Monday, February 13th, 2012
9:30am - 10:30am, in Campus Center 2-2545

Brendon Rhoades

University of Southern California

Cyclic Sieving and Cluster Multicomplexes

Abstract: Let $X$ be a finite set, $C = \langle c \rangle$ be a finite cyclic group acting on $X$, and $X(q) \in N[q]$ be a polynomial with nonnegative integer coefficients. 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 $X(\zeta^d)$, where $\zeta$ is a primitive $|C|^{th}$ root of unity. We explain how one can use representation theory to prove instances of the cyclic sieving phenomenon involving the action of tropical Coxeter elements on (complexes closely related to) cluster complexes. The representation theory involves cluster monomial bases of geometric realizations of finite type cluster algebras.