---
tag: A9AHHSV1EE
kind: definition
name: Coxeter group
msc: ["20F55", "51F15"]
defines: G6FND7X69J
uses: [G5S1M9, 0JGEQXQ4AC, 08MG7H8JW5]
sources:
  - url: https://en.wikipedia.org/wiki/Coxeter_group
---

## Definition

Let $I$ be a set.
A *Coxeter matrix* on $I$ is a family $M = (m_{ij})_{i, j \in I}$ with values in $\set{1, 2, 3, \ldots} \cup \set{\infty}$ such that:

1. $m_{ii} = 1$ for all $i \in I$;
2. $m_{ij} = m_{ji}$ for all $i, j \in I$;
3. $m_{ij} \ge 2$ for all $i \ne j$.

The *Coxeter group* of $M$ is the group with the [[presentation|0JGEQXQ4AC]]
$$W(M) = \langle s_i \ (i \in I) \mid (s_i s_j)^{m_{ij}} = 1 \text{ for all } i, j \in I \text{ with } m_{ij} < \infty \rangle.$$
A group is a *Coxeter group* if it is [[isomorphic|08MG7H8JW5]] to $W(M)$ for some Coxeter matrix $M$.
The pair $(W(M), \set{s_i})$ is a *Coxeter system*.

## Remarks

The relation for $i = j$ says $s_i^2 = 1$.
For $m_{ij} = 2$ the relation says that $s_i$ and $s_j$ commute.
In $W(M)$ each $s_i$ has order exactly $2$, and $s_i s_j$ has order exactly $m_{ij}$.
The matrix is often drawn as a Coxeter diagram: a vertex for each $i$, and an edge labelled $m_{ij}$ when $m_{ij} \ge 3$.

Example: the [[symmetric group|0515A6D2Y1]] $S_n$ is a Coxeter group.
Take $s_i = (i \ i{+}1)$ for $1 \le i \le n - 1$, with $m_{i, i+1} = 3$ and $m_{ij} = 2$ for $\abs{i - j} \ge 2$.
The relations hold: $(i \ i{+}1)(i{+}1 \ i{+}2)$ is a $3$-cycle, and disjoint transpositions commute.
A theorem of Moore says that these relations define $S_n$.

Non-example: the cyclic group $\Z/4$ is not a Coxeter group.
A Coxeter group is generated by elements of order $2$, so in its abelianization every element has order at most $2$.
But $\Z/4$ is abelian, so it equals its abelianization, and it has an element of order $4$.