  Clusters of galaxies are sensitive to the most nonlinear peaks in the cosmic
density field. The weak gravitational lensing of background galaxies by
clusters can allow us to infer their masses. However, galaxies associated with
the local environment of the cluster can also be intrinsically aligned due to
the local tidal gradient, contaminating any cosmology derived from the lensing
signal. We measure this intrinsic alignment in Dark Energy Survey (DES) Year 1
redMaPPer clusters. We find evidence of a non-zero mean radial alignment of
galaxies within clusters between redshift 0.1-0.7. We find a significant
systematic in the measured ellipticities of cluster satellite galaxies that we
attribute to the central galaxy flux and other intracluster light. We attempt
to correct this signal, and fit a simple model for intrinsic alignment
amplitude ($A_{\textrm{IA}}$) to the measurement, finding
$A_{\textrm{IA}}=0.15\pm 0.04$, when excluding data near the edge of the
cluster. We find a significantly stronger alignment of the central galaxy with
the cluster dark matter halo at low redshift and with higher richness and
central galaxy absolute magnitude (proxies for cluster mass). This is an
important demonstration of the ability of large photometric data sets like DES
to provide direct constraints on the intrinsic alignment of galaxies within
clusters. These measurements can inform improvements to small-scale modeling
and simulation of the intrinsic alignment of galaxies to help improve the
separation of the intrinsic alignment signal in weak lensing studies.
