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Machine Learning Clifford invariants of ADE Coxeter elements

Chen, S., Dechant, P. P., He, Y. H. , Heyes, E., Hirst, E. & Riabchenko, D. ORCID: 0009-0005-8099-8628 (2024). Machine Learning Clifford invariants of ADE Coxeter elements. Advances in Applied Clifford Algebras, 34(3), article number 20. doi: 10.1007/s00006-024-01325-y


There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for A8, D8 and E8 for a choice of basis of simple roots and compute their invariants, using high-performance computing. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science such as supervised and unsupervised machine learning. In this paper we focus on neural network classification and principal component analysis. Since the output – the invariants – is fully determined by the choice of simple roots and the permutation order of the corresponding reflections in the Coxeter element, we expect huge degeneracy in the mapping. This provides the perfect setup for machine learning, and indeed we see that the datasets can be machine learned to very high accuracy. This paper is a pump-priming study in experimental mathematics using Clifford algebras, showing that such Clifford algebraic datasets are amenable to machine learning, and shedding light on relationships between these novel and other well-known geometric invariants and also giving rise to analytic results.

Publication Type: Article
Additional Information: This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record will be available online at:
Publisher Keywords: Exceptional symmetries, invariants, Cayley-Hamilton theorem, Clifford algebras, Coxeter groups, root systems, Platonic solids.
Subjects: Q Science > QA Mathematics
Departments: School of Science & Technology
School of Science & Technology > Mathematics
SWORD Depositor:
[thumbnail of ML_Cliff_Inv.pdf] Text - Accepted Version
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