Coxeter Groups, Cayley Graphs and Lattices

A group $W$ is a Coxeter group if $W$ has a set $S$ of generators subject only to the relations of the form $(s,s')^{m(s,s')}=1$ with $m(s,s)=1$ for any $s \in S$ and $m(s,s') \geq 2$ for all $s \neq s'$ in $S$. An example of Coxeter group is the symmetric group $S_n$ with $n-1$ generators (which are the transpositions $i(i+1)$ for $i \in \{1,2,...,n-1\}$). In the talk, we will present some simple examples of Coxeter groups. We will also present a combinatorial approach of these groups as follows: we give the definition of the Cayley graph associated with a Coxeter group and an order relation defined from this graph, called the weak order. In the finite case, the weak order on a Coxeter group is a lattice (Bjorner, 1984). We will then focus on the relations between the algebraic properties of the groups and the properties of the weak order.