On equivariant embeddings of G-bundles

For a compact group G, we give a sufficient condition for embedding one G-equivariant vector bundle into another one and for a stable isomorphism between two such bundles to imply an isomorphism. Our criteria involve multiplicities of irreducible representations of stabiliser groups. We also apply our result to ordinary nonequivariant vector bundles over the fields of quaternions, real and complex numbers and to real'' and quaternionic’’ vector bundles. Our results apply to the classification of symmetry-protected topological phases of matter, providing computable bounds on the number of energy bands required to distinguish robust from fragile topological phases.