Sponsored by the Rutgers University Department of Mathematics and the

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

**Co-organizers:****Drew Sills**, Rutgers University, asills {at} math [dot] rutgers [dot] edu**Doron Zeilberger**, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Partitioning the Permutahedron

Speaker: **Etienne Rassart**, Institute for Advanced Study, Princeton

Date: October 28, 2004 4:30-5:30pm

Location: DIMACS Center, CoRE Bldg, Room 431, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

A permutahedron is a polytope obtained by taking the convex hull of the orbit of a point under the action of the symmetric group. Permutahedra appear in algebra because the weight diagram for a certain group representation is the set of all the lattice points inside a permutahedron, together with a function that associates an integer (called multiplicity) to each lattice point. I will explain how this function partitions the permutahedron into subpolytopes over which it is expressed by polynomials. With the help of the computer, in three dimensions, we can count these regions, not only for a given permutahedron but for all permutahedra at once. The multiplicity function has a continuous analogue called the Duistermaat-Heckman function, and the computer proves (with a little help) that these two functions partition the permutahedron in the exact same way in 3D.