Group equivariant neural networks is an area of deep learning that looks at how symmetries can be encoded in neural network architectures as an inductive bias. Historically, many group equivariant neural networks have been designed using either group convolutions or tensor power representations that have been decomposed into irreducible representations. Whilst these approaches have value, they come with drawbacks. In the former, a change of basis into the Fourier domain is often required, which can be computationally expensive. In the latter, the irreducible decomposition of a tensor power representation for most groups is unknown.

We instead establish weight tying as a fundamental paradigm for constructing group equivariant neural networks. We show that, for many important groups, including the symmetric, alternating, orthogonal, special orthogonal, and symplectic groups, we can determine the weight matrices that appear in group equivariant neural networks having tensor power representations of Rn as their layer spaces without needing to decompose them into irreducible representations. We study the combinatorics of set partition diagrams that are associated with each group to achieve a full characterisation of these weight matrices. We create a deeper structure for understanding the group equivariant neural networks themselves by developing a monoidal category theoretic framework and use it to construct a fast multiplication algorithm for the linear layer functions for four of these groups. We extend this framework to characterise the equivariant weight matrices for the automorphism group of a graph. Finally, we show that the equivariant neural network paradigm can be broadened to quantum groups, which often describe symmetries in non-commutative geometries. In particular, we use Woronowicz’s version of Tannaka–Krein duality to derive Compact Matrix Quantum Group Equivariant Neural Networks. We hope that our work will inspire others to develop neural network architectures that encode different algebraic structures as an inductive bias.