# Ehrhart positivity and Demazure characters

## TL;DR

This paper explores the relationship between Gelfand-Tsetlin polytopes and Ehrhart polynomials through Demazure characters, proposing conjectures on coefficient non-negativity based on computational evidence.

## Contribution

It provides an overview of the geometric and combinatorial interplay and introduces new conjectures on Ehrhart polynomial coefficients inspired by computational findings.

## Key findings

- Evidence suggests Ehrhart polynomial coefficients may be non-negative.
- Conjectures proposed for Ehrhart positivity of Gelfand-Tsetlin polytopes.
- Highlights the connection between Demazure characters and polyhedral geometry.

## Abstract

Demazure characters, also known as key polynomials, generalize the classical Schur polynomials. In particular, when all variables are set equal to $1$, these polynomials count the number of integer points in a certain class of Gelfand--Tsetlin polytopes. This property highlights the interaction between the corresponding polyhedral and combinatorial structures via Ehrhart theory. In this paper, we give an overview of results concerning the interplay between the geometry of Gelfand-Tsetlin polytopes and their Ehrhart polynomials. Motivated by strong computer evidence, we propose several conjectures about the non-negativity of the coefficients of such polynomials.