  The fundamental building blocks in band theory are band representations
(BRs): bands whose infinitely-numbered Wannier functions are generated (by
action of a space group) from a finite number of symmetric Wannier functions
centered on a point in space. This work aims to simplify questions on a
multi-rank BR by splitting it into unit-rank bands, via the following
crystallographic splitting theorem: being a rank-$N$ BR is equivalent to being
splittable into a finite sum of bands indexed by $\{1,2,\ldots,N\}$, such that
each band is spanned by a single, analytic Bloch function of $k$, and any
symmetry in the space group acts by permuting $\{1,2,\ldots,N\}$. Applying this
theorem, we develop computationally efficient methods to determine whether a
given energy band (of a tight-binding or Schr\"odinger Hamiltonian) is a BR,
and, if so, how to numerically construct the corresponding symmetric Wannier
functions. Thus we prove that rotation-symmetric topological insulators in
class AI are fragile, meaning that the obstruction to symmetric Wannier
functions is removable by addition of BRs. An implication of fragility is that
its boundary states, while robustly covering the bulk energy gap in finite-rank
tight-binding models, are unstable if the Hilbert space is expanded to include
all symmetry-allowed representations. These fragile insulators have photonic
analogs that we identify; in particular, we prove that an existing photonic
crystal built by Yang et al. [Nature 565, 622 (2019)] is fragile topological
with removable surface states, which disproves a widespread perception of
'topologically-protected' surface states in time-reversal-invariant, gapped
photonic/phononic crystals. Our theorem is finally applied to derive various
symmetry obstructions on the Wannier functions of topological insulators, and
to prove their equivalence with the nontrivial holonomy of Bloch functions.
