Integral Formulas for Vector Spherical Tensor Products

We derive integral formulas that simplify the Vector Spherical Tensor Product recently introduced by Xie et. al, which generalizes the Gaunt Tensor Product to anti-symmetric couplings. In particular, we obtain explicit closed-form expressions for the antisymmetric analogues of the Gaunt coefficients. This enables us to simulate the Clebsch-Gordan tensor product using a single Vector Spherical Tensor Product, yielding a $9\times$ reduction in the required tensor product evaluations. Our results enable efficient and practical implementations of the Vector Spherical Tensor Product, paving the way for applications of this generalization of Gaunt tensor products in $\mathrm{SO}(3)$-equivariant neural networks. Moreover, we discuss how the Gaunt and the Vector Spherical Tensor Products allow to control the expressivity-runtime tradeoff associated with the usual Clebsch-Gordan Tensor Products. Finally, we investigate low rank decompositions of the normalizations of the considered tensor products in view of their use in equivariant neural networks.