Efficient construction of Lie group-equivariant and permutation-invariant spaces

We introduce a practical construction of group-equivariant and permutation-invariant functions of $N$ variables given a finite-dimensional space stable with respect to the group action. The construction applies to any connected linear Lie group and relies on leveraging the Lie algebra to build a matrix $M$ whose kernel is in one-to-one correspondence with the subspace with desired equivariance and invariance properties, removing the need for prior knowledge of Clebsch--Gordan coefficients. A similar construction is proposed for group-equivariant functions alone, without imposing permutation-invariance. For the groups $SO(3)$ and $SU(2)$, we further exploit the structure of the Lie algebra to demonstrate the sparsity pattern and rank of the matrix $M$, which yields the exact dimension of the group-equivariant and permutation-invariant space, as well as the dimension of the group-equivariant space alone. We demonstrate analytically and verify numerically that the proposed method scales linearly with respect to the dimensionality of the basis, offering a high computational gain compared to existing methods in the literature which typically scale exponentially. We finally perform a dimensionality comparison, showing that for large values of~$N$, the dimension of group-equivariant and permutation-invariant spaces is of comparable order as the dimension of permutation-invariant spaces, while pre-asymptotically, the first dimensionality is orders of magnitude lower than the second. Hence a substantial computational gain can be achieved by explicitly enforcing group-equivariance on top of permutation-invariance when approximating such functions.