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### Cyclic Sieving, Promotion, and Representation Theory

**Algebra & Discrete Mathematics**

Speaker: | Brendon Rhoades, U. Minnesota |

Location: | 2112 MSB |

Start time: | Fri, May 16 2008, 1:10PM |

Let X be a finite set, C = ⟨c&rang be a finite cyclic group acting on X, and X(q) &isinZ[q] be a polynomial over the integers. Following Reiner, Stanton, and White, we say that the triple (X, C, X(q)) exhibits thecyclic sieving phenomenonif for any integerd&ge 0, the number of fixed points ofcis equal to X(ζ^{d}^{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 irreducibleGL(_{n}C)-representations constructed using the dual canonical basis and the Kazhdan-Lusztig cellular representations.