ScalingStacks

Let $(W,S)$ be a finite Coxeter group. Let $V$ be the geometric representation of $W$ over $k={\mathbf{C}}$: it comes with a basis $\{e_{s}\}_{s\in S}$. Given $s\in S$, we denote by $\alpha_{s}$ the linear form on $V$ such that $s(x)-x=\alpha_{s}(x)e_{s}$ for all $x\in V$. The set $\{\alpha_{s}\}_{s\in S}$ is a basis of $V^{*}$. Let $P=P_{S}=P_{(W,S)}=k[V]$ (we will denote by $X_{S}$ or $X_{(W,S)}$ a given object constructed from $(W,S)$).

The braid group $B_{S}=B_{(W,S)}$ associated to $(W,S)$ is the group generated by $\{\sigma_{s}\}_{s\in S}$ with relations

for any $s,t\in S$ such that the order $m_{st}$ of $st$ is finite.

We denote by $l:B_{S}\to{\mathbf{Z}}$ the length function. It is the morphism of groups defined by $l(\sigma_{s})=1$ for $s\in S$.