Piecewise Linear Equivariant Maps for Compact Groups

• URL: http://arxiv.org/abs/2608.21645v1
• Date: Fri, 21 Aug 2026 21:31:26 GMT
• Title: Piecewise Linear Equivariant Maps for Compact Groups
• Abstract summary: We show that all genuinely non-linear piecewise linear behaviour is confined to the subspaces on which the identity component of the group acts trivially.<n>We obtain a compact-group analogue of the finite-group existence criterion of Gibson--Tubbenhauer--Williamson for non-zero equivariant piecewise linear maps between irreducible representations.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Motivated by equivariant neural networks, we study piecewise linear equivariant maps between finite-dimensional real representations of compact groups. We show that all genuinely non-linear piecewise linear behaviour is confined to the subspaces on which the identity component of the group acts trivially, while equivariance forces linearity on the corresponding orthogonal complements. As a consequence, we obtain a compact-group analogue of the finite-group existence criterion of Gibson--Tubbenhauer--Williamson for non-zero equivariant piecewise linear maps between irreducible representations, with the identity component giving rise to a rigidity phenomenon absent from the finite-group case.

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