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Algebras and Representation Theory

, Volume 21, Issue 5, pp 1119–1132 | Cite as

Dual Numbers, Weighted Quivers, and Extended Somos and Gale-Robinson Sequences

  • Valentin OvsienkoEmail author
  • Serge Tabachnikov
Article
  • 46 Downloads

Abstract

We investigate a general method that allows one to construct new integer sequences extending existing ones. We apply this method to the classic Somos-4 and Somos-5, and the Gale-Robinson sequences, as well as to more general class of sequences introduced by Fordy and Marsh, and produce a great number of new sequences. The method is based on the notion of “weighted quiver”, a quiver with a \(\mathbb {Z}\)-valued function on the set of vertices that obeys very special rules of mutation.

Keywords

Somos sequences Gale-Robinson sequences Weighted quivers Cluster algebras Cluster superalgebras 

Mathematics Subject Classification (2010)

13F60 16G20 11B37 

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Notes

Acknowledgements

This paper was initiated by discussions with Michael Somos; we are indebted to him for many fruitful comments and a computer program. We are grateful to Gregg Musiker and Michael Shapiro for enlightening discussions. This paper was completed when the first author was a Shapiro visiting professor at Pennsylvania State University, V.O. is grateful to Penn State for its hospitality. S.T. was partially supported by the NSF Grant DMS-1510055.

References

  1. 1.
    Bousquet-Mélou, M., Propp, J., West, J.: Perfect matchings for the three-term Gale-Robinson sequences. Electron. J. Combin. 16(1), 1–37 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cruz Morales, J.A., Galkin, S.: Upper bounds for mutations of potentials, SIGMA Symmetry Integrability. Geom. Methods Appl. 9(005), 13 (2013)zbMATHGoogle Scholar
  3. 3.
    Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15, 497–529 (2002)CrossRefGoogle Scholar
  4. 4.
    Fomin, S., Zelevinsky, A.: The Laurent phenomenon. Adv. Appl. Math. 28, 119–144 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fordy, A., Hone, A.: Discrete integrable systems and Poisson algebras from cluster maps. Commun. Math. Phys. 325, 527–584 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fordy, A., Marsh, R.: Cluster mutation-periodic quivers and associated Laurent sequences. J. Algebraic Combin. 34, 19–66 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gale, D.: The strange and surprising saga of the Somos sequences. Math. Intell. 13, 40–42 (1991)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Galkin, S., Usnich, A.: Mutations of potentials, Preprint IPMU 10-0100 (2010)Google Scholar
  9. 9.
    Gross, M., Hacking, P., Keel, S.: Birational geometry of cluster algebras. Algebr. Geom. 2, 137–175 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Marsh, R.: Lecture notes on cluster algebras, Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2013)Google Scholar
  11. 11.
    Morier-Genoud, S., Ovsienko, V., Tabachnikov, S.: Introducing supersymmetric frieze patterns and linear difference operators. Math. Z. 281, 1061–1087 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ovsienko, V.: A step towards cluster superalgebras. arXiv:1503.01894
  13. 13.
    Speyer, D.: Perfect matchings and the octahedron recurrence. J. Algebraic Combin. 25, 309–348 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Study, E.: Geometrie der Dynamen. Leipzig (1903)Google Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.CNRS, Laboratoire de Mathématiques U.F.R. Sciences Exactes et Naturelles Moulin de la HousseREIMS Cedex 2France
  2. 2.Department of MathematicsPennsylvania State University, University ParkState CollegeUSA

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