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For a chosen positive system of a root system $\Pi$, the inversion set isFor a finite Weyl group, $|N(w)|=\ell (w)$.

A root system in the real inner product space $V$ is a finite spanning set $\Phi \subset V\setminus \{0\}$ such thatwhere the orthogonal reflectionFor a crystallographic root system one additionally requires $2(\beta ,\alpha )/(\alpha ,\alpha )\in \mathbb{Z}$; that condition is not needed for general finite reflection groups.

A fundamental system of a root system is a basis $\Delta \subset \Phi$ such that every root has either all nonnegative or all nonpositive coordinates in this basis. Its associated positive system of a root system isand $\Phi =\Pi \bigsqcup (-\Pi )$.

Solved by gpt-5.6-sol high.

A fundamental system of a root system is a subset $\Delta \subset \Phi$ which is a basis of $V$ and for which every $\beta \in \Phi$ has an expansionwhose coefficients are either all nonnegative or all nonpositive. Its associated positive system of a root system isThus $\Phi =\Pi \bigsqcup (-\Pi )$, and the elements of $\Delta$ are the simple roots.

Solved by gpt-5.6-sol high.

By the assumed transitivity on 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 givesFor every $w\in W$, its inversion set is contained in $\Pi$, so $\ell (w)\le |\Pi |$ 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.