On Volumes of Permutation Polytopes

TL;DR

This paper investigates the volumes of permutation polytopes linked to various groups using triangulations and Ehrhart polynomials, and explores their theta body hierarchies to advance understanding in geometric combinatorics.

Contribution

It introduces new methods for calculating volumes of permutation polytopes for specific groups and analyzes their theta body hierarchies, expanding the theoretical framework.

Findings

Volumes computed for permutation polytopes of cyclic, dihedral, automorphism, and Frobenius groups.

Triangulation techniques effectively determine polytope volumes.

Results on the theta body hierarchy provide insights into the geometric structure.

Abstract

This paper focuses on determining the volumes of permutation polytopes associated to cyclic groups, dihedral groups, groups of automorphisms of tree graphs, and Frobenius groups. We do this through the use of triangulations and the calculation of Ehrhart polynomials. We also present results on the theta body hierarchy of various permutation polytopes.