Longest parabolic elements and opposition permutation

PDF p. 5: Notation 2.5; (2.6)

Let $(W,S)$ be a Coxeter system with $S=\{s_i:i\in {\mathrm{\Gamma }}_0\}$, length $\mathrm{\ell }$, and simple roots ${\alpha }_i$ in its standard geometric representation. Let $A$ be the Artin–Tits group with generators ${\sigma }_i$, and for $J\subseteq {\mathrm{\Gamma }}_0$ write $W_J=⟨s_j:j\in J⟩$ and $A_J=⟨{\sigma }_j:j\in J⟩$. Suppose that $W_J$ is finite.

The longest element $w_J$ is the unique element of $W_J$ whose $\mathrm{\ell }$-length is maximal. Its positive lift is ${\mathrm{\Delta }}_J={\sigma }_{w_J}\in A_J$: if $w_J=s_{j_1}\cdots s_{j_r}$ is a shortest expression in simple generators, set ${\sigma }_{w_J}={\sigma }_{j_1}\cdots {\sigma }_{j_r}$, with the empty product equal to the identity. The opposition permutation ${\iota }_J:J\to J$ is the unique permutation satisfying $w_J{\alpha }_j=-{\alpha }_{{\iota }_J(j)}$ for every $j\in J$.

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The Coxeter-data definition supplies the length and geometric action, and the parabolic-subgroup definition supplies $W_J$, $A_J$ and the Artin generators. Since distinct simple generators are distinct elements of $W_J$, finiteness of $W_J$ makes $J$ finite. The arbitrary-rank parabolic theorem identifies the Coxeter system on the restricted matrix with $W_J$ and identifies its intrinsic length with ambient length. The finite-opposition interface applied to this finite-rank system gives the unique maximum-length element and a permutation sending each simple root to its negative as displayed. Its restricted geometric action agrees with the ambient action on the span of the ${\alpha }_j$, by the defining reflection formulas. The simple roots are distinct basis vectors, so the displayed equations determine the permutation uniquely.

A shortest $J$-word for $w_J$ is also shortest in $W$, by length preservation, and its Artin word lies in $A_J$. To compare it with any other shortest ambient expression, choose a finite set of simple generators containing $J$ and all letters of that expression. The arbitrary-rank parabolic theorem embeds its Coxeter group in $W$ with lengths preserved. The positive-lift interface applied to this finite-rank group equates the two Artin products; its braid relations also hold in $A$. Thus ${\mathrm{\Delta }}_J$ is independent of the chosen shortest expression. When $J=\mathrm{\varnothing }$, $W_J$ and $A_J$ are trivial, $w_J={\mathrm{\Delta }}_J=1$, and ${\iota }_J$ is the empty permutation.