0196-8858Google Scholar
  8. 8.
    Hone, A.N.W., Ward, C.: A family of linearizable recurrences with the Laurent property. Bull. Lond. Math. Soc. 46(3), 503–516 (2014). ISSN: 0024-6093Google Scholar
  9. 9.
    Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497–529 (electronic) (2002). ISSN: 0894-0347Google Scholar
  10. 10.
    Hietarinta, J., Viallet, C.: Singularity confinement and chaos in discrete systems. Phys. Rev. Lett. 81(2), 325–328 (1998)Google Scholar
  11. 11.
    Maeda, S.: Completely integrable symplectic mapping. Proc. Japan Acad. Ser. A Math. Sci. 63(6), 198–200 (1987). ISSN: 0386-2194Google Scholar
  12. 12.
    Fordy, A.P.: Mutation-periodic quivers, integrable maps and associated Poisson algebras. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 369(1939) 1264–1279 (2011). ISSN: 1364-503XGoogle Scholar
  13. 13.
    Fordy, A.P., Hone, A.: Discrete integrable systems and Poisson algebras from cluster maps. Comm. Math. Phys. 325(2), 527–584 (2014). ISSN: 0010-3616Google Scholar
  14. 14.
    Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster algebras and Poisson geometry. Mosc. Math. J. 3(3) 899–934, 1199 (2003). Dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthday. ISSN: 1609-3321Google Scholar
  15. 15.
    Hone, A.N.W.: Laurent polynomials and superintegrable maps. In: SIGMA Symmetry Integrability Geometry Methods and Applications, vol. 3, Paper 022, 18 (2007). ISSN: 1815-0659Google Scholar
  16. 16.
    Hone, A.N.W.: Diophantine non-integrability of a third-order recurrence with the Laurent property. J. Phys. A 39(12) L171–L177 (2006). ISSN: 0305-4470Google Scholar
  17. 17.
    Baragar, A.: The Markoff–Hurwitz equations over number fields. Rocky Mountain J. Math. 35(3), 695–712 (2005). ISSN: 0035-7596Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsMITCambridgeUSA

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