Cyclic sieving, promotion, and representation theory

الأصل · EN

We prove a collection of conjectures of D. White WComm, as well as some related conjectures of Abuzzahab-Korson-Li-Meyer AKLM and of Reiner and White ReinerComm, WComm, regarding the cyclic sieving phenomenon of Reiner, Stanton, and White RSWCSP as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan-Lusztig theory and a characterization of the dual canonical basis of C[x₁₁,..., xnn] due to Skandera SkanNNDCB. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on rectangular tableaux generated by promotion and evacuation, suggesting a possible sieving phenomenon for dihedral groups. Finally, we give applications of this theory to cyclic sieving phenomena involving reduced words for the long elements of hyperoctohedral groups and noncrossing partitions.