ScalingStacks

We refer to [Hu, §5 and §7.1–7.3] for basic properties of Coxeter groups and Hecke algebras. Recall that a Coxeter group $(W,S)$ is the data of a group $W$ with a subset $S\subset W$ such that $W$ has a presentation with generating set $S$ and relations

A reduced expression of an element $w\in W$ is a decomposition $w=s_{i_{1}}\cdots s_{i_{l}}$ such that $s_{i_{r}}\in S$ for $r=1,\ldots,l$ and such that $l$ is minimal with this property. The integer $l$ is the length $\ell(w)$ of $w$.

The Chevalley-Bruhat (partial) order on $W$ is defined as follows. Let $w^{\prime},w\in W$ and let $w=s_{i_{1}}\cdots s_{i_{l}}$ be a reduced decomposition. We say that $w^{\prime}\leq w$ if there is $l^{\prime}\leq l$ and an increasing injection $f:\{1,\ldots,l^{\prime}\}\to\{1,\ldots,l\}$ such that $w^{\prime}=s_{i_{f(1)}}\cdots s_{i_{f(l^{\prime})}}$. This is independent of the choice of the reduced decomposition of $w$.