For a quiver with potential (Q,W) with an action of a finite cyclic group
G, we study the skew group algebra ΛG of the Jacobian algebra
Λ=P(Q,W). By a result of Reiten and Riedtmann, the quiver
QG​ of a basic algebra η(ΛG)η Morita equivalent to ΛG is known. Under some assumptions on the action of G, we explicitly
construct a potential WG​ on QG​ such that η(ΛG)η≅P(QG​,WG​). The original quiver with potential can then be
recovered by the skew group algebra construction with a natural action of the
dual group of G. If Λ is self-injective, then ΛG is as well,
and we investigate this case. Motivated by Herschend and Iyama's
characterisation of 2-representation finite algebras, we study how cuts on
(Q,W) behave with respect to our construction.Comment: 34 pages, comments welcome. Final version, to appear in Journal of
Algebr