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Auslander-Reiten quiver of type A and generalized quantum affine Schur-Weyl duality


Author: Se-jin Oh
Journal: Trans. Amer. Math. Soc. 369 (2017), 1895-1933
MSC (2010): Primary 05E10, 16T30, 17B37; Secondary 81R50
DOI: https://doi.org/10.1090/tran6704
Published electronically: May 3, 2016
MathSciNet review: 3581223
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Abstract: The quiver Hecke algebra $ R$ can be also understood as a generalization of the affine Hecke algebra of type $ A$ in the context of the quantum affine Schur-Weyl duality by the results of Kang, Kashiwara and Kim. On the other hand, it is well known that the Auslander-Reiten (AR) quivers $ \Gamma _Q$ of finite simply-laced types have a deep relation with the positive roots systems of the corresponding types. In this paper, we present explicit combinatorial descriptions for the AR-quivers $ \Gamma _Q$ of finite type $ A$. Using the combinatorial descriptions, we can investigate relations between finite dimensional module categories over the quantum affine algebra $ U'_q(A_n^{(i)})$ $ (i=1,2)$ and finite dimensional graded module categories over the quiver Hecke algebra $ R_{A_n}$ associated to $ A_n$ through the generalized quantum affine Schur-Weyl duality functor.


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Additional Information

Se-jin Oh
Affiliation: School of Mathematics, Korea Institute for Advanced Study Seoul 130-722, Korea
Address at time of publication: Department of Mathematics, Ewha Womans University, Seoul 120-750, Korea
Email: sejin092@gmail.com

DOI: https://doi.org/10.1090/tran6704
Received by editor(s): May 13, 2014
Received by editor(s) in revised form: November 14, 2014, March 10, 2015, and March 13, 2015
Published electronically: May 3, 2016
Article copyright: © Copyright 2016 American Mathematical Society