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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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Mirković–Vilonen basis in type $A_1$
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by Pierre Baumann and Arnaud Demarais PDF
Represent. Theory 25 (2021), 780-806 Request permission

Abstract:

Let $G$ be a connected reductive algebraic group over $\mathbb C$. Through the geometric Satake equivalence, the fundamental classes of the Mirković–Vilonen cycles define a basis in each tensor product $V(\lambda _1)\otimes \cdots \otimes V(\lambda _r)$ of irreducible representations of $G$. We compute this basis in the case $G=\mathrm {SL}_2(\mathbb C)$ and conclude that in this case it coincides with the dual canonical basis at $q=1$.
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Additional Information
  • Pierre Baumann
  • Affiliation: Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France
  • MR Author ID: 633326
  • ORCID: 0000-0002-6947-0778
  • Email: p.baumann@unistra.fr
  • Arnaud Demarais
  • Affiliation: Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France
  • Address at time of publication: 15 allée du puits, 01290 Crottet, France
  • Email: arnaud.demarais@ac-dijon.fr
  • Received by editor(s): December 21, 2020
  • Received by editor(s) in revised form: April 15, 2021, and May 24, 2021
  • Published electronically: September 29, 2021
  • Additional Notes: The first author was supported by the ANR (project GeoLie ANR-15-CE40-0012).
  • © Copyright 2021 American Mathematical Society
  • Journal: Represent. Theory 25 (2021), 780-806
  • MSC (2020): Primary 22E46; Secondary 14M15
  • DOI: https://doi.org/10.1090/ert/582
  • MathSciNet review: 4319510