Species with potential arising from surfaces with orbifold points of order 2, Part I: one choice of weights

Abstract

We present a definition of mutations of species with potential that can be applied to the species realizations of any skew-symmetrizable matrix B over cyclic Galois extensions E/F whose base field F has a primitive [E:F]-th root of unity. After providing an example of a globally unfoldable skew-symmetrizable matrix whose species realizations do not admit non-degenerate potentials, we present a construction that associates a species with potential to each tagged triangulation of a surface with marked points and orbifold points of order 2. Then we prove that for any two tagged triangulations related by a flip, the associated species with potential are related by the corresponding mutation (up to a possible change of sign at a cycle), thus showing that these species with potential are non-degenerate. In the absence of orbifold points, the constructions and results specialize to previous work by the second author. The species constructed here for each triangulation $\tau$ is a species realization of one of the several matrices that Felikson-Shapiro-Tumarkin have associated to $\tau$, namely, the one that in their setting arises from choosing the number 1/2 for every orbifold point.