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

Published electronically:
December 1, 2011

MathSciNet review:
2869015

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Abstract | References | Similar Articles | 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.

**1.**Arkady Berenstein and David Kazhdan,*Geometric and unipotent crystals*, Geom. Funct. Anal.**Special Volume**(2000), 188–236. GAFA 2000 (Tel Aviv, 1999). MR**1826254**, 10.1007/978-3-0346-0422-2_8**2.**Arkady Berenstein and David Kazhdan,*Geometric and unipotent crystals. II. From unipotent bicrystals to crystal bases*, Quantum groups, Contemp. Math., vol. 433, Amer. Math. Soc., Providence, RI, 2007, pp. 13–88. MR**2349617**, 10.1090/conm/433/08321**3.**Seok-Jin Kang, Masaki Kashiwara, Kailash C. Misra, Tetsuji Miwa, Toshiki Nakashima, and Atsushi Nakayashiki,*Affine crystals and vertex models*, Infinite analysis, Part A, B (Kyoto, 1991) Adv. Ser. Math. Phys., vol. 16, World Sci. Publ., River Edge, NJ, 1992, pp. 449–484. MR**1187560****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**, 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**, 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**, 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

Received by editor(s):
September 6, 2010

Received by editor(s) in revised form:
June 29, 2011

Published electronically:
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 28 years after publication.