Symmetry and Geometric Structure for the Worpitzky identity

Meeting Details

Abstract: The classical Worpitzky identity for the symmetric group $S_n$ decomposes a cubical lattice into $n!$ simplices of different sizes, each with a multiplicity counted by the number of permutations of $n$ with a fixed number of descents. It is well-known in combinatorics that the Eulerian numbers can be represented as volumes of suitably normalized hypersimplices. We show how the Worpitzky identity encodes localization data for a system of simplicial polyhedral cones which emerge from the $A_n$ simple root system, and becomes now an isomorphism between two new graded, simplicial symmetric group representations.