NDLI: Cluster mutation-periodic quivers and associated Laurent sequences

Content Provider	Springer Nature Link
Author	Fordy, Allan P. Marsh, Robert J.
Copyright Year	2010
Abstract	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 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.
Starting Page	19
Ending Page	66
Page Count	48
File Format	PDF
ISSN	09259899
Journal	Journal of Algebraic Combinatorics
Volume Number	34
Issue Number	1
e-ISSN	15729192
Language	English
Publisher	Springer US
Publisher Date	2010-11-04
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Cluster algebra Quiver mutation Periodic quiver Somos sequence Integer sequences Pell’s equation Laurent phenomenon Integrable map Linearisation Seiberg duality Supersymmetric quiver gauge theory Group Theory and Generalizations Order, Lattices, Ordered Algebraic Structures Combinatorics Convex and Discrete Geometry Computer Science
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Algebra and Number Theory

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