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The longest element $w_0$ is the unique element of maximal Coxeter length in a finite Coxeter group. It is an involution and satisfies $\ell (w_{0}w)=\ell (w_0)-\ell (w)$.

We induct on the Coxeter length $\ell (w)$. There is nothing to prove when $\ell (w)=0$. Otherwise choose a simple root $\alpha$ such thatequivalently, $w\alpha$ is negative. Since $\lambda$ and $\mu =w\lambda$ lie in the Closed dominant Weyl chamber,Therefore $⟨\lambda ,{\alpha }^{\vee }⟩=0$, and the simple reflection $s_{\alpha }$ fixes $\lambda$. Moreover,The induction hypothesis writes $w{s}_{\alpha }$ as a product of simple reflections that fix $\lambda$. Multiplying on the right by $s_{\alpha }$ gives the required expression for $w$. This is the Weyl stabilizer of a dominant point lemma.

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.

Let $R^{+}$ be the positive roots, $W$ the Weyl group, $\ell (w)$ its Coxeter length, $\rho$ the half-sum of positive roots, and ${\alpha }^{\vee }$ a coroot. For a dominant integral highest weight $\lambda$, the Weyl character formula isTaking the value at the identity gives the Weyl dimension formula

For the q-character convention relevant to the Principal sl2 subalgebra, set $h_{\mathrm{pr}}=2{\rho }^{\vee }$, so ${\alpha }_i(h_{\mathrm{pr}})=2$ for every simple root, and defineThe q-character formula is the principal specialization of the Weyl character formula:

Solved by gpt-5.6-sol high.

Let $C$ be the fundamental chamber of a root system. The chambers $wC$ and $w{s}_{\alpha }C$ are adjacent across the reflecting hyperplane orthogonal to $w(\alpha )$. The chamber $wC$ lies on the side on which $w(\alpha )$ is positive. If $w(\alpha )\in \Pi$, then $C$ lies on that same side, so crossing this wall moves one step farther from $C$; if $w(\alpha )\in -\Pi$, it moves one step nearer. The gallery distance from $C$ to $wC$ is the Coxeter length $\ell (w)$, and adjacent chamber distances differ by one. Consequently

Solved by gpt-5.6-sol high.

Write $N(w)$ for the Inversion set of a Weyl-group element. Part b shows that $s_{\alpha }$ permutes $\Pi \setminus \{\alpha \}$. It follows that right multiplication by $s_{\alpha }$ changes the size of the inversion set byIndeed, all roots other than $\alpha$ are merely relabelled, while $w{s}_{\alpha }(\alpha )=-w(\alpha )$. Part c gives exactly the same recursion for the Coxeter length. Both quantities vanish at the identity, so induction along any word in the simple reflections gives

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.

A reduced expression for $w\in W$ is a product of the fewest possible simple generators representing $w$. The number of factors is its Coxeter length $\ell (w)$.