  We compare resonant electronic Raman scattering and photoluminescence
measurements for the characterization of a spin-polarized two-dimensional
electron gas embedded in $\text{Cd}_{1-x}\text{Mn}_x\text{Te}$ single quantum
wells. From Raman scattering by single-particle excitations in a zero magnetic
field, we measure the Fermi velocity and then obtain the Fermi energy (as well
as the electron density), which is comparable to that extracted from
photoluminescence for moderate electron densities, assuming a bare band-edge
mass. At large electron densities, the Fermi energies derived from Raman
scattering and photoluminescence differ. For an applied in-plane magnetic field
and zero wave vector transferred to the electron gas, Raman scattering spectra
show peaks at both the Zeeman energy $Z$, resulting from collective excitations
of the spin-polarized electron gas, and the one electron spin-flip energy
$Z^*$. Magneto-photoluminescence spectra show conduction band splitting that
are equivalent to $Z$, suggesting that collective effects are present in the
photoluminescence spectra. Assuming (as before) an uncorrected mass, the degree
of spin polarization $\zeta$ determined from the magneto-photoluminescence
lineshape is found to differ from that derived from the magnetic field
dependent Raman scattering measurements for large electron densities. We
attribute the discrepancy in measuring $\zeta$ and the Fermi energy to the
renormalized mass resulting from many-body electron-electron interactions.
