Let $ \mathfrak{g}$ be a complex semisimple Lie algebra, $ \Delta$ its root system contained in $ \mathfrak{t}^{\vee}$ for a Cartan sub-algebra $ \mathfrak{t}$ of $ \mathfrak{g}$ . Let $ W$ be its Weyl group. Then we (might) have the following short exact sequence of groups: $ $ 1\rightarrow W \rightarrow Aut(\Delta) \rightarrow E \rightarrow 1$ $ where $ Aut(\Delta)\subset GL(\mathfrak{t}^{\vee})$ is the group of symmetries of $ \Delta$ , and $ E$ (should) be the group of symmetries of the associated Dynkin diagram.

It is possible to see how the map $ Aut(\Delta) \rightarrow E$ is defined from the perspective of simple roots (let them be $ \{\alpha_1,\ldots,\alpha_k\}$ ): let $ \sigma\in Aut(\Delta)$ , then $ \{\sigma(\alpha_1),\ldots,\sigma(\alpha_k)\}$ is a set of simple roots with respect to a choice of positive roots which corresponds to its dominant Weyl chamber. By composing $ \sigma$ with the unique Weyl element mapping this chamber to the original dominant Weyl chamber, we get a permutation of $ \{\alpha_1,\ldots,\alpha_k\}$ . One can see that this permutation preserves the Cartan matrix and thus defines an element of $ E$ .

What I would like to know is whether there is an alternative way to define the above map from the perspective of dominant fundamental weights, namely, let $ \{\lambda_1,\ldots,\lambda_k\}$ be the corresponding fundamental weights, then I observe that any element $ \sigma\in Aut(\Delta)$ induces a permutation of the Weyl orbits generated by the $ \lambda_i$ ‘s. So we almost get the desired map except we haven’t checked this permutation preserves the Cartan matrix. This raises the following question:

Can these Weyl orbits themselves (as polyhedra in the Euclidean space) tell us the Cartan matrix?

As an example, $ A_3$ , these orbits consist of two congruent tetrahedra and one octahedron. We know that there is no edge joining the two tetrahedra in the Dynkin diagram. Can we see this fact from their configuration?