Abstract
A direct connection between two sequences of points, one of which is generated by seed mutations of the cluster algebra of type and the other by time evolutions of the periodic discrete Toda lattice, is explicitly given. In this construction, each of them is realized as an orbit of a QRT map, and specialization of the parameters in the maps and appropriate choices of the initial points relate them. The connection with the periodic discrete Toda lattice enables us a geometric interpretation of the seed mutations of the cluster algebra of type as an addition of points on an elliptic curve.
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