"The excedance statistic for the affine symmetric group"

Eric Clark
University of Kentucky

Monday, October 26, 2009
4:00 pm, 845 Patterson Office Tower

Abstract:

The symmetric group has many interesting permutation statistics such as descent, inversion, excedance, and the major index. These have been classically studied. Lusztig proved that the Coxeter group $\widetilde{A}_{n-1}$ can be realized as a group of ``infinite permutations,'' that is, a group of bijections on the integers with certain properties. This is called the affine symmetric group. Bj\"orner and Brenti began a study of affine permutation statistics by extending the notion of descent and inversion to the affine symmetric group. In this talk, I will generalize the excedance statistic to the affine symmetric group and calculate its generating function. It turns out this is equivalent to counting integer lattice points on the boundary of dilations of the root polytope. This is joint work with Richard Ehrenborg.