representation_theory

Representation-theoretic utilities for symmetric vector spaces, including isotypic decomposition, homomorphism-basis parameterizations of \(\mathbb{G}\)-equivariant linear maps, and irreducible decomposition helpers.

For the canonical isotypic decomposition and a practical Icosahedral example, see Isotypic Decomposition.

For the structure of equivariant linear maps and how GroupHomomorphismBasis implements Proposition I.13 and Eq. (40) from main_vk.pdf, see Leveraging the structure of Equivariant Linear maps.

GroupHomomorphismBasis(in_rep, out_rep[, ...])	Basis handler for \(\operatorname{Hom}_\mathbb{G}(\rho_{\mathcal{X}}, \rho_{\mathcal{Y}})\).
isotypic_decomp_rep(rep)	Return an equivalent representation disentangled into isotypic subspaces.
direct_sum(reps[, name, change_of_basis])	Return the direct sum \(\rho=\bigoplus_i \rho_i\) of representations.
permutation_matrix(oneline_notation)	Generate a permutation matrix from one-line notation.
irreps_stats(irreps_ids)	Compute summary statistics for a sequence of irrep identifiers.
escnn_representation_form_mapping(group, rep)	Reconstruct a representation from a map \(\mathbb{G}\to\mathrm{GL}(\mathcal{X})\).
is_complex_irreducible(group, rep)	Check complex irreducibility of \(\rho:\mathbb{G}\to\mathrm{GL}(\mathcal{X})\).
decompose_representation(G, rep)	Block-diagonalize \(\rho:\mathbb{G}\to\mathrm{GL}(\mathcal{X})\) into invariant subspaces.