Abstract
We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type A. Using geometric objects called rhombic tilings we derive a “Crossing Formula” for the action of the crystal operators on Lusztig data for an arbitrary reduced word of the longest Weyl group element. We provide the following three applications of this result. Using the tropical Chamber Ansatz of Berenstein–Fomin–Zelevinsky we prove an enhanced version of the Anderson–Mirković conjecture for the crystal structure on MV polytopes. We establish a duality between Kashiwara’s string and Lusztig’s parametrization, revealing that each of them is controlled by the crystal structure of the other. We identify the potential functions of the unipotent radical of a Borel subgroup of \(SL_n\) defined by Berenstein–Kazhdan and Gross–Hacking–Keel–Kontsevich, respectively, with a function arising from the crystal structure on Lusztig data.
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Acknowledgements
We are very grateful to Jacinta Torres for sharing an observation that led to this collaboration. This project was initiated during G.K.’s visit at the MPIM Bonn which he thanks for financial support. The project was finalized during B.S.’s visit at the University of Tokyo and V.G.’s and B.S.’s visit at the Laboratoire J.-V. Poncelet. V.G. and B.S. would like to thank Yoshihisa Saito for explanations about MV-polytopes that inspired Sect. 4. We would also like to thank Xin Fang and Hironori Oya for very helpful comments. V.G. was partially supported by the Laboratoire J.-V. Poncelet. G.K. was supported by the grant RSF 21-11-00283. B.S. was supported by a JSPS Postdoctoral Fellowship for Foreign Researchers, partially supported by the DFG Priority program Darstellungstheorie 1388 and the SFB/TRR 191 ‘Symplectic Structures in Geometry, Algebra and Dynamics’, funded by the DFG.
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Genz, V., Koshevoy, G. & Schumann, B. Combinatorics of canonical bases revisited: type A. Sel. Math. New Ser. 27, 67 (2021). https://doi.org/10.1007/s00029-021-00658-x
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DOI: https://doi.org/10.1007/s00029-021-00658-x
Keywords
- Rhombic tilings
- Lusztig’s parametrization
- Crystal bases
- String parametrization
- MV-polytopes
- Double Bruhat cells
- Cluster algebras
- Berenstein–Kazhdan potential
- Gross–Hacking–Keel–Kontsevich potential