Quivers with potentials for Grassmannian cluster algebras

Abstract

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 Geiß-Labardini-Schröer. 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.

24pages,16 figures