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Affine geometric crystals in unipotent loop groups
Authors:
Thomas Lam and Pavlo Pylyavskyy
Journal:
Represent. Theory 15 (2011), 719-728
MSC (2010):
Primary 17B37, 17B67, 22E65
Posted:
December 1, 2011
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Additional Information
Abstract: We study products of the affine geometric crystal of type  corresponding to symmetric powers of the standard representation. The quotient of this product by the  -matrix action is constructed inside the unipotent loop group. This quotient crystal has a semi-infinite limit, where the crystal structure is described in terms of limit ratios previously appearing in the study of total positivity of loop groups.
References
- 1.
A. BERENSTEIN AND D. KAZHDAN: Geometric and unipotent crystals. Geom. Funct. Anal. 2000, Special Volume, Part I, 188-236. MR 1826254 (2003b:17013)
- 2.
A. BERENSTEIN AND D. KAZHDAN: Geometric and unipotent crystals. II. From unipotent bicrystals to crystal bases. Quantum groups, 13-88, Contemp. Math., 433, Amer. Math. Soc., Providence, RI, 2007. MR 2349617 (2009b:17030)
- 3.
S.-J. KANG, M. KASHIWARA, K.C. MISRA, T. MIWA, T. NAKASHIMA, AND A. NAKAYASHI: Affine crystals and vertex models, in Infinite analysis Part A (Kyoto 1991), 449-484, Adv. Ser. Math. Phys., 16, World Sci. Publishing, River Edge, NJ, 1992. MR 1187560 (94a:17008)
- 4.
Masaki Kashiwara, Toshiki Nakashima, and Masato Okado, Affine geometric crystals and limit of
perfect crystals, Trans. Amer. Math. Soc.
360 (2008), no. 7,
3645–3686. MR 2386241
(2009e:17020), 10.1090/S0002-9947-08-04341-9
- 5.
Masaki Kashiwara, Toshiki Nakashima, and Masato Okado, Tropical 𝑅 maps and affine
geometric crystals, Represent. Theory
14 (2010), 446–509. MR 2661518
(2011k:17028), 10.1090/S1088-4165-2010-00379-9
- 6.
T. LAM AND P.PYLYAVSKYY: Total positivity in loop groups I: whirls and curls, preprint, 2009; arXiv:0812.0840.
- 7.
T. LAM AND P.PYLYAVSKYY: Intrinsic energy is a loop Schur function, preprint, 2009; arXiv:1003.3948.
- 8.
Yasuhiko Yamada, A birational representation of Weyl group,
combinatorial 𝑅-matrix and discrete Toda equation, Physics and
combinatorics, 2000 (Nagoya), World Sci. Publ., River Edge, NJ, 2001,
pp. 305–319. MR 1872262
(2002k:20021), 10.1142/9789812810007_0014
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Additional Information
Thomas Lam
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email:
tfylam@umich.edu
Pavlo Pylyavskyy
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Address at time of publication:
Department of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, Minnesota 55455
Email:
ppylyavs@umn.edu
DOI:
http://dx.doi.org/10.1090/S1088-4165-2011-00410-6
PII:
S 1088-4165(2011)00410-6
Received by editor(s):
September 6, 2010
Received by editor(s) in revised form:
June 29, 2011
Posted:
December 1, 2011
Additional Notes:
The first author was supported by NSF grant DMS-0652641 and DMS-0901111, and by a Sloan Fellowship.
The second author was supported by NSF grant DMS-0757165.
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
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