A band is a band representation exactly when it splits into analytic, topologically trivial unit-rank bands permuted by symmetry, which is used to prove that specific photonic crystals are fragile topological with removable boundary states.
A finite group G is solvable if there exists a series of normal groups, i.e., C1 =G0◁G 1◁G 2...◁G k =G (F1) for a k ≥ 1, such that Gj+1/Gj is abelian for all j = 1,...,k − 1
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Crystallographic splitting theorem for band representations and fragile topological photonic crystals
A band is a band representation exactly when it splits into analytic, topologically trivial unit-rank bands permuted by symmetry, which is used to prove that specific photonic crystals are fragile topological with removable boundary states.