Reduction of cluster iteration maps to symplectic maps

TL;DR

This paper demonstrates how cluster iteration maps from mutation periodic quivers can be reduced to symplectic maps using tools from cluster algebra and symplectic geometry, especially when the quiver matrix is singular.

Contribution

It provides a method to reduce complex cluster iteration maps to symplectic maps, with explicit computations for new periodic quivers.

Findings

Reduction of cluster iteration maps to symplectic maps when the quiver matrix is singular

Explicit computation of reduced maps for several new periodic quivers

Integration of cluster algebra theory with symplectic geometry techniques

Abstract

We study iteration maps of recurrence relations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and (pre)symplectic geometry, we show that these cluster iteration maps can be reduced to symplectic maps on a lower dimensional submanifold, provided the matrix representing the quiver is singular. The reduced iteration map is explicitly computed for several new periodic quivers.