Cluster mutation-periodic quivers and associated laurent sequences

Allan Fordy (University of Leeds)

Tuesday 13th October, 2009 15:00-16:00 Mathematics Building, room 204

Abstract

The Somos 4 recurrence relation has (at least!) two remarkable properties: it has the Laurent property and defines an integrable map. Its Laurent property is derived from the fact that it is the cluster mutation relation for a special class of cluster exchange matrix (or equivalently the related quiver), with a periodicity property. In our preprint arXiv:0904.0200v2 [math.CO], Robert Marsh and I consider the problem of classifying all such "mutation-periodic quivers", the results of which will be the basis of this talk. 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.

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