A definition of the crystal commutor using Kashiwara's involution
Joel Kamnitzer1
and Peter Tingley2
1American Institute of Mathematics Palo Alto CA USA
2UC Berkeley Department of Mathematics Berkeley CA USA
2UC Berkeley Department of Mathematics Berkeley CA USA
DOI: 10.1007/s10801-008-0136-1
Abstract
Henriques and Kamnitzer defined and studied a commutor for the category of crystals of a finite dimensional complex reductive Lie algebra. We show that the action of this commutor on highest weight elements can be expressed very simply using Kashiwara's involution on the Verma crystal.
Pages: 261–268
Keywords: keywords coboundary category; crystals; crystal commutor
Full Text: PDF
References
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2. Kamnitzer, J.: The crystal structure on the set of Mirković-Vilonen polytopes. Adv. Math. 215(1), 66- 93 (2007).
3. Kashiwara, M.: The crystal base and Littelmann's refined Demazure character formula. Duke Math. J. 71(3), 839-858 (1993)
4. Kashiwara, M.: On crystal bases. In: Representations of Groups, Banff, AB,
1994. CMS Conf. Proc., vol. 16, pp. 155-197. Amer. Math. Soc., Providence (1995)
5. Littelmann, P.: Cones, crystals, and patterns. Transform. Groups 3(2), 145-179 (1998)
6. Savage, A.: Crystals, quiver varieties and coboundary categories for Kac-Moody algebras. 1This question has recently been answered in the affirmative by Savage [6, Theorem 6.4].