  We use unrestricted Hartree-Fock, density matrix renormalization group, and
variational projected entangled pair state calculations to investigate the
ground state phase diagram of the triangular lattice Hubbard model at "half
doping" relative to single occupancy, i.e. at a filling of $(1\pm \frac{1}{2})$
electrons per site. The electron-doped case has a nested Fermi surface in the
non-interacting limit, and hence a weak-coupling instability towards
density-wave orders whose wavevectors are determined by Fermi surface nesting
conditions. We find that at moderate to strong interaction strengths other
spatially-modulated orders arise, with wavevectors distinct from the nesting
vectors. In particular, we identify a series closely-competing itinerant
long-wavelength magnetically ordered states, yielding to uniform ferromagnetic
order at the largest interaction strengths. For half-hole doping and a similar
range of interaction strengths, our data indicate that magnetic orders are most
likely absent.
