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- W3100205598 abstract "The coordinate ring of the Grassmannian variety of $k$-dimensional subspaces in $mathbb{C}^n$ has a cluster algebra structure with Plucker 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." @default.
- W3100205598 created "2020-11-23" @default.
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- W3100205598 date "2020-11-18" @default.
- W3100205598 modified "2023-09-23" @default.
- W3100205598 title "Rigid Indecomposable Modules in Grassmannian Cluster Categories" @default.
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- W3100205598 doi "https://doi.org/10.48550/arxiv.2011.09227" @default.
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