Persistent Subdivisions of Coxeter Permutahedra

We investigate the realizations of Coxeter permutahedra which are also Coxeter matroid polytopes; these are polytopes of the form $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}(W\cdot \mathbf{a})$ where $W$ is a finite Coxeter group acting on ${\mathbb{R}}^n$ and $\mathbf{a}$ is generic. Our main focus is how the geometric properties of $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}(W\cdot \mathbf{a})$ change as $\mathbf{a}$ changes, with particular attention to persistent simplices, triangulations, and subdivisions.