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Abstract: We prove a collection of conjectures of D. White \cite{WComm}, as well assome related conjectures of Abuzzahab-Korson-Li-Meyer \cite{AKLM} and of Reinerand White \cite{ReinerComm}, \cite{WComm}, regarding the cyclic sievingphenomenon of Reiner, Stanton, and White \cite{RSWCSP} as it applies tojeu-de-taquin promotion on rectangular tableaux. To do this, we useKazhdan-Lusztig theory and a characterization of the dual canonical basis of$\mathbb{C}x {11},

., x {nn}$ due to Skandera \cite{SkanNNDCB}. Afterwards,we extend our results to analyzing the fixed points of a dihedral action onrectangular tableaux generated by promotion and evacuation, suggesting apossible sieving phenomenon for dihedral groups. Finally, we give applicationsof this theory to cyclic sieving phenomena involving reduced words for the longelements of hyperoctohedral groups and noncrossing partitions.



Author: Brendon Rhoades

Source: https://arxiv.org/



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