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Geometric Realization of Dynkin Quiver Type Quantum Affine Schur–Weyl Duality

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For a Dynkin quiver $Q$ of type ADE and a sum $\beta$ of simple roots, we construct a bimodule over the quantum loop algebra and the quiver Hecke algebra of… Click to show full abstract

For a Dynkin quiver $Q$ of type ADE and a sum $\beta$ of simple roots, we construct a bimodule over the quantum loop algebra and the quiver Hecke algebra of the corresponding type via equivariant K-theory, imitating Ginzburg-Reshetikhin-Vasserot's geometric realization of the quantum affine Schur-Weyl duality. Our construction is based on Hernandez-Leclerc's isomorphism between a certain graded quiver variety and the space of representations of the quiver $Q$ of dimension vector $\beta$. We identify the functor induced from our bimodule with Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor. As a by-product, we verify a conjecture by Kang-Kashiwara-Kim on the simpleness of some poles of normalized R-matrices for any quiver $Q$ of type ADE.

Keywords: weyl duality; affine schur; quiver type; quantum affine; schur weyl

Journal Title: International Mathematics Research Notices
Year Published: 2018

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