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 c...

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Bibliographic Details
Published in:Journal of algebra 2025-03, Vol.666, p.94-148
Main Author: Ladkani, Sefi
Format: Article
Language:English
Subjects:
Finite-dimensional algebra
Jacobian algebra
Quiver mutation
Quiver with potential
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Summary: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.
ISSN:0021-8693
DOI:10.1016/j.jalgebra.2024.11.007