On equivariant embeddings of G-bundles

TL;DR

This paper establishes criteria for embedding G-equivariant vector bundles and relates stable isomorphism to actual isomorphism, with applications to topological phases of matter.

Contribution

It provides new sufficient conditions for embeddings and isomorphisms of G-bundles based on irreducible representation multiplicities, extending to various types of vector bundles.

Findings

Criteria for embedding G-bundles based on representation multiplicities

Stable isomorphism implies isomorphism under certain conditions

Applications to classification of topological phases of matter

Abstract

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.