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We show how the Schur-Weyl duality that exists between the partition algebraand the symmetric group results in a stronger theoretical foundation forcharacterising all of the possible permutation equivariant neural networkswhose layers are some tensor power of the permutation representation $M_n$ ofthe symmetric group $S_n$. In doing so, we unify two separate bodies ofliterature, and we correct some of the major results that are now widely quotedby the machine learning community. In particular, we find a basis of matricesfor the learnable, linear, permutation equivariant layer functions between suchtensor power spaces in the standard basis of $M_n$ by using an elegantgraphical representation of a basis of set partitions for the partition algebraand its related vector spaces. Also, we show how we can calculate the number ofweights that must appear in these layer functions by looking at certain pathsthrough the McKay quiver for $M_n$. Finally, we describe how our approachgeneralises to the construction of neural networks that are equivariant tolocal symmetries.

Connecting Permutation Equivariant Neural Networks and Partition Diagrams