Mathematics Colloquia and Seminars

Rational Slope Parking Functions and Affine Permutations

Algebra & Discrete Mathematics

 Speaker: Mikhail Mazin, Kansas State University Location: 1147 MSB Start time: Mon, Oct 28 2013, 11:00AM

Pak and Stanley constructed a map from the set of connected components of the complement to the k-Shi arrangement to the set of k-parking functions. It follows from the work of Fishel and Vazirani that these connected components are in bijection with kn+1-stable affine permutations on n elements (i.e. permutations with no inversions of height kn+1). We generalize Pak-Stanley labeling by constructing a map from the set of m-stable affine permutations to the set of rational slope parking functions. It follows from the work of Stanley that the map is a bijection for m=kn+1. We extend this argument to cover the case m=kn-1 and conjecture that the map is a bijection for all relatively prime (m,n). We also show that m-stable permutations label the cells in a cell decomposition in an affine Springer fiber considered by Hikita, with the dimension of the corresponding cell equal to the sum of values of the corresponding parking function.

In addition, we construct another map between the same sets and prove that it is a bijection for all relatively prime (m,n). The resulting map from the set of rational parking functions to itself is a direct generalization of the map zeta from the set of Dyck paths to itself, originally constructed by Haglund. Therefore, this work can be viewed as a bridge between two previously unrelated areas in combinatorics.

The talk is based on a joint work in progress with Eugene Gorsky and Monica Vazirani.