Advances in Theoretical and Mathematical Physics

Volume 27 (2023)

Number 3

Clustering cluster algebras with clusters

Pages: 797 – 828

DOI: https://dx.doi.org/10.4310/ATMP.2023.v27.n3.a5

Authors

Man-Wai Cheung (School of Mathematics, University of Tokyo, Kashiwa, Chiba, Japan)

Pierre-Philippe Dechant (School of Mathematics, University of Leeds, United Kingdom; Department of Mathematics, University of York, United Kingdom; and York Cross-disciplinary Centre for Systems Analysis, University of York, York, United Kingdom)

Yang-Hui He (London Institute for Mathematical Sciences, Royal Institution London, United Kingdom; Department of Mathematics, City University of London; Merton College, University of Oxford, United Kingdom; and School of Physics, Nankai University, Tianjin, China)

Elli Heyes (Department of Mathematics, City University of London, United Kingdom; and London Institute for Mathematical Sciences, Royal Institution London, United Kingdom)

Edward Hirst (Department of Mathematics, City, University of London, London, United Kingdom; and London Institute for Mathematical Sciences, Royal Institution London, United Kingdom)

Jian-Rong Li (Faculty of Mathematics, University of Vienna, Austria)

Abstract

Classification of cluster variables in cluster algebras (in particular, Grassmannian cluster algebras) is an important problem, which has direct applications to computations of scattering amplitudes in physics. In this paper, we apply the tableaux method to classify cluster variables in Grassmannian cluster algebras $\mathbb{C}[\mathrm{Gr}(k, n)]$ 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.

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Published 6 June 2024