Dimension vectors of $τ$-rigid modules and $f$-vectors of cluster monomials from triangulated surfaces
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929522333450240 |
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| author | Yurikusa, Toshiya |
| author_facet | Yurikusa, Toshiya |
| contents | For the cluster algebra $\mathcal{A}$ associated with a triangulated surface, we give a characterization of the triangulated surface such that different non-initial cluster monomials in $\mathcal{A}$ have different $f$-vectors. Similarly, for the associated Jacobian algebra $J$, we give a characterization of the triangulated surface such that different $τ$-rigid $J$-modules have different dimension vectors. Moreover, we also show that different basic support $τ$-tilting $J$-modules have different dimension vectors. Our main ingredient is a notion of intersection numbers defined by Qiu and Zhou. As an application, we show that the denominator conjecture holds for $\mathcal{A}$ if the marked surface is a closed surface with exactly one puncture, or the given tagged triangulation has neither loops nor tagged arcs connecting punctures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00133 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dimension vectors of $τ$-rigid modules and $f$-vectors of cluster monomials from triangulated surfaces Yurikusa, Toshiya Representation Theory Combinatorics For the cluster algebra $\mathcal{A}$ associated with a triangulated surface, we give a characterization of the triangulated surface such that different non-initial cluster monomials in $\mathcal{A}$ have different $f$-vectors. Similarly, for the associated Jacobian algebra $J$, we give a characterization of the triangulated surface such that different $τ$-rigid $J$-modules have different dimension vectors. Moreover, we also show that different basic support $τ$-tilting $J$-modules have different dimension vectors. Our main ingredient is a notion of intersection numbers defined by Qiu and Zhou. As an application, we show that the denominator conjecture holds for $\mathcal{A}$ if the marked surface is a closed surface with exactly one puncture, or the given tagged triangulation has neither loops nor tagged arcs connecting punctures. |
| title | Dimension vectors of $τ$-rigid modules and $f$-vectors of cluster monomials from triangulated surfaces |
| topic | Representation Theory Combinatorics |
| url | https://arxiv.org/abs/2410.00133 |