Ehrhart Series of Polytopes Related to Symmetric Doubly-Stochastic Matrices

TL;DR

This paper investigates the Ehrhart $h^*$-vectors of polytopes related to symmetric doubly-stochastic matrices, providing insights into their combinatorial and geometric properties.

Contribution

It introduces a study of $h^*$-vectors for polytopes of symmetric doubly-stochastic matrices, extending understanding beyond the classical Birkhoff polytope.

Findings

Analysis of $h^*$-vector properties for symmetric doubly-stochastic matrix polytopes

Identification of conditions affecting unimodality of $h^*$-vectors

Insights into the combinatorial structure of these polytopes

Abstract

In Ehrhart theory, the $h^*$-vector of a rational polytope often provide insights into properties of the polytope that may be otherwise obscured. As an example, the Birkhoff polytope, also known as the polytope of real doubly-stochastic matrices, has a unimodal $h^*$-vector, but when even small modifications are made to the polytope, the same property can be very difficult to prove. In this paper, we examine the $h^*$-vectors of a class of polytopes containing real doubly-stochastic symmetric matrices.