  As a fundamental concept of all crystals, space groups are partitioned into
symmorphic groups and nonsymmorphic groups. Each nonsymmorphic group contains
glide reflections or screw rotations with fractional lattice translations,
which are absent in symmorphic groups. Although nonsymmorphic groups
ubiquitously exist on real-space lattices, on the reciprocal lattices in
momentum space, the ordinary theory only allows symmorphic groups. In this
work, we develop a novel theory for momentum-space nonsymmorphic space groups
(k-NSGs), utilizing the projective representations of space groups. The theory
is quite general: Given any k-NSGs in any dimensions, it can identify the
real-space symmorphic space groups (r-SSGs) and construct the corresponding
projective representation of the r-SSG that leads to the k-NSG. To demonstrate
the broad applicability of our theory, we show these projective representations
and therefore all k-NSGs can be realized by gauge fluxes over real-space
lattices. Our work fundamentally extends the framework of crystal symmetry, and
therefore can accordingly extend any theory based on crystal symmetry, for
instance the classification crystalline topological phases.
