Non-degenerate potentials on the quiver X₇

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 X₇. We confirm a conjecture of Geiss-Labardini-Schröer by presenting a computer-assisted proof that over a ground field of characteristic 2, the Jacobian algebra of one member W₀ of this family is infinite-dimensional, whereas that of another member W₁ 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 X₇. We also show that when the characteristic of the ground field differs from 2, the Jacobian algebras of W₀ and W₁ are both finite-dimensional. Thus W₀ 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.