Cluster mutation-periodic quivers and associated Laurent sequences
Cluster mutation-periodic quivers and associated Laurent sequences
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DOI：
10.1007/s10801-010-0262-4
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发表时间：
2009-04
影响因子：
0.8
通讯作者：
A. Fordy;Robert J. Marsh
中科院分区：
文献类型：
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作者：
A. Fordy;Robert J. Marsh
我们考虑箭图/反对称矩阵的作用下的突变（在集群代数意义上）。我们对那些通过置换所有顶点的循环而同构于它们自身突变的箭图进行了分类，并给出了具有更高周期性的箭图族，周期性意味着由递归关系给出的序列以自然的方式产生于相应的簇代数。我们提出了一些有趣的新的家庭的非线性递归，必然与洛朗性质，无论是真实的线和平面，包含可积映射作为特殊情况。特别是，我们表明，这些递归可以线性化，并在一定的初始条件下，给出整数序列，其中包含一些特定的佩尔方程的所有解决方案。我们将我们的结构扩展到包括参数递归，解释了Gale的一些观察，最后，我们指出了在我们的分类中出现的颤动和在规范理论中出现的颤动之间的联系。
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 withparameters, 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.