We provide a full characterisation of all of the possible alternating group (A_n) equivariant neural networks whose layers are some tensor power of 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 ℝ^n. We also describe how our approach generalises to the construction of neural networks that are equivariant to local symmetries.…