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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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A quantum Mirković-Vybornov isomorphism
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by Ben Webster, Alex Weekes and Oded Yacobi PDF
Represent. Theory 24 (2020), 38-84 Request permission

Abstract:

We present a quantization of an isomorphism of Mirković and Vybornov which relates the intersection of a Slodowy slice and a nilpotent orbit closure in $\mathfrak {gl}_N$ to a slice between spherical Schubert varieties in the affine Grassmannian of $PGL_n$ (with weights encoded by the Jordan types of the nilpotent orbits). A quantization of the former variety is provided by a parabolic W-algebra and of the latter by a truncated shifted Yangian. Building on earlier work of Brundan and Kleshchev, we define an explicit isomorphism between these non-commutative algebras and show that its classical limit is a variation of the original isomorphism of Mirković and Vybornov. As a corollary, we deduce that the W-algebra is free as a left (or right) module over its Gelfand-Tsetlin subalgebra, as conjectured by Futorny, Molev, and Ovsienko.
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Additional Information
  • Ben Webster
  • Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada; and Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
  • MR Author ID: 794563
  • Email: ben.webster@uwaterloo.ca
  • Alex Weekes
  • Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada
  • MR Author ID: 1074676
  • Email: weekesal@math.ubc.ca
  • Oded Yacobi
  • Affiliation: School of Mathematics and Statistics, University of Sydney, Camperdown, New South Wales 2006, Australia
  • MR Author ID: 770448
  • Email: oded.yacobi@sydney.edu.au
  • Received by editor(s): October 6, 2017
  • Received by editor(s) in revised form: January 14, 2019, November 12, 2019, and December 23, 2019
  • Published electronically: January 16, 2020
  • Additional Notes: This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.
    The first author was supported in part by the NSF Division of Mathematical Sciences and the Alfred P. Sloan Foundation.
    The second author was supported in part by NSERC and the Ontario Ministry of Training, Colleges and Universities.
    The third author was supported by the Australian Reseach Council.
  • © Copyright 2020 American Mathematical Society
  • Journal: Represent. Theory 24 (2020), 38-84
  • MSC (2010): Primary 16S80, 17B37, 20C99
  • DOI: https://doi.org/10.1090/ert/536
  • MathSciNet review: 4052554