Auslander-Reiten quiver of type A and generalized quantum affine Schur-Weyl duality.(English)Zbl 1351.05230
Summary: 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 ofS.-J. Kang et al. [Proc. Lond. Math. Soc. (3) 111, No. 2, 420–444 (2015;Zbl 1322.81056)]. 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 \( U^\prime_q(A_n^{(i)})\) \( (i=1,2)\) 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.
MSC:
| 05E10 | Combinatorial aspects of representation theory |
| 16T30 | Connections of Hopf algebras with combinatorics |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
| 20C08 | Hecke algebras and their representations |
| 16G20 | Representations of quivers and partially ordered sets |
Keywords:
quiver Hecke algebraCitations:
Zbl 1322.81056Cite
References:
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