We consider (iced) quiver with potential $(\bar{ Q}(D), F(D), \bar{ W}(D))$ associated to a Postnilov Diagram $D$ and prove the mutation of the quiver with potential $(\bare{ Q}(D), F(D), \bar{ W}(D))$ is compatible with the geometric exchange of the Postnikov diagram $D$. This ensures we may define a quiver with potential for a Grassmannian cluster algebra. We show such quiver with potential is always rigid (thus non-degenerate) and Jacobian-finite. And in fact, it is the unique non-degenerate (thus unique rigid) quiver with potential associated to the Grassmannian cluster algebra up to right-equivalence, by using a general result of Gei\ss-Labardini-Schr\"oer. As an application, we verify that the auto-equivalence group of the generalized cluster category ${ \mathcal{ C}}_{ (Q, W)}$ is isomorphic to the cluster automorphism group of the associated Grassmannian cluster algebra ${ { \mathcal{ A}}_{ (Q, W)}}$ with trivial coefficients.
Categories: math.RT