Abstract
A nilpotent endomorphism of a quiver representation induces a linear transformation on the vector space at each vertex. Generically among all nilpotent endomorphisms, there is a well-defined Jordan form for these linear transformations, which is an interesting new invariant of a quiver representation. If Q is a Dynkin quiver and m is a minuscule vertex, we show that representations consisting of direct sums of indecomposable representations all including m in their support, the category of which we denote by \(\mathcal C_{Q,m}\), are determined up to isomorphism by this invariant. We use this invariant to define a bijection from isomorphism classes of representations in \(\mathcal C_{Q,m}\) to reverse plane partitions whose shape is the minuscule poset corresponding to Q and m. By relating the piecewise-linear promotion action on reverse plane partitions to Auslander–Reiten translation in the derived category, we give a uniform proof that the order of promotion equals the Coxeter number. In type \(A_n\), we show that special cases of our bijection include the Robinson–Schensted–Knuth and Hillman–Grassl correspondences.
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Acknowledgements
AG was suppported by NSERC grant RGPIN/05999-2014 and the Canada Research Chairs program. AG thanks Gabe Frieden for useful discussions and helpful comments on an earlier version of the paper. BP was supported by NSERC, CRM-ISM, and the Canada Research Chairs program. HT was supported by an NSERC Discovery Grant and the Canada Research Chairs program. He thanks Arkady Berenstein and Steffen Oppermann for helpful discussions, and Guillaume Chapuy for an inspiring explanation of Robinson–Schensted–Knuth. He also thanks the Université Paris VII for the excellent working conditions under which part of the research was carried out. The authors thank Bernhard Keller for a comment on an earlier version of the paper and Robin Sulzgruber for sharing an early version of his paper [33], which was inspirational to the authors.
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Garver, A., Patrias, R. & Thomas, H. Minuscule reverse plane partitions via quiver representations. Sel. Math. New Ser. 29, 37 (2023). https://doi.org/10.1007/s00029-023-00831-4
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DOI: https://doi.org/10.1007/s00029-023-00831-4