  We theoretically investigate one-dimensional (1D) SU($\kappa$) fermions in
the regime of spin-incoherent Luttinger liquid. We specifically focus on the
Tonks-Girardeau gas limit where its density is sufficiently low that effective
repulsions between atoms become infinite. In such case, spin exchange energy of
1D SU($\kappa$) fermions vanishes and all spin configurations are degenerate,
which automatically puts them into spin-incoherent regime. In this limit, we
are able to express the single-particle density matrices in terms of those of
anyons. This allows us to numerically simulate the number of particles up to
$N=32$. We numerically calculate single-particle density matrices in two cases:
(1) equal populations for each spin components (balanced) and (2) all $S_z$
manifolds included. In contrast to noninteracting multi-component fermions, the
momentum distributions are broadened due to strong interactions. As $\kappa$
increases, the momentum distributions are less broadened for fixed $N$, while
they are more broadened for fixed number of particle per spin component. We
then compare numerically calculated high momentum tails with analytical
predictions which are proportional to $1/p^4$, in good agreement. Thus, our
theoretical study provides a comparison with the experiments of repulsive
multicomponent alkaline-earth fermions with a tunable SU($\kappa$)
spin-symmetry in the spin-incoherent regime.
