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Sums of Weighted Lattice Points of Polytopes

Jesús A. De Loera, Laura Escobar, Nathan Kaplan, Chengyang Wang

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 Michèle 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.

Sums of Weighted Lattice Points of Polytopes

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 Michèle 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.
Paper Structure (15 sections, 12 theorems, 46 equations, 1 figure, 2 tables)

This paper contains 15 sections, 12 theorems, 46 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Michèle and Lattice Points in Polytopes
  3. Proofs of Theorem \ref{['polytopelifting']} and sketch of other results
  4. Applications
  5. Weighted Ehrhart in number theory
  6. Simultaneous core partitions.
  7. Numerical semigroups.
  8. Weighted Ehrhart in combinatorial representation theory
  9. Maximizing Kostka numbers.
  10. Robinson-Schensted-Knuth (RSK) identity
  11. Littlewood--Richardson Coefficients.
  12. Newell-Littlewood Polytopes.
  13. Experiments
  14. Integration over polytopes
  15. Weights of numerical semigroups

Key Result

Theorem 1.1

Let $P$ be a rational convex polytope in the form $\{\mathbf{x}\mid \mathbf{Ax} = \mathbf{b}, \mathbf{x}\geq 0\}$, where $\mathbf{A} \in \mathbb{Z}^{s \times n}, \mathbf{b} \in \mathbb{Z}^{s}$. Let $Q(x_{1},\ldots,x_{n})$ be the parametric family of rational convex polytopes parameterized by $x_{1}, where $\mathbf{C} \in \mathbb{Z}^{r \times m}, \mathbf{d_{i}},\mathbf{e} \in \mathbb{Z}^{r}$. Defin

Figures (1)

  • Figure 1: Curve plot of the quotient of average weight and genus square.

Theorems & Definitions (28)

  • Theorem 1.1: The existence of weight lifting polytopes
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • proof : Proof of Theorem \ref{['polytopelifting']}
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • ...and 18 more