Cyclic sieving, promotion, and representation theory

Let C be a finite cyclic group of order m with generator c which acts on a finite set X. Let X(q) be a polynomial with integer coefficients. Following Reiner, Stanton, and White, we say that the triple (X, C, X(q)) exhibits the cyclic sieving phenomenon if for any nonnegative integer d, the number of fixed points of c^d is equal to X(z^d), where z is a primitive m'th root of unity. We prove a pair of conjectures of Reiner et. al. concerning cyclic sieving phenomena where X is the set of standard tableaux of a fixed rectangular shape or the set of semistandard tableaux with fixed rectangular shape and uniformly bounded entries and C acts by jeu-de-taquin promotion. Our proofs involve modeling the action of promotion via irreducible GL(n)-representations constructed using the dual canonical basis and the Kazhdan-Lusztig cellular representations.