Gröbner Cones for Finite Type Cluster Algebras
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910782012260352 |
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| author | Ilten, Nathan So, Karolyn |
| author_facet | Ilten, Nathan So, Karolyn |
| contents | Let $\mathcal{A}$ be a cluster algebra of finite cluster type. We study the Gröbner cone $\mathcal{C}_{\mathcal{A}}$ parametrizing term orders inducing an initial degeneration of the ideal $I_{\mathcal{A}}$ of relations among the cluster variables of $\mathcal{A}$ to the ideal generated by products of incompatible cluster variables. We show that for any cluster variable $v$, the weight induced by taking compatibility degrees with $v$ belongs to $\mathcal{C}_{\mathcal{A}}$. This allows us to construct an explicit circular term order and prove a conjecture of Ilten, Nájera Chávez, and Treffinger. Furthermore, we give explicit descriptions of the rays and lineality spaces of $\mathcal{C}_{\mathcal{A}}$ in terms of combinatorial models for cluster algebras of types $A_n$, $B_n$, $C_n$, $D_n$ with a special choice of frozen variables, and in the case of no frozen variables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07065 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gröbner Cones for Finite Type Cluster Algebras Ilten, Nathan So, Karolyn Commutative Algebra Combinatorics 13F60, 13P10 Let $\mathcal{A}$ be a cluster algebra of finite cluster type. We study the Gröbner cone $\mathcal{C}_{\mathcal{A}}$ parametrizing term orders inducing an initial degeneration of the ideal $I_{\mathcal{A}}$ of relations among the cluster variables of $\mathcal{A}$ to the ideal generated by products of incompatible cluster variables. We show that for any cluster variable $v$, the weight induced by taking compatibility degrees with $v$ belongs to $\mathcal{C}_{\mathcal{A}}$. This allows us to construct an explicit circular term order and prove a conjecture of Ilten, Nájera Chávez, and Treffinger. Furthermore, we give explicit descriptions of the rays and lineality spaces of $\mathcal{C}_{\mathcal{A}}$ in terms of combinatorial models for cluster algebras of types $A_n$, $B_n$, $C_n$, $D_n$ with a special choice of frozen variables, and in the case of no frozen variables. |
| title | Gröbner Cones for Finite Type Cluster Algebras |
| topic | Commutative Algebra Combinatorics 13F60, 13P10 |
| url | https://arxiv.org/abs/2501.07065 |