MIT-Harvard-MSR Combinatorics Seminar: Weighted Ehrhart Theory and why you should care!


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May 3, 2023 4:15 pm - 5:15 pm
Science Center 232

Jesús A. De Loera - UC Davis

A great tool in the arsenal of combinatorialists is modeling problems as counting  the lattice points of some convex polytope. Let $P\subseteq\R^d$ be a rational  convex polytope, that is, a polytope with vertices in $\mathbb{Q}^d$, then the Ehrhart function of the polytope $i(P,n)$ counts the number of integer lattice of the dilation $nP$ (here $nP$ denotes the polytope obtained from dilating $P$ by a factor n). Ehrhart functions have a rich history and many wonderful properties (e.g., Ehrhart himself proved that when $P$ is a lattice polytope, then $i(P,n)$ is a polynomial of degree $dim(P)$. The connections to Hilbert series are legendary). This topic has appeared in algebraic combinatorics, representation theory, algebraic geometry and others areas. But what if we count the integer lattice points with *weights*? Say  $w: \R^d \to \R$   a  function, often called a  \emph{weight function}. We can consider the, \emph{weighted Ehrhart} function: \[ i(P,w,n)=\sum_ {x\in nP \cap \Z^d} w(x).  \] (Here $w(x) := w(x_1,\dots,x_d)$ runs over the set of integer points belonging to $P$)

In this lecture I review what we know about weighted Ehrhart functions.

Some basic things remain true, other classical results have delicate variations and extensions. I will discuss several new theorems:

1) We generalized R. Stanley's theorem that the $h^\ast$-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes.  We show that these results continue to hold for \emph{weighted} Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope.

2) We also investigated  nonnegativity of the $h^\ast$-polynomial as a real-valued function for  a larger family of weights. In fact, discuss the case of  counting lattice points of a polytope that are weighted not by a simple polynomial, but by an Ehrhart quasi-polynomial of a family of parametric polytopes. As applications one can compute integrals and maximum values of such quasi-polynomials. We obtain new identities in representation theory and semigroup theory similar to RSK.

This work comprises 3 papers joint work with subsets of the following wonderful people: Esme Bajo, Rob Davis, Laura Escobar, Alexey Garber, Katharina Jochemko, Nathan Kaplan, Sofia Garzon-Mora, Josephine Yu, Rafael Villarreal, and Chengyang Wang.


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