Machine Learning Clifford invariants of ADE Coxeter elements

The well-known classification of the Clifford algebras $Cl(r,s)$ leads to canonical forms of complex and real representations which are essentially unique by virtue of the Wedderburn theorem. For $s\ge 1$ representations of $Cl(r,s)$ on $R^{2N}$ are obtained from representations on $R^N$ by adding two new generators while in passing from a representation of $Cl(p,0)$ on $R^N$ to a representation of $Cl(r,0)$ on $R^{2N}$ the number of generators that can be added is either 1, ...

In the recent paper (Casselman, 2001) I described how a number of ideas due to Fokko du Cloux and myself could be incorporated into a reasonably efficient program to carry out multiplication in arbitrary Coxeter groups. At the end of that paper I discussed how this algorithm could be used to build the reflection table of minimal roots, which could in turn form the basis of a much more efficient multiplication algorithm. In this paper, following a suggestion of Bob Howlett, I ...

One of the central problems in the interface of deep learning and mathematics is that of building learning systems that can automatically uncover underlying mathematical laws from observed data. In this work, we make one step towards building a bridge between algebraic structures and deep learning, and introduce \textbf{AIDN}, \textit{Algebraically-Informed Deep Networks}. \textbf{AIDN} is a deep learning algorithm to represent any finitely-presented algebraic object with a s...

Modern advances in general-purpose computer algebra systems offer solutions to a variety of problems, which in the past required substantial time investments by trained mathematicians. An excellent example of such development are the Clifford algebras. The main objective of the paper is to demonstrate an utterly algorithmic construction of a Clifford algebra matrix algebra representation of a non-degenerate signature (p, q). While this is not the most economical way of implem...

Let $W$ be a rank $n$ irreducible finite reflection group and let $p_1(x),\ldots,p_n(x)$, $x\in\mathbb{R}^n$, be a basis of algebraically independent $W$-invariant real homogeneous polynomials. The orbit map $\overline p:\mathbb{R}^n\to\mathbb{R}^n:x\to (p_1(x),\ldots,p_n(x))$ induces a diffeomorphism between the orbit space $\mathbb{R}^n/W$ and the set ${\cal S}=\overline p(\mathbb{R}^n)\subset\mathbb{R}^n$. The border of ${\cal S}$ is the $\overline p$ image of the set of r...

The characterization of orbits of roots under the action of a Coxeter element is a fundamental tool in the study of finite root systems and their reflection groups. This paper develops the analogous tool in the affine setting, adding detail and uniformity to a result of Dlab and Ringel.

This paper presents a novel, interdisciplinary study that leverages a Machine Learning (ML) assisted framework to explore the geometry of affine Deligne-Lusztig varieties (ADLV). The primary objective is to investigate the nonemptiness pattern, dimension and enumeration of irreducible components of ADLV. Our proposed framework demonstrates a recursive pipeline of data generation, model training, pattern analysis, and human examination, presenting an intricate interplay betwee...

Recent work has applied supervised deep learning to derive continuous symmetry transformations that preserve the data labels and to obtain the corresponding algebras of symmetry generators. This letter introduces two improved algorithms that significantly speed up the discovery of these symmetry transformations. The new methods are demonstrated by deriving the complete set of generators for the unitary groups U(n) and the exceptional Lie groups $G_2$, $F_4$, and $E_6$. A thir...

Recent work has constructed neural networks that are equivariant to continuous symmetry groups such as 2D and 3D rotations. This is accomplished using explicit Lie group representations to derive the equivariant kernels and nonlinearities. We present three contributions motivated by frontier applications of equivariance beyond rotations and translations. First, we relax the requirement for explicit Lie group representations with a novel algorithm that finds representations of...

Quantum Machine Learning (QML) models are aimed at learning from data encoded in quantum states. Recently, it has been shown that models with little to no inductive biases (i.e., with no assumptions about the problem embedded in the model) are likely to have trainability and generalization issues, especially for large problem sizes. As such, it is fundamental to develop schemes that encode as much information as available about the problem at hand. In this work we present a s...