The fundamental building blocks in band theory are band representations -bands whose infinitelynumbered 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 band representation by splitting it into unit-rank bands, via the following crystallographic splitting theorem: being a rank-N band representation is equivalent to being splittable into a finite sum of bands indexed by {1, 2, . . . , 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, . . . , N }. We prove this theorem for all band representations (of crystallographic space groups) whose Wannier functions transform in the integer-spin representation; in the half-integer-spin case, the only exceptions to the theorem exist for three-spatial-dimensional space groups with cubic point groups. Applying this theorem, we develop computationally efficient methods to determine whether a given energy band (of a tight-binding or Schrödinger-type Hamiltonian) is a band representation, and, if so, how to numerically construct the corresponding symmetric Wannier functions. Thus we prove that rotation-symmetric topological insulators in Wigner-Dyson class AI are fragile, meaning that the obstruction to symmetric Wannier functions can be removed by addition of band representations to the filled-band subspace. An implication of fragility is that its boundary states, while robustly covering the bulk energy gap in finite-rank tight-binding models, can be destabilized 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 Yihao Yang et al. [Nature 565, 622 (2019)] is fragile topological with removable boundary states, which disproves a widespread perception of 'topologically-protected' boundary states in timereversal-invariant, gapped photonic/phononic crystals. As a final application of our theorem, we derive various symmetry obstructions on the Wannier functions of topological insulators; for certain space groups, these obstructions are proven to be equivalent to the nontrivial holonomy of Bloch functions.CONTENTS 46 G. Tightly-bound BRs and the existence of the symmetric tight-binding limit 47 1. G-vector bundles and tight-binding lattice models 47 2. BRs and tightly-bound BRs as G-vector bundles 48 3. Existence of symmetric tight-binding limit 48 H. Lemma for Zak phases of tightly-bound band representations 48 I. Proof of localization obstruction lemma 49 References 50

Dirac materials, unlike the Weyl materials, have not been found in experiments to support intrinsic topological surface states, as the surface arcs in existing systems are unstable against symmetrypreserving perturbations. Utilizing the proposed glide and time-reversal symmetries, we theoretically design and experimentally verify an acoustic crystal of two frequency-isolated three-dimensional Dirac points with Z2 monopole charges and four gapless helicoid surface states. arXiv:2001.06205v1 [cond-mat.mtrl-sci]