Quivers with potentials for Grassmannian cluster algebras

We consider a quiver with potential (QP) and an iced quiver with potential (IQP) associated with a Postnikov Diagram D and prove that their mutations are compatible with the geometric exchanges of D. This ensures that we may define a QP and an IQP for a Grassmannian cluster algebra up to mutation equivalence. It shows that is always rigid (thus nondegenerate) and Jacobi-finite. Moreover, in fact, we show that it is the unique nondegenerate (thus rigid) QP by using a general result of Geiβ, Labardini-Fragoso, and Schröer (2016, Advances in Mathematics 290, 364-452). Then we show that, within the mutation class of the QP for a Grassmannian cluster algebra, the quivers determine the potentials up to right equivalence. As an application, we verify that the auto-equivalence group of the generalized cluster category is isomorphic to the cluster automorphism group of the associated Grassmannian cluster algebra with trivial coefficients.