The Birkhoff polytope of the groups $\mathsf{F}_4$ and $\mathsf{H}_4$

TL;DR

This paper computes the facets of the convex hull of Coxeter groups F4 and H4, revealing new orbit counts that challenge previous results and conjectures in the field.

Contribution

It provides the first detailed computation of the facets of these polytopes, offering counterexamples to earlier assumptions and conjectures.

Findings

F4 has 2 orbits of facets, contradicting previous results

H4 has 1063 orbits of facets, countering the conjecture

New insights into the structure of Coxeter group polytopes

Abstract

We compute the set of facets of the polytope which is the convex hull of the Coxeter groups $\mathsf{F}_4$ or $\mathsf{H}_4$: For the group $\mathsf{F}_4$ we found $2$ orbits of facets which contradicts previous results published in \cite{birkhoff}. For the group $\mathsf{H}_4$ we found $1063$ orbits of facets which provides a counterexample to the conjecture of \cite{birkhoff}.