  Galaxy cluster centring is a key issue for precision cosmology studies using
galaxy surveys. Mis-identification of central galaxies causes systematics in
various studies such as cluster lensing, satellite kinematics, and galaxy
clustering. The red-sequence Matched-filter Probabilistic Percolation
(redMaPPer) estimates the probability that each member galaxy is central from
photometric information rather than specifying one central galaxy. The
redMaPPer estimates can be used for calibrating the off-centring effect,
however, the centring algorithm has not previously been well-tested. We test
the centring probabilities of redMaPPer cluster catalog using the projected
cross correlation between redMaPPer clusters with photometric red galaxies and
galaxy-galaxy lensing. We focus on the subsample of redMaPPer clusters in which
the redMaPPer central galaxies (RMCGs) are not the brightest member galaxies
(BMEM) and both of them have spectroscopic redshift. This subsample represents
nearly 10% of the whole cluster sample. We find a clear difference in the
cross-correlation measurements between RMCGs and BMEMs, and the estimated
centring probability is 74$\pm$10% for RMCGs and 13$\pm$4% for BMEMs in the
Gaussian offset model and 78$\pm$9% for RMCGs and 5$\pm$5% for BMEMs in the NFW
offset model. These values are in agreement with the centring probability
values reported by redMaPPer (75% for RMCG and 10% for BMEMs) within 1$\sigma$.
Our analysis provides a strong consistency test of the redMaPPer centring
probabilities. Our results suggest that redMaPPer centring probabilities are
reliably estimated. We confirm that the brightest galaxy in the cluster is not
always the central galaxy as has been shown in previous works.
