Graph Neural Networks for Tensor Product Decompositions of Lie Algebra Representations

Max Vargas, Helen Jenne, Davis Brown, Henry Kvinge

Keywords: Graph neural networks, representation theory, tensor decomposition, lie algebras

TL;DR: We use graph neural networks to predict the decomposition numbers of tensor products of irreducible representations of Lie algebras

Abstract: Advances in AI promise to accelerate progress in mathematics by automating the process of pattern recognition within large mathematically-motivated datasets. In this extended abstract, we report on work-in-progress using graph neural networks (GNNs) to predict properties of tensor products of Lie algebra representations. First, we impose a graph structure on the weight lattice associated to a finite-dimensional semisimple Lie algabra $\mathfrak{g}$. This structure is used to generate datasets for predicting decomposition factors of tensor products between finite-dimensional irreducible representations of $\mathfrak{g}$. We find that while this problem quickly grows in complexity, GNNs have the potential to learn algorithmic rules for predicting the structure of tensor products.

Submission Number: 88