Gr\"obner Cones for Finite Type Cluster Algebras

TL;DR

This paper investigates the Gr"obner cone associated with finite type cluster algebras, providing explicit descriptions, constructing a specific term order, and confirming a related conjecture, thereby deepening understanding of their algebraic and combinatorial structure.

Contribution

It introduces an explicit circular term order for finite type cluster algebras and describes the structure of their Gr"obner cones, confirming a conjecture and linking algebraic and combinatorial models.

Findings

Constructed an explicit circular term order.

Proved a conjecture of Ilten, Nájera Chávez, and Treffinger.

Described the rays and lineality spaces of the Gr"obner cone for various types.

Abstract

Let $\mathcal{A}$ be a cluster algebra of finite cluster type. We study the Gr\"obner 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, N\'ajera Ch\'avez, 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.