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Cluster mutation-periodic quivers and associated Laurent sequences. (English) Zbl 1272.13020

Let \(Q\) be a quiver. The mutation of \(Q\) at node \(k\) is a new quiver \(\mu_{k}Q\) obtained from \(Q\) via first reversing all arrows originating or terminating at \(k\); then adding \(pq\) arrows from node \(i\) to node \(j\) in addition to these arrows from node \(i\) to node \(j\) existing in \(Q\), if there are \(p\) arrows from node \(i\) to node \(k\) and \(q\) arrows from node \(k\) to node \(j\) in the quiver \(Q\); and finally removing any two-cycles created in the previous steps. Let us order the nodes of \(Q\) as \(1, 2, \dots, N\) and define a permutation of the nodes of \(Q\) as \(\rho: (1, 2, \dots, N)\rightarrow (N, 1, \dots, N-1)\). Let us denote by \(Q(1)\) the quiver \(\mu_{1}Q\) and by \(Q(k+1)\) the quiver \(\mu_{k}Q(k)\). A quiver \(Q\) is said to have period \(m\) if \(Q(m+1)=\rho^{m}Q(1)\). A quiver \(Q\) is said to be a period \(m\) sink-type quiver if it is of period \(m\) and for \(1\leq i\leq m\), node \(i\) of \(Q(i)\) is a sink.
In this paper, the authors study some periodic quivers. First of all, the authors classify all periodic quivers of sink-type; and the results are stated in Sections \(3, 4\) and \(5\). Then, they completely classify all period \(1\) quivers and construct some specific examples in Section \(6\). The authors also classify all quivers of period \(2\) with at most five nodes in Section \(7\). In the rest of the paper, the authors study the recurrences which can be associated to periods \(1\) and \(2\) quivers. In particular, they prove that the recurrences associated to period \(1\) primitive quivers can be linearized. They also extend the construction of mutation periodic quivers to quivers involving frozen cluster variables and thus introduce parameters into recurrences. Moreover, they study the connections with supersymmetric quiver gauge theories.

MSC:

13F60 Cluster algebras
16G20 Representations of quivers and partially ordered sets
81T60 Supersymmetric field theories in quantum mechanics

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