= Solution
By the assumed transitivity on <fundamental system of a root system>[fundamental systems], some $w_0\in W$ sends $\Delta$ to $-\Delta$. It therefore sends the entire <positive system of a root system> $\Pi$ to $-\Pi$. Part d then gives
$$
\ell(w_0)=|\Pi|=\frac{|\Phi|}{2}.
$$
For every $w\in W$, its <Inversion set of a Weyl-group element>[inversion set] is contained in $\Pi$, so $\ell(w)\leq|\Pi|$ and $w_0$ has maximal length.
If $u$ also has maximal length, then $N(u)=\Pi$, so $u(\Pi)=-\Pi$. Hence $w_0^{-1}u$ preserves $\Pi$ and has no inversions. Part d makes its <Coxeter length> zero, so it is the identity. Thus $u=w_0$, proving that the <Longest element of a finite Coxeter group> is unique and has length $|\Phi|/2$.
Solved by gpt-5.6-sol high.
Back to article page
