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An insight into the description of the crystal structure for Mirković-Vilonen polytopes


Authors: Yong Jiang and Jie Sheng
Journal: Trans. Amer. Math. Soc. 369 (2017), 6407-6427
MSC (2010): Primary 05E10; Secondary 16G20, 17B20
DOI: https://doi.org/10.1090/tran/6918
Published electronically: March 1, 2017
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Abstract: We study the description of the crystal structure on the set of Mirković-Vilonen polytopes. Anderson and Mirković defined an operator and conjectured that it coincides with the Kashiwara operator. Kamnitzer proved the conjecture for type $ A$ and gave a counterexample for type $ C_{3}$. He also gave an explicit formula to calculate the Kashiwara operator for type $ A$. In this paper we prove that a part of the AM conjecture still holds in general, answering an open question of Kamnitzer (2007). Moreover, we show that although the formula given by Kamnitzer does not hold in general, it is still valid in many cases regardless of the type. The main tool is the connection between MV polytopes and preprojective algebras developed by Baumann and Kamnitzer.


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Additional Information

Yong Jiang
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, D-33501 Bielefeld, Germany
Address at time of publication: Leibniz-Institut für Pflanzengenetik und Kulturpflanzenforschung (IPK), Corrensstrasse 3, D-06466 Stadt Seeland OT Gatersleben, Germany
Email: jiang@ipk-gatersleben.de

Jie Sheng
Affiliation: Department of Applied Mathematics, China Agricultural University, 100083 Beijing, People’s Republic of China – and – Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: shengjie@amss.ac.cn

DOI: https://doi.org/10.1090/tran/6918
Keywords: Mirkovi\'c-Vilonen polytope, crystal, preprojective algebra, diagram automorphism
Received by editor(s): March 8, 2015
Received by editor(s) in revised form: September 27, 2015
Published electronically: March 1, 2017
Additional Notes: The first author was supported by the Sonderforschungsbereich 701 in Universität Bielefeld
The second author was supported by NSF of China (No. 11301533).
Article copyright: © Copyright 2017 American Mathematical Society

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