Efficient generation and explicit dimensionality of Lie group-equivariant and permutation-invariant bases

TL;DR

This paper introduces a practical, scalable method for constructing Lie group-equivariant and permutation-invariant functions using a matrix derived from the Lie algebra, avoiding complex coefficients.

Contribution

It provides a generic construction method for any linear Lie group that simplifies basis derivation without needing Clebsch--Gordan coefficients.

Findings

The method scales linearly, outperforming existing exponential-scaling approaches.

Explicit formulas for the dimension of invariant and equivariant spaces are derived for groups like SO(3) and SU(2).

Numerical simulations confirm the efficiency and scalability of the proposed approach.

Abstract

In this article, we propose a practical construction of Lie group-equivariant and permutation-invariant functions of $N$ variables from the knowledge of a one-particle basis that is stable with respect to the group action. The construction is generic for any linear Lie group and relies on building a matrix constructed from the Lie algebra whose kernel is spanned by a group-equivariant and permutation-invariant basis. In particular, this construction does not require the knowledge of Clebsch--Gordan coefficients and instead directly builds generalized Clebsch--Gordan coefficients. For specific groups such as $SO(3)$ and $SU(2)$, we exploit the Lie algebra structure to simplify the matrix, which then allows us to derive an explicit formula for the exact dimension of the group-equivariant and permutation-invariant space. Numerical simulations are provided to show that the proposed method scales linearly instead of exponentially for existing methods in the literature. We also show that for large values of $N$, the number of rotation-equivariant and permutation-invariant basis functions is of a comparable order as the number of permutation-invariant basis functions, while pre-asymptotically, a large gain can be achieved by explicitly enforcing rotation-equivariance on top of permutation-invariance.