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 Labardini-Fragoso (Proc Lond Math Soc 98(3):797–839, 2009. arXiv:0803.1328 ; Sel Math New Ser 22(1):145–189, 2016. doi: 10.1007/s00029-015-0188-8 . arXiv:1206.1798 ). 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 $$\tau $$ in (Adv Math 231(5):2953–3002, 2012. arXiv:1111.3449 ), namely, the one that in their setting arises from choosing the number $$\frac{1}{2}$$ for every orbifold point.