A differential characterization of volume polynomials of permutohedra

TL;DR

This paper characterizes volume polynomials of permutohedra via a differential approach, linking algebraic properties to geometric structures, and introduces a criterion called geometricity for their linear combinations.

Contribution

It provides a differential characterization of volume polynomials of permutohedra and introduces the concept of geometricity for expressing these polynomials.

Findings

The polynomial space coincides with one defined by directional derivatives.

The space is finite-dimensional if all principal minors are nonzero.

Dimension in each degree equals a binomial coefficient, total dimension is a power of two.

Abstract

We study a graded vector space of polynomials associated to a square matrix, defined by a finite difference condition along the rows. We show this space coincides with one defined by directional derivatives, and prove it is finite-dimensional precisely when all principal minors are nonzero. In that case, its dimension in each degree equals a binomial coefficient, giving total dimension a power of two. For Cartan matrices of irreducible root systems, we construct an explicit basis of volume polynomials of faces of the associated permutohedra, yielding an elementary criterion, which we call geometricity, for expressing a polynomial as a linear combination of these volume polynomials.