  The category of Cohen-Macaulay modules of an algebra $B_{k,n}$ is used
[JKS16] to give an additive categorification of the cluster algebra structure
on the homogeneous coordinate ring of the Grassmannian of $k$-planes in
$n$-space. In this paper, we find canonical Auslander--Reiten sequences and
study the Auslander--Reiten translation periodicity for this category.
Furthermore, we give an explicit construction of Cohen-Macaulay modules of
arbitrary rank. We then use our results to establish a correspondence between
rigid indecomposable modules of rank 2 and real roots of degree 2 for the
associated Kac-Moody algebra in the tame cases.
