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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 $1$).

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  • $\begingroup$ Hint: Look at the automorphism group of the extended Dynkin diagram. $\endgroup$ – Moishe Kohan Mar 25 at 21:00
  • $\begingroup$ 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... $\endgroup$ – Bugs Bunny Mar 26 at 7:10
  • $\begingroup$ @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? $\endgroup$ – user154087 Mar 26 at 11:14
  • $\begingroup$ I guess this is no longer a comment. So I post it as an answer. $\endgroup$ – Bugs Bunny Mar 26 at 13:34
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    $\begingroup$ 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. $\endgroup$ – Sam Hopkins Mar 26 at 13:57
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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.

No, I don't know where this calculation has been carried out. Please, compute the length of all the edges of this simplex yourself, and put it here for the benefit of the whole humankind:-))

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    $\begingroup$ Upvoted for the last sentence. Carrying out routine calculations and putting them somewhere publically accessible is an underestimated service to humanity. $\endgroup$ – LSpice Mar 26 at 14:25

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