Machine Learning Clifford invariants of ADE Coxeter elements

TL;DR

This work addresses the problem of understanding Clifford geometric invariants for Coxeter transformations in the ADE root systems by exhaustively generating Coxeter elements for $A_8$, $D_8$, and $E_8$ and extracting nine invariants (the SOCM) per element. The authors combine high‑performance algebra with data‑driven analysis, showing that only 128 distinct SOCMs arise from $40320$ permutations per algebra and applying neural networks and PCA to reveal structure, degeneracies, and symmetries. They demonstrate near‑perfect predictive capabilities for invariants from permutation data, interpretability via gradient saliency, and informative low‑dimensional structure through PCA, all while linking bivector subinvariants to graph representations whose spectra reflect Dynkin geometry. The findings pave the way for analytic conjectures and theorems in Clifford‑algebraic invariant theory and illustrate a productive interplay between exhaustive algebraic computation and data science in experimental mathematics.

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.

Figures (16)

• Figure 1: The diagrams of the 8-dimensional simply-laced root systems $A_8$, $D_8$ and $E_8$ (vertically downwards respectively), along with our labelling for the simple roots and a bipartite colouring.
• Figure 2: Sorted multiplicities of the 128 unique SOCMs, for each root system considered: $A_8$, $D_8$, $E_8$ respectively. $A_8$ is mostly quadruplets, $E_8$ mostly doublets and $D_8$ half and half. See https://github.com/DimaDroid/ML_Clifford_Invariants.git for the full list of values.
• Figure 3: Distributions of the maximum eigenvalues for 282240 random connected matrices (of which 282086 are unique matrices, overall having 9741 unique eigenvalues).
• Figure 4: Distributions of the maximum eigenvalues for each of the bivector subinvariants for each of the considered algebras: $A_8$, $D_8$, $E_8$ respectively. Data includes all 282240 non-empty bivector subinvariants, coloured according to which order invariant they correspond to.
• Figure 5: The multiplicities that each of the 8 graph nodes (ie. simple roots $\alpha_i$) exists as the most central node in a bivector subinvariant graph, for all graphs across all invariant orders for each of the considered root systems: $A_8$, $D_8$, $E_8$ respectively.