Permutohedral plates: new geometric structures for the classical Worpitzky identity

Using a new combinatorial model for hyperplane arrangements which was discovered A. Ocneanu and developed further in our Ph.D. thesis, we present a new generalization of the classical Worpitzky identity to an identity involving symmetric group characters. The S_n modules involved are spanned by certain piecewise-constant functions on hypersimplices, and replace the known unimodular triangulations which were studied by Stanley, Sturmfels, and Lam-Postnikov.