## Rank 2 affine MV polytopes

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- by Pierre Baumann, Thomas Dunlap, Joel Kamnitzer and Peter Tingley
- Represent. Theory
**17**(2013), 442-468 - DOI: https://doi.org/10.1090/S1088-4165-2013-00438-7
- 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.## References

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## Bibliographic 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
- MR Author ID: 676374
- Email: jkamnitz@math.toronto.edu
**Peter Tingley**- Affiliation: Department of Mathematics and Statistics, Loyola University, Chicago, Illinois 60660
- MR Author ID: 679482
- Email: ptingley@luc.edu
- 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: https://doi.org/10.1090/S1088-4165-2013-00438-7
- MathSciNet review: 3084418