We characterize the luminosity distribution, halo mass dependence, and redshift evolution of red galaxies in galaxy clusters using the SDSS Data Release 8 RedMaPPer cluster sample. We propose a simple prescription for the relationship between the luminosity of both central and satellite galaxies and the mass of their host halos, and show that this model is well-fit by the data. Using a larger galaxy cluster sample than previously employed in the literature, we find that the luminosities of central galaxies scale as $langle log L rangle propto A_L log (M_{200b})$, with $A_L=0.39pm0.04$, and that the scatter of the central--galaxy luminosity at fixed $M_{200b}$ ( $sigma_{log L|M}$) is $0.23 ^{+0.05}_{-0.04}$ dex, with the error bar including systematics due to miscentering of the cluster finder, photometry, and photometric redshift estimation. Our data prefers a positive correlation between the luminosity of central galaxies and the observed richness of clusters at a fixed halo mass, with an effective correlation coefficient $d_{rm{eff}}=0.36^{+0.17}_{-0.16}$. The characteristic luminosity of satellites becomes dimmer from $z=0.3$ to $z=0.1$ by $sim 20%$ after accounting for passive evolution. We estimate the fraction of galaxy clusters where the brightest galaxy is not the central to be $P_{rm{BNC}} sim 20%$. We discuss implications of these findings in the context of galaxy evolution and the galaxy--halo connection.
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