Brendon Rhoades

Abstract:   Let X be a finite set, C = ⟨ c &rang be a finite cyclic group acting on X, and X(q) &isin Z[q] be a polynomial over the integers. Following Reiner, Stanton, and White, we say that the triple (X, C, X(q)) exhibits the cyclic sieving phenomenon if for any integer d ≥ 0, the number of fixed points of c^d is equal to X(ζ^d), where &zeta is a primitive |C|^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 actio of promotion via irreducible GL_n(C)-representations constructed using the dual canonical basis and the Kazhdan-Lusztig cellular representations.
 

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