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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Cyclic sieving and plethysm coefficients

Author: David B. Rush
Journal: Trans. Amer. Math. Soc. 371 (2019), 923-947
MSC (2010): Primary 05E05, 05E10
Published electronically: July 31, 2018
MathSciNet review: 3885166
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Abstract: A combinatorial expression for the coefficient of the Schur function $ s_{\lambda }$ in the expansion of the plethysm $ p_{n/d}^d \circ s_{\mu }$ is given for all $ d$ dividing $ n$ for the cases in which $ n=2$ or $ \lambda $ is rectangular. In these cases, the coefficient $ \langle p_{n/d}^d \circ s_{\mu }, s_{\lambda } \rangle $ is shown to count, up to sign, the number of fixed points of an $ \langle s_{\mu }^n, s_{\lambda } \rangle $-element set under the $ d$th power of an order-$ n$ cyclic action. If $ n=2$, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if $ \lambda $ is rectangular, the action is a certain power of Schützenberger and Shimozono's jeu-de-taquin promotion.

This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case $ n=2$ is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function.

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Additional Information

David B. Rush
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

Received by editor(s): August 27, 2014
Received by editor(s) in revised form: December 28, 2014, March 8, 2017, March 9, 2017, March 14, 2017, and March 24, 2017
Published electronically: July 31, 2018
Additional Notes: This research was undertaken at the University of Michigan, Ann Arbor, under the direction of Professor David Speyer and with the financial support of the U.S. National Science Foundation via grant DMS-1006294. Throughout his graduate studies, the author was supported by the NSF Graduate Research Fellowship Program.
Article copyright: © Copyright 2018 American Mathematical Society