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I will give the definition of what I mean by reflection which is in Suzuki's group theory I.

Let $\Sigma $ be an apartment of a building that contains adjacent chambers $C$ and $C'$. Then there are foldings $\phi, \phi'$ of $\Sigma$ such that $\phi(C')=C$ and $\phi'(C)=C'$ and we have $\Sigma = \phi(\Sigma ) \cup \phi'(\Sigma) $ with $\phi(\Sigma)$ and $\phi'(\Sigma)$ having no chamber in common. We define

$$ \sigma (A)= \left\{ \begin{array}{ll} \phi(A) & \text{ if } A \in \phi(\Sigma) \\ \phi'(A) & \text{ if } A \in \phi'(\Sigma). \\ \end{array} \right. $$

This is an automorphism of $\Sigma$ of order 2.

This is called the reflection of $\Sigma$ with respect to the wall of $C$ and $C'$.

Then the Weyl group $W$ is defined as the group of automorphisms generated by these reflections. Let the generating set be $S$ so $W=\langle S \rangle$.

With this definition, I am having a hard time seeing why if we have any reflection $s \in S$ and any chamber $C \in \Sigma$, why are the chambers $C$ and $s(C)$ adjacent?

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    $\begingroup$ They are not in general adjacent even in the $A_2$ case. $\endgroup$
    – Moishe Kohan
    Commented Sep 12, 2023 at 14:46
  • $\begingroup$ Your two cases for $\sigma(A)$ had the same condition "if $A \in \phi'(\Sigma)$". I changed one of them to "if $A \in \phi(\Sigma)$", but I'm not sure if it was the right one. $\endgroup$
    – LSpice
    Commented Sep 12, 2023 at 17:33
  • $\begingroup$ I also would like to add that I find this definition of reflections in a Coxeter complex (aka an apartment) ridiculously convoluted. $\endgroup$
    – Moishe Kohan
    Commented Sep 13, 2023 at 0:54
  • $\begingroup$ How would you define these reflections? $\endgroup$
    – Anonmath101
    Commented Sep 13, 2023 at 10:56
  • $\begingroup$ Consider reading a different book on Coxeter groups or buildings. For instance, by Brown or Ronan, or the chapter in Bourbaki on Coxeter groups. $\endgroup$
    – Moishe Kohan
    Commented Sep 13, 2023 at 12:43

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