  We calculate the frequency of collective modes of a one-dimensional
repulsively interacting Fermi gas with high-spin symmetry confined in harmonic
traps at zero temperature. This is a system realizable with fermionic
alkaline-earth-metal atoms such as $^{173}$Yb, which displays an exact
SU($\kappa$) spin symmetry with $\kappa\geqslant2$ and behaves like a spinless
interacting Bose gas in the limit of infinite spin components
$\kappa\rightarrow\infty$, namely high-spin bosonization. We solve the
homogeneous equation of state of the high-spin Fermi system by using Bethe
ansatz technique and obtain the density distribution in harmonic traps based on
local density approximation. The frequency of collective modes is calculated by
exactly solving the zero-temperature hydrodynamic equation. In the limit of
large number of spin-components, we show that the mode frequency of the system
approaches to that of a one-dimensional spinless interacting Bose gas, as a
result of high-spin bosonization. Our prediction of collective modes is in
excellent agreement with a very recent measurement for a Fermi gas of
$^{173}$Yb atoms with tunable spin confined in a two-dimensional tight optical
lattice.
