Gröbner cones for finite-type cluster algebras

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.