Clustering Cluster Algebras with Clusters

TL;DR

This paper classifies cluster variables in Grassmannian cluster algebras using tableaux methods and machine learning, providing datasets and conjectures to understand their structure and enumeration.

Contribution

It introduces a novel combination of tableaux classification and machine learning to analyze cluster variables in Grassmannian cluster algebras, with datasets and conjectures.

Findings

Datasets of classified cluster variables are made available on GitHub.

Machine learning methods reveal structures related to tableaux and cluster variables.

Conjectures on tableaux enumeration and structure are proposed.

Abstract

Classification of cluster variables in cluster algebras (in particular, Grassmannian cluster algebras) is an important problem, which has direct application to computations of scattering amplitudes in physics. In this paper, we apply the tableaux method to classify cluster variables in Grassmannian cluster algebras $\mathbb{C}[Gr(k,n)]$ up to $(k,n)=(3,12), (4,10)$, or $(4,12)$ up to a certain number of columns of tableaux, using HPC clusters. These datasets are made available on GitHub. Supervised and unsupervised machine learning methods are used to analyse this data and identify structures associated to tableaux corresponding to cluster variables. Conjectures are raised associated to the enumeration of tableaux at each rank and the tableaux structure which creates a cluster variable, with the aid of machine learning.