D4 diagram and generator conventions

PDF p. 27: §7 opening diagram and paragraph

For this example, let $\mathrm{\Gamma }=D_4$ denote the Coxeter diagram with vertices $\{1,2,3,4\}$ and three unlabelled edges $\{1,2\}$, $\{2,3\}$ and $\{2,4\}$. Thus vertex $2$ is trivalent: it is incident to exactly three edges. Equivalently, the Coxeter matrix has $m(s_i,s_i)=1$, $m(s_2,s_i)=m(s_i,s_2)=3$ for $i\in \{1,3,4\}$, and $m(s_i,s_j)=2$ for distinct $i,j\in \{1,3,4\}$.

Let $(W,S)$ be the associated Coxeter system, with $S=\{s_1,s_2,s_3,s_4\}$, and set $J=\{2\}$. Write $s=s_1$, $t=s_2$, $u=s_3$ and $v=s_4$; in particular $t$ is the simple reflection at the trivalent vertex.

Let $V={⨁}_{i=1}^4\mathbb{R}{\alpha }_i$ be the standard geometric representation: its Coxeter form satisfies $B({\alpha }_i,{\alpha }_i)=1$, $B({\alpha }_2,{\alpha }_i)=-1\mathrm{/}2$ for $i\in \{1,3,4\}$, and $B({\alpha }_i,{\alpha }_j)=0$ for distinct $i,j\in \{1,3,4\}$; the generator $s_i$ acts by $x\mapsto x-2B(x,{\alpha }_i){\alpha }_i$. Let $\mathrm{\Theta }=V^{\ast }$ be the space of real linear functionals on $V$, with the contragredient action $(wf)(x)=f(w^{-1}x)$ for $w\in W$, $f\in \mathrm{\Theta }$ and $x\in V$.

Coxeter-system, roots, length and weak-order conventions

Real Coxeter representation and contragredient notation

By Coxeter-system, roots, length and weak-order conventions, an unlabelled edge records matrix entry $3$ and absence of an edge records entry $2$; hence the stated diagram and matrix specify the same data. The matrix is symmetric, has diagonal entries $1$ and allowed off-diagonal entries, so the associated presented Coxeter group and its simple generators are defined. The set $J$ is a subset of the vertex set, and the letters $s,t,u,v$ are aliases for those generators.

The finite-rank construction in Real Coxeter representation and contragredient notation supplies the representation and dual action for this matrix. Its form has off-diagonal values $-\cos (\pi \mathrm{/}3)=-1\mathrm{/}2$ on the three edges and $-\cos (\pi \mathrm{/}2)=0$ on the remaining pairs, giving the displayed formulas. The same supplier ensures that the reflection maps define a group action and that the contragredient formula defines an action on $V^{\ast }$.