# Cluster mutation-periodic quivers and associated Laurent sequences

@article{Fordy2009ClusterMQ, title={Cluster mutation-periodic quivers and associated Laurent sequences}, author={Allan P Fordy and Robert J. Marsh}, journal={Journal of Algebraic Combinatorics}, year={2009}, volume={34}, pages={19-66} }

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…

## 90 Citations

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The thesis deals with a range of questions in cluster algebras and the representation theory of quivers. In particular, we provide solutions to the following problems:
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- Mathematics
- 2011

We consider two kinds of periodicities of mutations in cluster algebras. For any sequence of mutations under which exchange matrices are periodic, we define the associated T- and Y-systems. When the…

### Mutation-periodic quivers, integrable maps and associated Poisson algebras

- MathematicsPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2011

This paper considers a class of map, recently derived in the context of cluster mutation, whose bi-Hamiltonian structure is derived and used to construct a sequence of Poisson-commuting functions and hence show complete integrability.

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- 2019

We consider a class of map, recently derived in the context of cluster mutation. In this paper we start with a brief review of the quiver context, but then move onto a discussion of a related Poisson…

### Difference equations arising from cluster algebras

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- 2020

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We consider two kinds of periodicities of mutations in cluster algebras. For any sequence of mutations under which exchange matrices are periodic, we define the associated T- and Y-systems. When the…

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This paper considers a class of map, recently derived in the context of cluster mutation, whose bi-Hamiltonian structure is derived and used to construct a sequence of Poisson-commuting functions and hence show complete integrability.

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