Chamber paths, concatenation and wall words

PDF p. 13: §3.4 before Proposition 3.29

Work in the finite-rank relative chamber arrangement with fixed $J\subseteq {\mathrm{\Gamma }}_0$ and Coxeter group $W$ of the adjacent-wall formula. Its chambers are labelled by pairs $(x,I)\in \mathrm{C}\mathrm{h}\mathrm{a}\mathrm{m}(J)$: $W_{J}x=x{W}_I$ and $x$ has minimum word length in $x{W}_I$. A chamber is a connected component of the arrangement complement; two distinct chambers are adjacent when they share a wall, meaning a codimension-one intersection of a chamber closure with an arrangement hyperplane.

A path is a finite nonempty sequence $p=((x_1,I_1),\dots ,(x_m,I_m))$ of chamber labels such that each consecutive pair labels adjacent chambers. Its source is $s(p)=(x_1,I_1)$, its target is $t(p)=(x_m,I_m)$, and its length is $m-1$. If $q=((y_1,K_1),\dots ,(y_n,K_n))$ and $t(p)=s(q)$, their concatenation is $p\ast q=((x_1,I_1),\dots ,(x_m,I_m),(y_2,K_2),\dots ,(y_n,K_n))$.

For each consecutive pair, let $a_i\in {\mathrm{\Gamma }}_0\setminus I_i$ be the index supplied by the adjacent-wall formula, so that ${\omega }_{a_i,I_i}(x_i,I_i)=(x_{i+1},I_{i+1})$. The length-one path consisting of this pair is called a simple wall crossing, and is denoted ${\omega }_{a_i,I_i}:(x_i,I_i)\to (x_{i+1},I_{i+1})$. Thus $p$ is the ordered concatenation of its simple wall crossings when $m>1$. Define its wall word element by $v_p=v_{a_1,I_1}\cdots v_{a_{m-1},I_{m-1}}\in W$, where each $v_{a_i,I_i}$ is the simple wall-crossing element. When $m=1$, the empty product is the identity of $W$.

Tits cone intersections, relative roots and arrangements

Independent relative chamber labelling input

Chambers, walls and adjacency

Algebraic chamber-label set

Simple wall-crossing elements and maps

Adjacent chambers and inverse wall-crossing elements

The relative-cone definition specifies the relative arrangement in its finite-dimensional real Coxeter setting. The relative chamber-labelling theorem identifies its chambers bijectively with the algebraic labels, and the chamber and label definitions specify the objects and adjacency relation. The adjacent-wall formula supplies a unique crossing index for each ordered adjacent pair and ensures that the corresponding crossing element is defined. Hence the finite sequence determines its ordered list of crossing elements and their product in $W$ uniquely. When two endpoints match, omitting the repeated endpoint leaves the original consecutive adjacencies on both sides of the join, so concatenation is again a path. The one-chamber case has no crossings and its product is the group identity.

The article reference attributes the terminology and construction. The well-definedness argument uses the declared local definitions and adjacent-wall formula.