Jacobian matrices of Y-seed mutations
Combinatorics
2020-01-07 v5 Geometric Topology
Representation Theory
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 -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.
Keywords
Cite
@article{arxiv.1805.00044,
title = {Jacobian matrices of Y-seed mutations},
author = {Yuma Mizuno},
journal= {arXiv preprint arXiv:1805.00044},
year = {2020}
}