Nonlinear Maps Between Irreducible Representations in Graph Neural Networks

Equivariant linear maps simplify after decomposing feature spaces into irreducible representations, but nonlinear activations can mix those components. We develop a degree-resolved description of these nonlinear interactions through the spaces $$\text{Hom}_G(\text{Sym}^kL_i,L_j)$$ which separate the representation-theoretic availability of a route from the degrees supplied by the activation. We use this structure to construct Specht Triple, a representation-aware residual on unordered graph triples. On structural graph regression, Specht Triple reduces mean standardised test error by 11.05% relative to a parameter-matched Projector Control with the same irreducible decomposition but without explicit nonlinear source-to-target processing. This provides empirical evidence that organising nonlinear equivariant computation by irreducible components can provide a useful inductive bias.