  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 $Sp(n)$.
