The field of two-dimensional ferromagnets has been reinvigorated by the discovery of ${\mathrm{VSe}}_{2}$ monolayer grown on van der Waals substrates, which is reported to be ferromagnetic with a Curie point higher than 330 K. However, the ferromagnetic and nonmagnetic states of pristine monolayer ${\mathrm{VSe}}_{2}$ are highly debated. Here, employing density functional theory, Wannier function calculations, and the band unfolding method, we explore the electronic structure of monolayer ${\mathrm{VSe}}_{2}$ with a $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{7}$ charge density wave (CDW). Certain qualitative aspects of the calculated unfolded band dispersion and unfolded Fermi surface of monolayer ${\mathrm{VSe}}_{2}$ with the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{7}$ CDW in the nonmagnetic state agree well with previous angle-resolved photoemission spectroscopy results, albeit with uncertainty about whether these experiments probed single or multiple domains. Specifically, we find that an isolated CDW domain naturally induces a strong breaking of the threefold symmetry of the electronic structure. In addition we find that, relative to the undistorted structure, the CDW structure shows a strong competition between nonmagnetic and various magnetic states, with an energy difference less than 5 meV/f.u. For the CDW structure in the antiferromagnetic state, the band dispersions and Fermi surface are similar to those in the nonmagnetic state, while the unfolded bands of the ferromagnetic CDW state display a sizable exchange splitting. These results indicate the possibility of various antiferromagnetic fluctuations in ${\mathrm{VSe}}_{2}$ to coexist and compete with ferromagnetic order and the experimentally reported CDW order. Our calculations build insights for exploring the interplay between magnetism and CDW behaviors more generally in transition metal dichalcogenides.

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