arXiv:2604.01975v3 Announce Type: replace Abstract: We give a practical construction of functions of $N$ variables that are simultaneously equivariant under a group action and invariant under permutations, or simply group-equivariant, valid for any finite-dimensional space stable under that action. The method applies to any connected linear Lie group: it uses the Lie algebra to build a matrix $M$ whose kernel corresponds exactly to the subspace with the desired symmetry properties, removing the need for Clebsch-Gordan coefficients and explicit symmetrization over permutations. For the groups $SO(3)$ and $SU(2)$, the special structure of the matrix $M$ yields a linear scaling method which significantly outperforms the typically exponential cost of existing methods in the literature. As a byproduct, we obtain the exact dimension of group-equivariant and permutation-invariant spaces. This allows us to compare dimensionalities and show that, for large $N$, this dimension is of comparable order to that of the permutation-invariant space alone, while pre-asymptotically it is orders of magnitude lower. Enforcing group-equivariance explicitly, on top of permutation-invariance, therefore yields a substantial computational gain when approximating such functions.