  In this paper, we develop a theory about the relationship between
$G$-invariant/equivariant functions and deep neural networks for finite group
$G$. Especially, for a given $G$-invariant/equivariant function, we construct
its universal approximator by deep neural network whose layers equip
$G$-actions and each affine transformations are $G$-equivariant/invariant. Due
to representation theory, we can show that this approximator has exponentially
fewer free parameters than usual models.
