Geodesic
Cluster mutation-periodic quivers and associated Laurent sequences
Journal of Algebraic Combinatorics, Tome 34 (2011) no. 1, pp. 19-66.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Keywords: keywords cluster algebra, quiver mutation, periodic quiver, somos sequence, integer sequences, pell's equation, Laurent phenomenon, integrable map, linearisation, Seiberg duality, supersymmetric quiver gauge theory 
@article{JAC_2011__34_1_a3,
     author = {Fordy, Allan P. and Marsh, Robert J.},
     title = {Cluster mutation-periodic quivers and associated {Laurent} sequences},
     journal = {Journal of Algebraic Combinatorics},
     pages = {19--66},
     publisher = {mathdoc},
     volume = {34},
     number = {1},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2011__34_1_a3/}
}
TY  - JOUR
AU  - Fordy, Allan P.
AU  - Marsh, Robert J.
TI  - Cluster mutation-periodic quivers and associated Laurent sequences
JO  - Journal of Algebraic Combinatorics
PY  - 2011
SP  - 19
EP  - 66
VL  - 34
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JAC_2011__34_1_a3/
LA  - en
ID  - JAC_2011__34_1_a3
ER  - 
%0 Journal Article
%A Fordy, Allan P.
%A Marsh, Robert J.
%T Cluster mutation-periodic quivers and associated Laurent sequences
%J Journal of Algebraic Combinatorics
%D 2011
%P 19-66
%V 34
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JAC_2011__34_1_a3/
%G en
%F JAC_2011__34_1_a3
Fordy, Allan P.; Marsh, Robert J. Cluster mutation-periodic quivers and associated Laurent sequences. Journal of Algebraic Combinatorics, Tome 34 (2011) no. 1, pp. 19-66. http://geodesic.mathdoc.fr/item/JAC_2011__34_1_a3/