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Gröbner Cones for Finite Type Cluster Algebras

Nathan Ilten, Karolyn So

TL;DR

This work analyzes the Gröbner cone $\mathcal{C}_{\mathcal{A}}$ associated to a cluster algebra $\mathcal{A}$ of finite cluster type, describing how weights coming from compatibility degrees of cluster variables sit inside this cone and yield circular term orders; it further provides explicit descriptions of the cone’s rays and lineality spaces in classical types with and without frozen variables. Central to the approach is the compatibility-degree framework tied to root systems and combinatorial polygon models, which translates cluster-exchange relations into geometric inequalities that place the relevant weight vectors in $\mathcal{C}_{\mathcal{A}}$. The paper proves a key inequality for all primitive exchanges, handles exceptional types via computation, and then derives concrete cone generators in types $A_n,B_n,C_n,D_n$ (with frozen variables and without) using exchange-quadrilateral geometry. These results yield a constructive, explicit description of circular term orders and confirm a conjecture on multidegrees of first-order deformations, with potential applications to tropicalizations and deformation theory of cluster varieties. The methods integrate combinatorial models, root-system techniques, and computational verification to produce a comprehensive, type-by-type characterization of $\mathcal{C}_{\mathcal{A}}$ and its geometric structure.

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.

Gröbner Cones for Finite Type Cluster Algebras

TL;DR

This work analyzes the Gröbner cone associated to a cluster algebra of finite cluster type, describing how weights coming from compatibility degrees of cluster variables sit inside this cone and yield circular term orders; it further provides explicit descriptions of the cone’s rays and lineality spaces in classical types with and without frozen variables. Central to the approach is the compatibility-degree framework tied to root systems and combinatorial polygon models, which translates cluster-exchange relations into geometric inequalities that place the relevant weight vectors in . The paper proves a key inequality for all primitive exchanges, handles exceptional types via computation, and then derives concrete cone generators in types (with frozen variables and without) using exchange-quadrilateral geometry. These results yield a constructive, explicit description of circular term orders and confirm a conjecture on multidegrees of first-order deformations, with potential applications to tropicalizations and deformation theory of cluster varieties. The methods integrate combinatorial models, root-system techniques, and computational verification to produce a comprehensive, type-by-type characterization of and its geometric structure.

Abstract

Let be a cluster algebra of finite cluster type. We study the Gröbner cone parametrizing term orders inducing an initial degeneration of the ideal of relations among the cluster variables of to the ideal generated by products of incompatible cluster variables. We show that for any cluster variable , the weight induced by taking compatibility degrees with belongs to . 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 in terms of combinatorial models for cluster algebras of types , , , with a special choice of frozen variables, and in the case of no frozen variables.
Paper Structure (25 sections, 35 theorems, 91 equations, 18 figures)

This paper contains 25 sections, 35 theorems, 91 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cluster basics
  4. The Gröbner Cone
  5. Compatibility Degree and Root Systems
  6. Combinatorial Models
  7. Notation
  8. Type $A_n$
  9. Types $B_n$ and $C_n$
  10. Type $D_n$
  11. Exchange Quadrilaterals
  12. Term Orders from Compatibility Degree
  13. General Approach
  14. Classical Types
  15. Exceptional Types
  16. ...and 10 more sections

Key Result

Theorem 2.1.6

A cluster algebra $\mathcal{A}$ is of finite cluster type if and only if it admits a seed $(\tilde{\mathbf{x}},\tilde{B})$ such that the Cartan matrix $A(B)$ is of finite type.

Figures (18)

  • Figure 1: The polygon $\mathbf{P}_8$
  • Figure 2: Exchange quadrilaterals
  • Figure 3: Opposite edges of exchange quadrilaterals
  • Figure 4: Possible positions of $l$ relative to $\mathbf{Q}$
  • Figure 5: Single and double lines in type 2 exchanges
  • ...and 13 more figures

Theorems & Definitions (72)

  • Definition 2.1.1: Seed
  • Definition 2.1.2: Mutation
  • Definition 2.1.4: Cluster algebra
  • Remark 2.1.5
  • Theorem 2.1.6: Fomin_IIFiniteTypeClassification
  • Definition 2.2.1
  • Remark 2.2.2
  • Definition 2.2.3
  • Lemma 2.2.4: cf. ilten2021deformation
  • Remark 2.3.1
  • ...and 62 more