The finite Coxeter groups can be realized as symmetry groups of (semi)-regular polytopes. Not all semi-regular polytopes can be realized this way, but all regular polytopes can.
Some examples of Coxeter groups that show up naturally are the Weyl groups of simple complex Lie algebras. Let $\mathfrak{g}$ be a simple complex Lie algebra, $\mathfrak{h}$ a Cartan subalgebra, and $\mathfrak{h}_{\mathbb{R}}$ the real span of the roots. $\mathfrak{h}_{\mathbb{R}}$ is a Euclidean space via the Killing form, and so it makes sense to talk about polytopes in there.
My question: Is there a "natural" way to embed the polytope associated to the Weyl group in $\mathfrak{h}_{\mathbb{R}}$?
My first guess was to look at the convex hull of the roots. This works for some examples, but it doesn't even work for $\mathfrak{sl}_3\mathbb{C}$. To be explicit, any answer to the question should give the following examples:
- $\mathfrak{sl}_{n+1}$ gives an $n$-simplex.
- $\mathfrak{so}_{2n+1}$, $\mathfrak{sp}_{2n}$ gives a hypercube or its dual.
- $\mathfrak{g}_2$ gives a hexagon.