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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types
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by Ryo Fujita PDF
Represent. Theory 26 (2022), 211-263 Request permission


For a Dynkin quiver $Q$ (of type $\mathrm {ADE}$), we consider a central completion of the convolution algebra of the equivariant $K$-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc’s monoidal category $\mathcal {C}_{Q}$ of modules over the quantum loop algebra $U_{q}(L\mathfrak {g})$ via Nakajima’s homomorphism. As an application, we show that Kang-Kashiwara-Kim’s generalized quantum affine Schur-Weyl duality functor gives an equivalence between the category of finite-dimensional modules over the quiver Hecke algebra associated with $Q$ and Hernandez-Leclerc’s category $\mathcal {C}_{Q}$, assuming the simpleness of some poles of normalized $R$-matrices for type $\mathrm {E}$.
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Additional Information
  • Ryo Fujita
  • Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Oiwake-Kitashir- akawa, Sakyo, Kyoto 606-8502, Japan; and Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université de Paris, F-75013 Paris, France
  • MR Author ID: 1243866
  • ORCID: 0000-0003-4905-2402
  • Email:
  • Received by editor(s): November 11, 2017
  • Received by editor(s) in revised form: August 9, 2019
  • Published electronically: March 18, 2022
  • Additional Notes: The work of the author was supported in part by the Kyoto Top Global University program. It was also supported by Grant-in-Aid for JSPS Research Fellow (No. 18J10669) and by JSPS Overseas Research Fellowships during the revision.
  • © Copyright 2022 American Mathematical Society
  • Journal: Represent. Theory 26 (2022), 211-263
  • MSC (2020): Primary 17B37; Secondary 17B67
  • DOI:
  • MathSciNet review: 4396615