Jacobian matrices of Y-seed mutations

TL;DR

This paper introduces a new pair of matrices linked to quiver mutation sequences, relating them to Jacobian matrices and cluster transformation properties, and applies this to show certain quivers lack specific mutation sequences.

Contribution

It establishes explicit relationships between mutation matrices, Jacobian matrices, and C-matrices, and applies these results to classify quivers from punctured surfaces.

Findings

Derived explicit formulas connecting matrices and Jacobians.

Proved quivers from once-punctured surfaces lack maximal green sequences.

Linked mutation matrices to cluster transformation properties.

Abstract

For any quiver mutation sequence, we define a pair of matrices that describe a fixed point equation of a cluster transformation determined from the mutation sequence. We give an explicit relationship between this pair of matrices and the Jacobian matrix of the cluster transformation. Furthermore, we show that this relationship reduces to a relationship between the pair of matrices and the $C$-matrix of the cluster transformation in a certain limit of cluster variables. As an application, we prove that quivers associated with once-punctured surfaces do not have maximal green or reddening sequences.