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arXiv · 2501.07065

Gröbner Cones for Finite Type Cluster Algebras

Nathan Ilten·Karolyn So
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Abstract

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.

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Nathan Ilten, Karolyn So. 2025-01-13. Gröbner Cones for Finite Type Cluster Algebras. https://arxiv.org/abs/2501.07065

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