Sums of Weighted Lattice Points of Polytopes

TL;DR

This paper introduces a method to convert weighted lattice point sums of polytopes into unweighted sums, enabling faster computations and new algebraic identities in combinatorics and number theory.

Contribution

A novel transformation that converts weighted sums of lattice points into unweighted sums, even with complex quasipolynomial weights.

Findings

Enables faster integration over polytopes.

Facilitates derivation of new algebraic identities.

Applies to a broad class of quasipolynomial weights.

Abstract

We study the problem of counting lattice points of a polytope that are weighted by an Ehrhart quasi-polynomial of a family of parametric polytopes. As applications one can compute integrals and maximum values of such quasi-polynomials, as well as obtain new identities in representation theory. These topics have been of great interest to Mich\`ele Vergne since the late 1980's. Our new contribution is a result that transforms weighted sums into unweighted sums, even when the weights are very general quasipolynomials. In some cases it leads to faster integration over a polytope. We can create new algebraic identities and conjectures in algebraic combinatorics and number theory.