Permutation Equivariant Neural Networks for Antisymmetric Tensors

Antisymmetric tensors change sign under index swaps, yet learning methods that exploit this structure remain largely unexplored. We characterise all linear permutation equivariant maps between antisymmetric power spaces of ${\mathbb{R}}^n$, showing that they are determined by a class of objects that we term antispherical bipartitions. As a consequence, we show that the dimension of the space of such maps is at most two, independently of the tensor orders and of $n$. We further introduce IO patterns, which replace large weight matrices with a representation whose storage is independent of $n$. This framework is well-suited to graph-structured data with signed or directed edges, where antisymmetry encodes the orientation of edges. We illustrate our approach on the Wiki-RfA dataset, showing that the characterised layers perform at least as well as general permutation equivariant maps at a lower parameter cost, and that IO patterns remain practical at sizes where the dense weight matrix exhausts available memory.