Abstract
Recently, Panyushev (2015) raised five conjectures concerning the structure of certain root posets arising from Z-gradings of simple Lie algebras. This paper aims to provide proofs for four of them. Our study also links these posets with Kostant-Macdonald identity, minuscule representations, Stembridge’s “t = −1 phenomenon”, and the cyclic sieving phenomenon due to Reiner et al. (2004).
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Acknowledgments
This work was supported by National Natural Science Foundation of China (Grant No. 11571097). The authors thank Doctor Bai, Doctor Wang, and Professor Stembridge for helpful discussions. A revision of the paper was carried out during Dong’s visit of Massachusetts Institute of Technology. He thanks the math department there sincerely for offering excellent working conditions. Finally, The authors express their sincere gratitude to the referees for giving their valuable suggestions.
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Dedicated to the Memory of Professor Bertram Kostant
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Dong, C., Weng, G. Minuscule representations and Panyushev conjectures. Sci. China Math. 61, 1759–1774 (2018). https://doi.org/10.1007/s11425-017-9136-y
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DOI: https://doi.org/10.1007/s11425-017-9136-y