Skew-symmetrizable cluster algebras from surfaces and symmetric quivers

Paper

Abstract

We study skew-symmetrizable cluster algebras

associated with unpunctured surfaces

endowed with an orientation-preserving involution

. Cluster variables of

correspond to

-orbits of arcs of

, while clusters are given by admissible

-invariant ideal triangulations. We provide a cluster expansion formula for any

-orbit

in terms of perfect matchings of some labeled modified snake graphs constructed from the arcs of

. Then, we associate a symmetric finite-dimensional algebra

to any seed of

, such that non-initial cluster variables bijectively correspond to orthogonal indecomposable

-modules. Finally, we exhibit a purely representation-theoretic map from the category of orthogonal

-modules to

. If

is a regular polygon, we recover the results proved in arXiv:2403.11308, arXiv:2405.14915, and arXiv:2502.06410 for cluster algebras of type B.