---
date: 2022-05-12 16:04
modification date: Thursday 12th May 2022 16:04:31
title: Coxeter group
aliases:
  - Coxeter group
  - Bruhat order
  - Coxeter
  - Coxeter fan
  - Coxeter system
  - Coxeter complex
  - Coxeter diagram
created: 2022-05-12T16:04
updated: 2024-06-09T19:31
---

# Coxeter diagram

![](2024-06-09.png)
![](2024-06-09-1.png)
![](2024-06-09-2.png)
![](2024-06-09-3.png)

# Coxeter group

![Pasted image 20211206015540.png](Pasted%20image%2020211206015540.png)

Let $(W, S)$ be a Coxeter system with Coxeter matrix $M=(m(s, t))_{s, t \in S}$. The canonical representation is given by a vector space $V$ with basis of formal symbols $\left(e_s\right)_{s \in S}$, which is equipped with the symmetric bilinear form $B\left(e_s, e_t\right)=-\cos \left(\frac{\pi}{m(s, t)}\right)$. In particular, $B\left(e_s, e_s\right)=1$. The action of $W$ on $V$ is then given by $s(v)=v-2 B\left(e_s, v\right) e_s$.

# Tits complex

This representation describes $W$ as a reflection group, with the caveat that $B$ might not be positive definite. It becomes important then to distinguish the representation $V$ from its dual $V^*$. The vectors $e_s$ lie in $V$ and have corresponding dual vectors $e_s^{\vee}$ in $V^*$ given by
$$
\left\langle e_s^{\vee}, v\right\rangle=2 B\left(e_s, v\right)
$$
where the angled brackets indicate the natural pairing between $V^*$ and $V$.
Now $W$ acts on $V^*$ and the action is given by
$$
s(f)=f-\left\langle f, e_s\right\rangle e_s^{\vee},
$$
for $s \in S$ and any $f \in V^*$. Then $s$ is a reflection in the hyperplane $H_s=\left\{f \in V^*:\left\langle f, e_s\right\rangle=0\right\}$. One has the fundamental chamber $\mathcal{C}=\left\{f \in V^*:\left\langle f, e_s\right\rangle>0 \forall s \in S\right\}$; this has faces the so-called walls, $H_s$. The other chambers can be obtained from $\mathcal{C}$ by translation: they are the $w \mathcal{C}$ for $w \in W$.

The Tits cone is $X=\bigcup_{w \in W} w \overline{\mathcal{C}}$. This need not be the whole of $V^*$. Of major importance is the fact that $X$ is convex. The closure $\overline{\mathcal{C}}$ of $\mathcal{C}$ is a fundamental domain for the action of $W$ on $X$.

The Coxeter complex $\Sigma(W, S)$ of $W$ with respect to $S$ is $\Sigma(W, S)=(X \backslash\{0\}) / \mathbb{R}_{+}$.

# Coxeter fan

Let $(W, S)$ be a finite Coxeter system acting by reflections on $\EE^n$. Let $\boldsymbol{a}$ be a point in the complement of the hyperplanes corresponding to the reflections in $W$. The convex hull of the $W$-orbit of $\boldsymbol{a}$ is a simple convex polytope known as a permutahedron, and denoted $\operatorname{Perm}^a(W)$. The normal fan of $\operatorname{Perm}^a(W)$ is the Coxeter fan $\mathcal{F}$.

![](attachments/Pasted%20image%2020220601225925.png)

![](attachments/Pasted%20image%2020220601230712.png)
![](attachments/Pasted%20image%2020220601230850.png)


Let $W$ be a [Weyl group](Unsorted/Weyl%20group.md). 

- The **Coxeter arrangement** $\mathcal{A}$ for $W$ is the collection of all reflecting hyperplanes for $W$. The complement $V \backslash(\bigcup \mathcal{A})$ of $\mathcal{A}$ consists of open cones. 
	- Their closures are called [chambers](Unsorted/chamber.md). 

- The chambers are in canonical bijective correspondence with the elements of $W$. The **fundamental chamber** $D:=\bigcap_{s \in S}\left\{v \in V \mid\left\langle v, \alpha_{s}\right\rangle \geq 0\right\}$ corresponds to the identity $e \in W$ and the chamber $w(D)$ corresponds to $w \in W$.

- Subsets above and below hyperplanes:
	- A subset $U$ of $V$ is **below** a hyperplane $H \in \mathcal{A}$ if every point in $U$ is on $H$ or on the same side of $H$ as $D$. 
	- The subset $U$ is **strictly below** $H \in \mathcal{A}$ if $U$ is below $H$ and $U \cap H=\varnothing$. 
	- Similarly, $U$ is **above** or **strictly above** a hyperplane $H \in \mathcal{A}$. The inversions of $w \in W$ are the reflections that correspond to the hyperplanes $H$ which $w(D)$ is above.
	- For a simple reflection $s \in S$, we have $\ell(s w)<\ell(w)$ if and only if $s \leq w$ in the weak order if and only if $w(D)$ is above $H_{s}$. To decide whether $w(D)$ is above or below $H_{s}$ is therefore a weak order comparison.

- See [Fans](Unsorted/toric.md#Fans).

- For a Coxeter arrangement $\mca$, the chambers and all their faces  $\mathcal{A}$ define the **Coxeter fan** $\mathcal{F}$.
	- The Coxeter fan $\mathcal{F}$ is known to be complete, essential, and simplicial.
	- The fundamental chamber $D \in \mathcal{F}$ is a (maximal) cone spanned by the (extremal) rays $\left\{\rho_{s} \mid s \in S\right\}$, where $\rho_{s}$ is the intersection of $D$ with the subspace orthogonal to the hyperplane spanned by $\left\{\alpha_{t} \mid t \in\langle s\rangle\right\}$.

- The rays of $\mathcal{F}$ decompose into $n$ orbits under the action of $W$ and each orbit contains exactly one $\rho_{s}, s \in S$. 
	- Thus, any ray $\rho \in \mathcal{F}^{(1)}$ is $w\left(\rho_{s}\right)$ for some $w \in W$ where $s \in S$ is uniquely determined by $\rho$ but $w$ is not unique. 
	- In fact, $w\left(\rho_{s}\right)=g\left(\rho_{s}\right)$ if and only if $w \in g W_{\langle s\rangle}$.

# Examples

![](attachments/Pasted%20image%2020220601224143.png)
![](attachments/Pasted%20image%2020220601224158.png)