Quivers with potentials for Grassmannian cluster algebras

TL;DR

This paper establishes a correspondence between quivers with potentials and Grassmannian cluster algebras, proving their compatibility with geometric exchanges, their rigidity, and their uniqueness, and relates auto-equivalences to cluster automorphisms.

Contribution

It constructs and analyzes quivers with potentials for Grassmannian cluster algebras, proving their rigidity, uniqueness, and compatibility with geometric mutations.

Findings

Quivers with potentials are compatible with Postnikov diagram mutations.

Such quivers are always rigid and Jacobian-finite.

Auto-equivalence groups match cluster automorphism groups.

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 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.