Algebraic chamber-label set

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Let $(W,S)$ be a Coxeter system with $S=\{s_i:i\in {\mathrm{\Gamma }}_0\}$, where the index set ${\mathrm{\Gamma }}_0$ may be infinite. Let $\mathrm{\ell }(w)$ be the least number of simple generators in a word representing $w\in W$. For $K\subseteq {\mathrm{\Gamma }}_0$, let $W_K=⟨s_k:k\in K⟩$ be the standard parabolic subgroup generated by those simple generators. Fix $J\subseteq {\mathrm{\Gamma }}_0$.

The algebraic chamber-label set is $\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{m}(J)=\{(x,I)\in W\times \mathcal{P}({\mathrm{\Gamma }}_0):W_{J}x=x{W}_I\text{ and }\mathrm{\ell }(x)\le \mathrm{\ell }(xu)\text{ for every }u\in W_I\}$. Here $\mathcal{P}({\mathrm{\Gamma }}_0)$ is the set of all subsets of ${\mathrm{\Gamma }}_0$, and $W_{J}x=\{vx:v\in W_J\}$ and $x{W}_I=\{xu:u\in W_I\}$. Thus the length condition says that $x$ has minimal length in the coset $x{W}_I$.

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

Standard parabolic subgroups and parabolic roots

The Coxeter-system definition supplies the group $W$ and its finite-valued word-length function. The standard parabolic definition supplies the subgroups $W_J$ and $W_I$ for every indicated subset, including the empty subset. Group multiplication therefore defines both displayed cosets, and the equality and length inequalities specify a property of each member of the set $W\times \mathcal{P}({\mathrm{\Gamma }}_0)$. The displayed comprehension consequently defines a set. It is nonempty: the identity element $1$ has length zero, $W_{J}1=1W_J$, and $(1,J)$ satisfies the length condition. No uniqueness of a minimal coset representative or geometric identification of these labels is needed for the definition.