Journal of Algebraic Combinatorics

, Volume 34, Issue 1, pp 19–66 | Cite as

Cluster mutation-periodic quivers and associated Laurent sequences

  • Allan P. Fordy
  • Robert J. MarshEmail author


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.

Finally, we point out a connection between quivers which arise in our classification and those arising in the context of quiver gauge theories.


Cluster algebra Quiver mutation Periodic quiver Somos sequence Integer sequences Pell’s equation Laurent phenomenon Integrable map Linearisation Seiberg duality Supersymmetric quiver gauge theory 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.School of MathematicsUniversity of LeedsLeedsUK

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