Cyclic sieving, promotion, and representation theory

We prove a collection of conjectures of White [D. White, personal communication, 2007], as well as some related conjectures of Abuzzahab, Korson, Li, and Meyer [O. Abuzzahab, M. Korson, M. Li, S. Meyer, Cyclic and dihedral sieving for plane partitions, U. Minnesota REU Report. 2005] and of Reiner and White [V. Reiner, personal communication, 2007; D. White, personal communication, 2007], regarding the cyclic sieving phenomenon of Reiner. Stanton and White [V. Reiner, D. Stanton, D. White, The cyclic sieving phenomenon, J. Combin. Theory Ser. A 108 (2004)] 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[x(11)...., x(nn)] due to Skandera [M. Skandera, On the dual canonical and Kazhdan-Lusztig bases and 3412, 4231-avoiding permutations. 2006, submitted for publication]. 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. (C) 2009 Elsevier Inc. All rights reserved.