Standard expressions and their lengths

PDF p. 17: Definition 4.9

Let $(W,S)$ be a Coxeter system with diagram vertex set ${\mathrm{\Gamma }}_0$, and fix $J\subseteq {\mathrm{\Gamma }}_0$. For $L\subseteq {\mathrm{\Gamma }}_0$, write $W_L=⟨s_i:i\in L⟩$; an associate of $J$ is a subset whose standard parabolic subgroup is conjugate to $W_J$. The Brink–Howlett groupoid $\mathcal{B}\mathcal{H}(J)$ has these associates as objects. A morphism $x:I\to K$ is labelled by an element $x\in W$ satisfying $\{x{\alpha }_k:k\in K\}=\{{\alpha }_i:i\in I\}$, and composable labels multiply from left to right. Write $\mathrm{\ell }(x)$ for the least number of simple generators in a word representing $x$.

For an associate $I$ and $a\in {\mathrm{\Gamma }}_0\setminus I$, let $D=[I,a]$ be the component containing $a$ in the diagram induced on $I\cup \{a\}$. If $W_D$ is finite, write $w_L$ for the longest element of a finite $W_L$ and ${\iota }_D$ for the permutation defined by $w_D{\alpha }_i=-{\alpha }_{{\iota }_D(i)}$. Put $v_{a,I}=w_{D\setminus \{a\}}{w}_D$ and $I^{\mathrm{\prime }}=(I\cup \{a\})\setminus \{{\iota }_D(a)\}$. This label is a morphism $I\to I^{\mathrm{\prime }}$.

For objects $I,K$ and a morphism $w:I\to K$, a standard expression is a finite sequence of such morphisms $v_{a_1,I_1},\dots ,v_{a_m,I_m}$, composable in that order from $I$ to $K$, whose product is $w$ and for which $\mathrm{\ell }(w)={\sum }_{i=1}^m\mathrm{\ell }(v_{a_i,I_i})$. For $m>0$ the source condition says $I_1=I$. The empty sequence is allowed when $I=K$ and $w$ is the identity; its product is the identity and its length sum is zero. The standard-expression length of the expression is its number $m$ of factors. When a morphism has a standard expression and every standard expression for it has the same number of factors, this common number is denoted ${\mathrm{\ell }}_{\mathcal{B}\mathcal{H}}(w)$ and called the standard-expression length of $w$.

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

Associates of a parabolic subset

Brink–Howlett groupoid

Connected component notation [I,a]

Longest parabolic elements and opposition permutation

Simple wall-crossing elements and maps

Wall-crossing elements are Brink–Howlett morphisms

The Coxeter length is defined as a minimum over finite representing words. The groupoid supplies the source, target, identity and composition of each finite composable sequence; the wall-crossing construction and its morphism lemma supply the displayed factors under the stated finiteness condition. Thus the product and the equality of lengths are well-defined conditions on a finite sequence. Its number of entries is a uniquely determined nonnegative integer. The notation for a morphism is introduced only under the explicitly stated existence and independence conditions, so no expression-independent value is assumed here.

Definition 4.9 attributes the terminology to section 2 of its reference [5]. Existence and independence are separate subsequent results in the article; they are not established by this definition.