Cluster mutation-periodic quivers and associated Laurent sequences

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 non-linear 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.

💡 Research Summary

The paper investigates quivers (or equivalently skew‑symmetric integer matrices) under the operation of cluster‑algebra mutation, focusing on those that are isomorphic to their own mutation after a cyclic permutation of all vertices. This property, called mutation‑periodicity, is first defined for the simplest case – a “cycle mutation” where every vertex is shifted simultaneously along a single directed cycle. The authors give a complete classification of all quivers that are invariant under such a cycle mutation. Three main families emerge: (i) circular chains, where a linear chain of n vertices is closed by an extra edge to form a single directed cycle; (ii) multi‑circular configurations, in which several cycles intersect or share vertices, leading to higher‑order periodicities; and (iii) lattice‑type quivers that exhibit genuine k‑periodicity (k ≥ 2) on a two‑dimensional grid. For each family the period k is precisely the number of successive mutations required to return to the original quiver.

The central consequence of mutation‑periodicity is that the associated cluster variables satisfy recurrence relations that are automatically Laurent polynomials in the initial variables. In one dimension the recurrence takes the form

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