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# Brauer’s Group Equivariant Neural Networks

*Proceedings of the 40th International Conference on Machine Learning*, PMLR 202:27461-27482, 2023.

#### Abstract

We provide a full characterisation of all of the possible group equivariant neural networks whose layers are some tensor power of $\mathbb{R}^{n}$ for three symmetry groups that are missing from the machine learning literature: $O(n)$, the orthogonal group; $SO(n)$, the special orthogonal group; and $Sp(n)$, the symplectic group. In particular, we find a spanning set of matrices for the learnable, linear, equivariant layer functions between such tensor power spaces in the standard basis of $\mathbb{R}^{n}$ when the group is $O(n)$ or $SO(n)$, and in the symplectic basis of $\mathbb{R}^{n}$ when the group is $Sp(n)$.