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Representation Theory
Representation Theory
ISSN 1088-4165

 

Rank 2 affine MV polytopes


Authors: Pierre Baumann, Thomas Dunlap, Joel Kamnitzer and Peter Tingley
Journal: Represent. Theory 17 (2013), 442-468
MSC (2010): Primary 05E10; Secondary 17B67, 52B20
Published electronically: August 5, 2013
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Abstract: 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
Email: p.baumann@unistra.fr

Thomas Dunlap
Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
Email: tdunlap@umich.edu

Joel Kamnitzer
Affiliation: Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4 Canada
Email: jkamnitz@math.toronto.edu

Peter Tingley
Affiliation: Department of Mathematics and Statistics, Loyola University, Chicago, Illinois 60660
Email: ptingley@luc.edu

DOI: http://dx.doi.org/10.1090/S1088-4165-2013-00438-7
PII: S 1088-4165(2013)00438-7
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.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.