Combinatorics and Representation Theory for Generalized Permutohedra I: Simplicial Plates

Abstract

In this paper, we announce results from our thesis, which studies for the first time the categorification of the theory of generalized permutohedra. The vector spaces in the categorification are tightly constrained by certain continuity relations which appeared in physics in the mid 20th century. We describe here the action of the symmetric group on the vector spaces in this categorification. Generalized permutohedra are replaced by vector spaces of characteristic functions of polyhedral cones about faces of permutohedra, called plates, due to A. Ocneanu. The symmetric group acts on plates by coordinate permutation. In combinatorics, the Eulerian numbers count the number of permutations with a given number of ascent and descents. The classical Worpitzky identity expands a power $r^p$ as a sum of Eulerian numbers, with binomial coefficients. In our thesis, for the main result we generalize the classical Worpitzky identity to an isomorphism of symmetric group modules, corresponding geometrically to the tiling of a scaled simplex by unit hypersimplices. In the categorification, the volume of a hypersimplex is replaced by the complex-linear dimension of a vector space associated to it. The main technical aspect of the proof of the character formula for the simplex involves a partition of unity of a commutative algebra of translations on a discrete torus, and a certain modular Diophantine equation. A detailed paper is in preparation.