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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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Equivariant multiplicities of simply-laced type flag minors
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by Elie Casbi PDF
Represent. Theory 25 (2021), 1049-1092 Request permission

Abstract:

Let $\mathfrak {g}$ be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism $\overline {D}$ on the coordinate ring $\mathbb {C}[N]$ related to Brion’s equivariant multiplicities via the geometric Satake correspondence. This map is known to take distinguished values on the elements of the MV basis corresponding to smooth MV cycles, as well as on the elements of the dual canonical basis corresponding to Kleshchev-Ram’s strongly homogeneous modules over quiver Hecke algebras. In this paper we show that when $\mathfrak {g}$ is of type $A_n$ or $D_4$, the map $\overline {D}$ takes similar distinguished values on the set of all flag minors of $\mathbb {C}[N]$, raising the question of the smoothness of the corresponding MV cycles. We also exhibit certain relations between the values of $\overline {D}$ on flag minors belonging to the same standard seed, and we show that in any $ADE$ type these relations are preserved under cluster mutations from one standard seed to another. The proofs of these results partly rely on Kang-Kashiwara-Kim-Oh’s monoidal categorification of the cluster structure of $\mathbb {C}[N]$ via representations of quiver Hecke algebras.
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Additional Information
  • Elie Casbi
  • Affiliation: Université de Paris, Sorbonne Université, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, F-75013 Paris, France
  • MR Author ID: 1379041
  • ORCID: 0000-0001-9064-2741
  • Email: elie.casbi@imj-prg.fr
  • Received by editor(s): June 22, 2020
  • Received by editor(s) in revised form: August 18, 2021, August 27, 2021, and September 7, 2021
  • Published electronically: December 16, 2021
  • Additional Notes: The author was supported by the European Research Council under the European Union’s Framework Programme H2020 with ERC Grant Agreement number 647353 Qaffine
  • © Copyright 2021 American Mathematical Society
  • Journal: Represent. Theory 25 (2021), 1049-1092
  • MSC (2020): Primary 16G10, 20G05
  • DOI: https://doi.org/10.1090/ert/589
  • MathSciNet review: 4353894