# Isometry type of alcoves in affine Coxeter complexes

Let $$W$$ be an irreducible affine Coxeter group (say of type $$\widetilde{X}_n$$), and let $$\Sigma$$ be the associated Coxeter complex. Thus, $$\Sigma$$ is an $$n$$-dimensional Euclidean space tesselated by isometric copies of a given simplex $$A$$ (namely, by the alcoves of $$\Sigma$$). I would like to know what the isometry type of $$A$$ is, depending on $$\widetilde{X}_n$$ (at least for the classical types $$\widetilde{A}_n$$, $$\widetilde{B}_n$$, $$\widetilde{C}_n$$ and $$\widetilde{D}_n$$).

More specifically, if $$1$$ is the length of the shortest edge of $$A$$, what are the other possible lengths for the edges of $$A$$?

(For instance, in type $$\widetilde{A}_2$$, all edges have length $$1$$).

• Hint: Look at the automorphism group of the extended Dynkin diagram. – Moishe Kohan Mar 25 at 21:00
• Did you compute them for $n\leq 4$? This should give you a clear picture what it is. All the necessary equations for the calculation are in Bourbaki, Lie Groups and Lie Algebras, ch 4-6... – Bugs Bunny Mar 26 at 7:10
• @BugsBunny Do you know of any reference where these calculations are made, or can you show on an example (eg $\widetilde{D}_4$) what has to be done? – user154087 Mar 26 at 11:14
• I guess this is no longer a comment. So I post it as an answer. – Bugs Bunny Mar 26 at 13:34
• Note that even in Type A, you do not get a regular simplex: the simplex has cyclic symmetry (as explained by the comment of Moishe Kohan), but not full symmetry. – Sam Hopkins Mar 26 at 13:57

Go to Bourbaki, Groupes et Algebres de Lie, Ch.4-6 where the fundamental weights $$\varpi_1, \ldots , \varpi_n$$ of $$X_n$$ are listed. The vertices of the simplex are $$0 \ \mbox{ and } \ x_k \varpi_k, \ k=1,\ldots,n$$ where $$x_k>0$$ is such that $$x_k \varpi_k$$ lies on the fixed hyperplane of the last Coxeter generator of the affine Weyl group. This hyperplane is given by the equation $$\langle z , \alpha_0^{\vee}\rangle =1$$ where $$\alpha_0^\vee$$ is highest coroot. The highest roots are also listed in the same book. Don't forget to swap $$B_n \leftrightarrow C_n$$, when looking for the highest coroot, because the coroot system is Langlands dual to the root system.