Machine Learning Clifford invariants of ADE Coxeter elements

TL;DR

This study explores Clifford geometric invariants of Coxeter transformations in root systems, using high-performance computing and machine learning to analyze their properties and relationships, opening new avenues in experimental mathematics.

Contribution

It introduces a computational framework combining algebraic invariants with machine learning techniques to analyze Coxeter transformations in root systems.

Findings

Datasets can be classified with high accuracy using neural networks.

Principal component analysis reveals structure in the invariants.

Clifford algebraic datasets are suitable for machine learning applications.

Abstract

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 $A_8$, $D_8$ and $E_8$ 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.