Spin-Weighted Spherical Harmonics Enable Complete and Scalable $\mathrm{E}(3)$-Equivariant Networks
Existing $\mathrm{E}(3)$-equivariant networks are constrained by the $O(L^6)$ computational complexity of Clebsch–Gordan tensor products, while efficient alternatives such as Gaunt tensor products sacrifice expressivity due to the absence of antisymmetric pathways. This work introduces spin-weighted spherical harmonics (SWSH) into equivariant learning for the first time and proposes the SpinGTP framework: a novel tensor product operator derived from the algebraic structure of SWSH that retains Gaunt-level computational efficiency while recovering full symmetry expressivity, including parity-odd components. Experiments demonstrate that SpinGTP achieves accuracy comparable to full Clebsch–Gordan tensor products on benchmarks such as Tetris, 3BPA, SPICE-MACE-OFF, and OC20, and further exhibits superior performance on tasks involving chiral materials and non-centrosymmetric structures.