Question: Which finite irreducible reflection groups $\Gamma\subseteq\mathrm O(\Bbb R^d)$ contain the central inversion $-\mathrm{Id}$, and how can this be spotted from the Coxeter diagram?
The following groups do not have central inversion:
- $I_2(n)$ for odd $n\ge 3$,
- $A_d$ for all $d\ge 3$,
- $D_d$ for odd $d\ge 3$, and
- $E_6$.
I am not sure about $E_7$ and $E_8$. All others do have central inversion. The emphasize of my question is on the second part: is there combinatorial data in the Coxeter diagram from which to infer whether $-\mathrm{Id}\in\Gamma$?
