Ramified Quivers with Potentials, Monodromy Characters, and Cluster Bases I
Abstract
We construct categorical models and cluster characters for skew-symmetrizable cluster algebras admitting good ramified realizations. Goodness is equivalent to nondegenerate realizability. When a reddening sequence is given, it is enough to check sign coherence of the arrow types along that sequence. Triangular extensions with arbitrary connecting bimodule types preserve goodness when both component matrices admit reddening sequences. Consequently, every matrix obtained from one vertex by triangular extensions and mutations admits a good Jacobi-finite nondegenerate realization for every compatible positive integral symmetrizer. For a good Jacobi-finite nondegenerate realization, the monodromy F-polynomials satisfy mutation for all finite decorated representations, without a rigidity hypothesis. A full initial order determines valued universal presentations and their cluster characters. Under the reachability condition, these characters form a basis of the middle cluster algebra for arbitrary geometric coefficients, with frozen variables inverted, and contain all cluster monomials. The same family is an upper-cluster-algebra basis when the middle and upper algebras coincide. For a fixed Jacobian algebra and δ-vector, the Newton polytope of the generic monodromy F-polynomial is independent of the order.
Related
- Ramified convolution and monodromy bases for symmetrizable Kac--Moody algebras
- Quasimaps to Quivers with Potentials
- Obstructions to Jacobi-Finiteness of Quivers with Potentials
- Categorified Cluster Structures on Coulomb Branches of Certain Star-Shaped Quivers
- Syzygy Transformations of Cluster Characters and Geometric Twists
BibTeX
@misc{fei2026ramified,
title = {{Ramified Quivers with Potentials, Monodromy Characters, and Cluster Bases I}},
author = {Jiarui Fei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09868},
eprint = {2610.09868},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09868},
}
RIS
TY - UNPB
TI - Ramified Quivers with Potentials, Monodromy Characters, and Cluster Bases I
AU - Jiarui Fei
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.09868
UR - https://arxiv.org/abs/2610.09868
AB - We construct categorical models and cluster characters for skew-symmetrizable cluster algebras admitting good ramified realizations. Goodness is equivalent to nondegenerate realizability. When a reddening sequence is given, it is enough to check sign coherence of the arrow types along that sequence. Triangular extensions with arbitrary connecting bimodule types preserve goodness when both component matrices admit reddening sequences. Consequently, every matrix obtained from one vertex by triangular extensions and mutations admits a good Jacobi-finite nondegenerate realization for every compatible positive integral symmetrizer. For a good Jacobi-finite nondegenerate realization, the monodromy F-polynomials satisfy mutation for all finite decorated representations, without a rigidity hypothesis. A full initial order determines valued universal presentations and their cluster characters. Under the reachability condition, these characters form a basis of the middle cluster algebra for arbitrary geometric coefficients, with frozen variables inverted, and contain all cluster monomials. The same family is an upper-cluster-algebra basis when the middle and upper algebras coincide. For a fixed Jacobian algebra and δ-vector, the Newton polytope of the generic monodromy F-polynomial is independent of the order.
ER -