Mathematics Colloquia and Seminars

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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 &ge 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 action of promotion via irreducible GL_n(C)-representations constructed using the dual canonical basis and the Kazhdan-Lusztig cellular representations.