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The coordinate ring of the Grassmannian variety of $k$-dimensional subspacesin $\mathbb{C}^n$ has a cluster algebra structure with Pl\"ucker relationsgiving rise to exchange relations. In this paper, we study indecomposablemodules of the corresponding Grassmannian cluster categories ${\rmCM}(B_{k,n})$. 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, rigidindecomposable modules correspond to roots. In general, the link between thecategory ${\rm CM}(B_{k,n})$ and the root system $J_{k,n}$ remains mysteriousand it is an open question whether indecomposables always give roots. In thispaper, we provide evidence for this association in the infinite types: we showthat every indecomposable rank 2 module corresponds to a root of the associatedroot system. We also show that indecomposable rank 3 modules in ${\rmCM}(B_{3,n})$ 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 underlyingprofiles are cyclic permutations of a certain canonical one. We alsocharacterize the rank 3 modules in ${\rm CM}(B_{3,n})$ corresponding toimaginary roots. By proving that there are exactly 225 profiles of rigidindecomposable rank 3 modules in ${\rm CM}(B_{3,9})$ we confirm the linkbetween the Grassmannian cluster category and the associated root system inthis 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 acanonical profile.

Rigid Indecomposable Modules in Grassmannian Cluster Categories