  Photon beam asymmetry $\Sigma$ measurements for $\omega$ photoproduction in
the reaction $\vec{\gamma} p \to \omega p$ are reported for photon energies
from 1.152 to 1.876 GeV. Data were taken using a linearly-polarized tagged
photon beam, a cryogenic hydrogen target, and the CLAS spectrometer in Hall B
at Jefferson Lab. The measurements obtained markedly increase the size of the
database for this observable, extend coverage to higher energies, and resolve
discrepancies in previously published data. Comparisons of these new results
with predictions from a chiral-quark-based model and from a dynamical
coupled-channels model indicate the importance of interferences between
$t$-channel meson exchange and $s$- and $u$-channel contributions, underscoring
sensitivity to the nucleon resonances included in those descriptions.
Comparisons with the Bonn-Gatchina partial-wave analysis indicate the $\Sigma$
data reported here help to fix the magnitudes of the interference terms between
the leading amplitudes in that calculation (Pomeron exchange and the resonant
portion of the $J^P=3/2^+$ partial wave), as well as the resonant portions of
the smaller partial waves with $J^P$= $1/2^-$, $3/2^-$, and $5/2^+$.
