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

PDF p. 4: §2.2

Let $S=\{s_i:i\in {\mathrm{\Gamma }}_0\}$ be indexed by a set ${\mathrm{\Gamma }}_0$, possibly infinite, and let $m:S\times S\to \{1,2,3,\dots \}\cup \{\mathrm{\infty }\}$ be symmetric, with $m(s,s)=1$ and $m(s,t)\ge 2$ for distinct $s,t$. The Coxeter diagram $\mathrm{\Gamma }$ has vertex set ${\mathrm{\Gamma }}_0$: distinct vertices $i,j$ are joined when $m(s_i,s_j)\ge 3$, with the edge labelled by $m(s_i,s_j)$ when that value is at least $4$ or infinity; an unlabelled edge denotes $3$, and no edge denotes $2$. The associated Coxeter group is $W=⟨S\mid s^2=1,(st{)}^{m(s,t)}=1\text{ for }s\mathrm{\ne }t\text{ with }m(s,t)<\mathrm{\infty }⟩$. The pair $(W,S)$ is its Coxeter system. The diagram is spherical, or of finite type, precisely when $W$ is finite.

Let $V={⨁}_{i\in {\mathrm{\Gamma }}_0}\mathbb{R}{\alpha }_i$, the real vector space of finite linear combinations of the basis vectors ${\alpha }_i$. Define the symmetric bilinear Coxeter form $B$ by $B({\alpha }_i,{\alpha }_i)=1$, $B({\alpha }_i,{\alpha }_j)=-\cos (\pi \mathrm{/}m(s_i,s_j))$ for distinct $i,j$ with finite label, and $B({\alpha }_i,{\alpha }_j)=-1$ when that label is infinity. For $s_i\in S$ put $r_i(v)=v-2B(v,{\alpha }_i){\alpha }_i$. The standard geometric representation is the homomorphism $\rho :W\to \mathrm{G}\mathrm{L}(V)$ with $\rho (s_i)=r_i$; write $wv$ for $\rho (w)v$. Its roots are $\mathrm{\Phi }=\{w{\alpha }_i:w\in W,i\in {\mathrm{\Gamma }}_0\}$. Put $V_{+}=\{{\sum }_i{a}_i{\alpha }_i:a_i\ge 0\}$, with finite support, and define the positive and negative roots by ${\mathrm{\Phi }}^{+}=\mathrm{\Phi }\cap V_{+}$ and ${\mathrm{\Phi }}^{-}=-{\mathrm{\Phi }}^{+}$.

The length $\mathrm{\ell }(w)$ is the least number of simple generators in a word representing $w$, with the empty word representing the identity. Define the prefix weak order by $u\le w$ when $w=ux$ for some $x\in W$ and $\mathrm{\ell }(w)=\mathrm{\ell }(u)+\mathrm{\ell }(x)$. Following the article, call such a $u$ a left divisor of $w$ and call this the left weak order; the finite-rank supplier calls the same prefix convention right weak order.

Let $\mathrm{\Theta }=V^{\ast }={\mathrm{Hom}}_{\mathbb{R}}(V,\mathbb{R})$, the algebraic dual. Its contragredient action is $(wf)(v)=f(\rho (w{)}^{-1}v)$ for $w\in W$, $f\in \mathrm{\Theta }$ and $v\in V$. No identification of $V$ with $V^{\ast }$ or nondegeneracy of $B$ is assumed.

The presentation and length are defined for arbitrary $S$ by Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order; in particular each group element has a finite word and the minimum length exists. The diagram simply records the matrix entries, and spherical type is a finiteness condition on this group.

The basis values extend uniquely to $B$ because vectors have finite support. For each finite $T\subseteq S$, the maps $r_s$ for $s\in T$ restrict to the finite-dimensional span of the corresponding basis vectors and are the simple reflections of Real Coxeter representation and contragredient notation. To verify any defining relation on an arbitrary $v\in V$, choose a finite $T$ containing the generators in that relation and the support of $v$. The finite-rank representation supplier shows the relation holds on that finite-dimensional span. Thus all defining relations hold on $V$, and the presentation gives the unique homomorphism $\rho$; in particular each $\rho (w)$ is invertible. The displayed root sets and the dual action are consequently well-defined. The inverse in the dual formula makes it a left action: $(xy)f=x(yf)$.

The prefix relation is reflexive, using the identity. If $w=ux$ and $z=wy$ are length-additive, then $z=u(xy)$; subadditivity of word length and the two displayed equalities force $\mathrm{\ell }(z)=\mathrm{\ell }(u)+\mathrm{\ell }(xy)$, proving transitivity. If $u\le w$ and $w\le u$, length additivity forces both factors to have length zero, hence $u=w$. Thus it is an order. In finite rank this is the prefix order of Bounded weak-order joins and finite parabolic criteria; no finite-rank lattice assertion is used here.