  The coordinate ring of the Grassmannian variety of $k$-dimensional subspaces
in $\mathbb{C}^n$ has a cluster algebra structure with Pl\"ucker relations
giving rise to exchange relations. In this paper, we study indecomposable
modules of the corresponding Grassmannian cluster categories ${\rm
CM}(B_{k,n})$. Jensen, King, and Su have associated a Kac-Moody root system
$J_{k,n}$ to ${\rm CM}(B_{k,n})$ and shown that in the finite types, rigid
indecomposable modules correspond to roots. In general, the link between the
category ${\rm CM}(B_{k,n})$ and the root system $J_{k,n}$ remains mysterious
and it is an open question whether indecomposables always give roots. In this
paper, we provide evidence for this association in the infinite types: we show
that every indecomposable rank 2 module corresponds to a root of the associated
root system. We also show that indecomposable rank 3 modules in ${\rm
CM}(B_{3,n})$ all give rise to roots of $J_{3,n}$. For the rank 3 modules in
${\rm CM}(B_{3,n})$ corresponding to real roots, we show that their underlying
profiles are cyclic permutations of a certain canonical one. We also
characterize the rank 3 modules in ${\rm CM}(B_{3,n})$ corresponding to
imaginary roots. By proving that there are exactly 225 profiles of rigid
indecomposable rank 3 modules in ${\rm CM}(B_{3,9})$ we confirm the link
between the Grassmannian cluster category and the associated root system in
this case. We conjecture that the profile of any rigid indecomposable module in
${\rm CM}(B_{k,n})$ corresponding to a real root is a cyclic permutation of a
canonical profile.

Representation Theory;Rings and Algebras