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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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Rank 2 affine MV polytopes
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by Pierre Baumann, Thomas Dunlap, Joel Kamnitzer and Peter Tingley PDF
Represent. Theory 17 (2013), 442-468 Request permission


We give a realization of the crystal $B(-\infty )$ for $\widehat {\mathrm {sl}}_2$ using decorated polygons. The construction and proof are combinatorial, making use of Kashiwara and Saito’s characterization of $B(-\infty )$, in terms of the $*$ involution. The polygons we use have combinatorial properties suggesting they are the $\widehat {\mathrm {sl}}_2$ analogues of the Mirković-Vilonen polytopes defined by Anderson and the third author in finite type. Using Kashiwara’s similarity of crystals we also give MV polytopes for $A_2^{(2)}$, the other rank 2 affine Kac-Moody algebra.
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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
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  • Thomas Dunlap
  • Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
  • Email:
  • Joel Kamnitzer
  • Affiliation: Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4 Canada
  • MR Author ID: 676374
  • Email:
  • Peter Tingley
  • Affiliation: Department of Mathematics and Statistics, Loyola University, Chicago, Illinois 60660
  • MR Author ID: 679482
  • Email:
  • Received by editor(s): May 9, 2012
  • Received by editor(s) in revised form: February 7, 2013
  • Published electronically: August 5, 2013
  • Additional Notes: The first author acknowledges support from the ANR, project ANR-09-JCJC-0102-01
    The second author acknowledges support from the ERC, project #247049(GLC)
    The third author acknowledges support from NSERC
    The fourth author acknowledges support from the NSF postdoctoral fellowship DMS-0902649.
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 17 (2013), 442-468
  • MSC (2010): Primary 05E10; Secondary 17B67, 52B20
  • DOI:
  • MathSciNet review: 3084418