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Use the positive-root criterion for Coxeter length: for a simple root $\alpha \in \Delta$,The hypothesis $w\Delta \subseteq \Pi$ therefore says that every simple generator is a right ascent of $w$. If $w\ne 1$, a reduced expression in a Coxeter group for $w$ has a final simple generator $s_{\alpha }$, and deleting it givesa contradiction. Hence $w=1$.

If $w$ stabilizes $\Delta$ setwise, then $w\Delta =\Delta \subseteq \Pi$, so the result just proved gives $w=1$. Thus the stabilizer of every fundamental system is trivial.

Solved by gpt-5.6-sol high.

If a finitely generated Coxeter group $W$ is finite, its integer-valued Coxeter length has a maximum. Conversely, if some $w_0$ has globally maximal length $N$, every group element has a word of length at most $N$. There are only finitely many words of bounded length in the finite set of simple generators, so $W$ is finite.

Realize the finite group as the reflection group of a root system with fundamental system $\Delta$ and positive system $\Pi$. Maximality and the fact that multiplication by a simple generator changes Coxeter length by one giveThe positive-root criterion for Coxeter length therefore gives $w_0\Delta \subseteq -\Pi$. Since $w_0\Delta$ is itself fundamental, it must be the simple system $-\Delta$ of the positive system $-\Pi$.

If $v_0$ is another maximal-length element, the same argument gives $v_0\Delta =-\Delta =w_0\Delta$. Hence $v_0^{-1}{w}_0$ stabilizes $\Delta$, and part c gives $v_0^{-1}{w}_0=1$. The Longest element of a finite Coxeter group is therefore unique.

Solved by gpt-5.6-sol high.

For $f\in C$, duality givesThe positive-root criterion for Coxeter length saysis a positive root. Its basis coefficients are nonnegative and not all zero, so $f(\sigma (w^{-1})e_i)>0$ and ${\sigma }^{\ast }(w)(C)\subseteq A_i$. If the length decreases, that root is negative and the same calculation gives ${\sigma }^{\ast }(w)(C)\subseteq A_i^{-}$.

If ${\sigma }^{\ast }(w)$ is the identity and $w\ne 1$, choose a left descent $x_i$ from the first letter of a reduced expression for $w$. The preceding result givesalthough $C\subseteq A_i$. The two open half-spaces are disjoint, a contradiction. Thus the Dual geometric representation of a Coxeter group is faithful.

Solved by gpt-5.6-sol high.