Coxeter group

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Definition

Symbol-free definition

A group is termed a Coxeter group if it can be equipped with a finite presentation given by the following relations:

The square of every generator
For some of the pairwise products of the generators, a certain power of that pairwise product

Definition with symbols

A group $G$ is termed a Coxeter group if it can be equipped with a finite presentation with generators $s_{i}$ and relations:

$s_{i}^{2} = 1$
$(s_{i} s_{j})^{m_{i j}} = 1$ where $m_{i j}$ is a function of $i$ and $j$ (for distinct $i$ and $j$ )

Alternatively we can consider a matrix $m_{i j}$ with the diagonal entries being $1$ and simply require that for each $i$ and $j$ (not necessarily distinct) $(s_{i} s_{j})^{m_{i j}} = 1$ .