Define fragile topology within the IBR-based symmetry analysis framework

Develop a precise and internally consistent definition of fragile topology within the symmetry analysis framework based on irreducible band representations and Wigner–Eckart–type selection rules for the Berry–Wilczek–Zee phase, and delineate the criteria by which bands in this framework should be regarded as fragile.

Background

The paper situates its analysis in an IBR-centric framework where selection rules derived from symmetry (via Wigner–Eckart-type arguments) can forbid BWZ phases, identifying topologically trivial phases. It discusses how certain IBR combinations (e.g., type 3 with associated type 2α) mimic aspects of fragile topology in terms of decomposition at high-symmetry points, yet remain symmetry-forbidden in BWZ phase.

Within this IBR-based approach, the author explicitly states that a clear definition of fragile topology is lacking, highlighting the need to reconcile or reformulate the fragile topology notion in terms consistent with the IBR and selection-rule framework.