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The learnable, linear neural network layers between tensor power spaces of$\mathbb{R}^{n}$ that are equivariant to the orthogonal group, $O(n)$, thespecial orthogonal group, $SO(n)$, and the symplectic group, $Sp(n)$, werecharacterised in arXiv:2212.08630. We present an algorithm for multiplying avector by any weight matrix for each of these groups, using category theoreticconstructions to implement the procedure. We achieve a significant reduction incomputational cost compared with a naive implementation by making use ofKronecker product matrices to perform the multiplication. We show that ourapproach extends to the symmetric group, $S_n$, recovering the algorithm ofarXiv:2303.06208 in the process.

An Algorithm for Computing with Brauer's Group Equivariant Neural Network Layers