Remote Access Representation Theory
Green Open Access

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
DOI: https://doi.org/10.1090/S1088-4165-2013-00438-7
Published electronically: August 5, 2013
MathSciNet review: 3084418
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Jared E. Anderson, A polytope calculus for semisimple groups, Duke Math. J. 116 (2003), no. 3, 567-588. MR 1958098 (2004a:20047), https://doi.org/10.1215/S0012-7094-03-11636-1
  • [2] Pierre Baumann, Joel Kamnitzer, and Peter Tingley. Affine Mirković-Vilonen polytopes. Preprint. arXiv:1110.3661.
  • [3] Jonathan Beck, Vyjayanthi Chari, and Andrew Pressley, An algebraic characterization of the affine canonical basis, Duke Math. J. 99 (1999), no. 3, 455-487. MR 1712630 (2000g:17013), https://doi.org/10.1215/S0012-7094-99-09915-5
  • [4] Ilaria Damiani, A basis of type Poincaré-Birkhoff-Witt for the quantum algebra of $ \widehat {\rm sl}(2)$, J. Algebra 161 (1993), no. 2, 291-310. MR 1247357 (94k:17021), https://doi.org/10.1006/jabr.1993.1220
  • [5] Thomas Rough Dunlap II, Combinatorial representation theory of affine sl2 via polytope calculus, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)-Northwestern University. MR 2736801
  • [6] Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, Providence, RI, 2002. MR 1881971 (2002m:17012)
  • [7] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219 (92k:17038)
  • [8] Joel Kamnitzer, The crystal structure on the set of Mirković-Vilonen polytopes, Adv. Math. 215 (2007), no. 1, 66-93. MR 2354986 (2009a:17021), https://doi.org/10.1016/j.aim.2007.03.012
  • [9] Joel Kamnitzer, Mirković-Vilonen cycles and polytopes, Ann. of Math. (2) 171 (2010), no. 1, 245-294. MR 2630039 (2011g:20070), https://doi.org/10.4007/annals.2010.171.245
  • [10] Masaki Kashiwara, The crystal base and Littelmann's refined Demazure character formula, Duke Math. J. 71 (1993), no. 3, 839-858. MR 1240605 (95b:17019), https://doi.org/10.1215/S0012-7094-93-07131-1
  • [11] Masaki Kashiwara, On crystal bases, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 155-197. MR 1357199 (97a:17016)
  • [12] Masaki Kashiwara, Similarity of crystal bases, Lie algebras and their representations (Seoul, 1995) Contemp. Math., vol. 194, Amer. Math. Soc., Providence, RI, 1996, pp. 177-186. MR 1395599 (97g:17013), https://doi.org/10.1090/conm/194/02393
  • [13] Masaki Kashiwara and Yoshihisa Saito, Geometric construction of crystal bases, Duke Math. J. 89 (1997), no. 1, 9-36. MR 1458969 (99e:17025), https://doi.org/10.1215/S0012-7094-97-08902-X
  • [14] George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston Inc., Boston, MA, 1993. MR 1227098 (94m:17016)
  • [15] Dinakar Muthiah, Double MV cycles and the Naito-Sagaki-Saito crystal, Adv. Math. 240 (2013), 268-290. MR 3046309
  • [16] Dinakar Muthiah and Peter Tingley, Affine PBW bases and MV polytopes in rank $ 2$, Selecta Mathematica, published on line January 24, 2013, DOI 10.1007/s0029-012-0117-z.
  • [17] Satoshi Naito, Daisuke Sagaki, and Yoshihisa Saito, Toward Berenstein-Zelevinsky data in affine type $ A$, I: Construction of affine analogs, Algebraic groups and quantum groups, 143-184, Contemp. Math., 565, Amer. Math. Soc., Providence, RI, 2012. MR 2932426

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 05E10, 17B67, 52B20

Retrieve articles in all journals with MSC (2010): 05E10, 17B67, 52B20


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: https://doi.org/10.1090/S1088-4165-2013-00438-7
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.

American Mathematical Society