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 $5/2^+$.
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