Non-degenerate potentials on the quiver $X_7$

Abstract

We develop a method to compute certain mutations of quivers with potentials and use this to construct an explicit family of non-degenerate potentials on the exceptional quiver $X_7$. We confirm a conjecture of Geiss-Labardini-Schroer by presenting a computer-assisted proof that over a ground field of characteristic 2, the Jacobian algebra of one member $W_0$ of this family is infinite-dimensional, whereas that of another member $W_1$ is finite-dimensional, implying that these potentials are not right equivalent. As a consequence, we draw some conclusions on the associated cluster categories, and in particular obtain a representation theoretic proof that there are no reddening mutation sequences for the quiver $X_7$. We also show that when the characteristic of the ground field differs from 2, the Jacobian algebras of $W_0$ and $W_1$ are both finite-dimensional. Thus $W_0$ seems to be the first known non-degenerate potential with the property that the finite-dimensionality of its Jacobian algebra depends upon the ground field.