  The symmetric splitting of two spin-wave branches in an antiferromagnetic
resonance (AFR) experiment has been an essential measurement of
antiferromagnets for over half a century. In this work, circularly polarized
time-domain THz spectroscopy experiments performed on the low symmetry
multiferroic h-HoMnO$_3$ reveal an AFR of the Mn sublattice to split
asymmetrically in applied magnetic field, with an $\approx$ 50\% difference in
$g$-factors between the high and low energy branches of this excitation. The
temperature dependence of the $g$-factors, including a drastic renormalization
at the Ho spin ordering temperature, reveals this asymmetry to unambiguously
stem from Ho-Mn interactions. Theoretical calculations demonstrate the AFR
asymmetry is not explained by conventional Ho-Mn exchange mechanisms alone and
are only reproduced if quartic spin interactions are also included in the spin
Hamiltonian. Our results provide a paradigm for the optical study of such novel
interactions in hexagonal manganites and low symmetry antiferromagnets in
general.
