  We provide a full characterisation of all of the possible alternating group
($A_n$) equivariant neural networks whose layers are some tensor power of
$\mathbb{R}^{n}$. In particular, we find a basis of matrices for the learnable,
linear, $A_n$-equivariant layer functions between such tensor power spaces in
the standard basis of $\mathbb{R}^{n}$. We also describe how our approach
generalises to the construction of neural networks that are equivariant to
local symmetries.
