  We present the small-scale (0.2h^{-1} to 7h^{-1} Mpc) cross-correlations
between 32,000 luminous early-type galaxies and a reference sample of 16
million normal galaxies from the Sloan Digital Sky Survey. Our method allows us
to construct the spherically averaged, real-space cross-correlation function
between the spectroscopic LRG sample and galaxies from the SDSS imaging. We
report the cross-correlation as a function of scale, luminosity, and redshift.
We find very strong luminosity dependences in the clustering amplitudes, up to
a factor of 4 over a factor of 4 in luminosity, and measure this dependence
with high signal-to-noise ratio. The luminosity dependence of bias is found to
depend on scale, with more variation on smaller scales. The clustering as a
function of scale is not a power law, but instead has a dip at 1h^{-1} Mpc and
an excess on small scales. The fraction of red galaxies within the L* sample
surrounding LRGs is a strong function of scale, as expected. However, the
fraction of red galaxies evolves in redshift similarly on small and large
scales, suggesting that cluster and field populations are changing in the same
manner. The results highlight the advantage on small scales of using
cross-correlation methods as a means of avoiding shot noise in samples of rare
galaxies.
