Clarify why elementary band representations are fundamental for classifying non-trivial phases

Determine the theoretical justification for treating elementary band representations, as used in topological quantum chemistry, as fundamental building blocks in the classification of non-trivial topological phases of crystalline solids, given the alternative framework in which irreducible band representations and Wigner–Eckart–based selection rules directly constrain Berry–Wilczek–Zee phases.

Background

The paper argues that irreducible band representations (IBRs), rather than elementary band representations (EBRs), are the appropriate building blocks for identifying symmetry-forbidden Berry–Wilczek–Zee (BWZ) phases and hence topologically trivial phases. This challenges the standard topological quantum chemistry paradigm that elevates EBRs to a fundamental status for classifying topological phases.

In this context, the author explicitly notes uncertainty about the rationale for privileging EBRs in classifying non-trivial phases, raising the need for a rigorous justification or clarification of their role relative to IBRs within symmetry-based analyses of band topology.