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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
MathSciNet review: 3084418
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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

Thomas Dunlap
Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

Joel Kamnitzer
Affiliation: Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4 Canada
MR Author ID: 676374

Peter Tingley
Affiliation: Department of Mathematics and Statistics, Loyola University, Chicago, Illinois 60660
MR Author ID: 679482

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.