An insight into the description of the crystal structure for Mirković-Vilonen polytopes
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- by Yong Jiang and Jie Sheng PDF
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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.References
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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
- MR Author ID: 1069205
- 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
- MR Author ID: 889511
- Email: shengjie@amss.ac.cn
- 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). - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 6407-6427
- MSC (2010): Primary 05E10; Secondary 16G20, 17B20
- DOI: https://doi.org/10.1090/tran/6918
- MathSciNet review: 3660227