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.

最近人们对线性变换的新颖克利福德几何不变量产生了兴趣。这激发了在根系、反射群、李群和李代数的背景下研究某种类型的感兴趣的几何变换的不变量：Coxeter 变换。我们对\(A_8\)、\(D_8\)和\(E_8\)的所有 Coxeter 变换进行详尽计算，以选择简单根的基础，并使用高性能计算计算它们的不变量。这种计算代数范式生成一个数据集，然后可以使用数据科学技术（例如监督和无监督机器学习）来挖掘该数据集。在本文中，我们重点关注神经网络分类和主成分分析。由于输出（不变量）完全由简单根的选择和 Coxeter 元素中相应反射的排列顺序决定，因此我们预计映射中会有巨大的简并性。这为机器学习提供了完美的设置，实际上我们看到数据集可以通过机器学习达到非常高的精度。本文是使用 Clifford 代数进行实验数学的泵启动研究，表明此类 Clifford 代数数据集适合机器学习，并揭示了这些新颖的和其他众所周知的几何不变量之间的关系，并产生了分析结果。