  A large variance exists in the amplitude of the Stellar Mass - Halo Mass
(SMHM) relation for group and cluster-size halos. Using a sample of 254
clusters, we show that the magnitude gap between the brightest central galaxy
(BCG) and its second or fourth brightest neighbor accounts for a significant
portion of this variance. We find that at fixed halo mass, galaxy clusters with
a higher magnitude gap have a higher BCG stellar mass. This relationship is
also observed in semi-analytic representations of low-redshift galaxy clusters
in simulations. This SMHM-magnitude gap stratification likely results from BCG
growth via hierarchical mergers and may link assembly of the halo with the
growth of the BCG. Using a Bayesian model, we quantify the importance of the
magnitude gap in the SMHM relation using a multiplicative stretch factor, which
we find to be significantly non-zero. The inclusion of the magnitude gap in the
SMHM relation results in a large reduction in the inferred intrinsic scatter in
the BCG stellar mass at fixed halo mass. We discuss the ramifications of this
result in the context of galaxy formation models of centrals in group and
cluster-sized halos.
