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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2023
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2306.03818 |
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| _version_ | 1866929620119453696 |
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| author | Ladkani, Sefi |
| author_facet | Ladkani, Sefi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_03818 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-degenerate potentials on the quiver $X_7$ Ladkani, Sefi Representation Theory 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. |
| title | Non-degenerate potentials on the quiver $X_7$ |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2306.03818 |