  The shapes of cluster central galaxies are not randomly oriented, but rather
exhibit coherent alignments with the shapes of their parent clusters as well as
with the surrounding large-scale structures. In this work, we aim to identify
the galaxy and cluster quantities that most strongly predict the central galaxy
alignment phenomenon among a large parameter space with a sample of 8237
clusters and 94817 members within 0.1<z<0.35, based on the redMaPPer cluster
catalog constructed from the Sloan Digital Sky Survey. We first quantify the
alignment between the projected central galaxy shapes and the distribution of
member satellites, to understand what central galaxy and cluster properties
most strongly correlate with these alignments. Next, we investigate the angular
segregation of satellites with respect to their central galaxy major axis
directions, to identify the satellite properties that most strongly predict
their angular segregation. We find that central galaxies are more aligned with
their member galaxy distributions in clusters that are more elongated and have
higher richness, and for central galaxies with larger physical size, higher
luminosity and centering probability, and redder color. Satellites with redder
color, higher luminosity, located closer to the central galaxy, and with
smaller ellipticity show a stronger angular segregation toward their central
galaxy major axes. Finally, we provide physical explanations for some of the
identified correlations, and discuss the connection to theories of central
galaxy alignments, the impact of primordial alignments with tidal fields, and
the importance of anisotropic accretion.
