Recently I’ve been looking at a family of groups called Coxeter Groups. A Coxeter group is generated by a set S and relations between elements of S, of the form
(st) m(s,t) = e
where s and t are elements in S, e is the identity element, and m(s,t) is a number greater than 1 which changes depending on what s and t are. A Coxeter group has the property that all of its generators are involutions, meaning m(s,s)=1. When you ‘multiply out’ the other relations you get equations that look like
st=ts (when m(s,t)=2), sts=tst (when m(s,t)=3), or in general ststst…sts=tststs…tst
and these are called braid relations. You can represent all of this information on a diagram! These are called Coxeter-Dynkin diagrams and allow you to easily see some properties of the groups without having to write lots of s’s and t’s on your blackboard.
In the diagram, we have a vertex (or blob) for every generator of the group (so every element of S). We put edges between the vertices s and t depending on m(s,t). If m(s,t)=2 then we don’t draw an edge between the vertices. If m(s,t)=3 then we draw an edge with no label, and if m(s,t) is bigger than 4 we draw an edge and label it with m(s,t). Let’s do an example:
Here the group is generated by three elements s, t, and u. The edges tell us that (st) 4 = e, (tu) 3 = e, and (su) 2 = e. This can also be written as stst=tsts, tut=utu, and su=us. I like the picture better than these equations!
Here is a slightly crazier example:
The infinity symbol here means that ststst…sts is never equal to tststs…tst!
Some examples of ‘standard’ Coxeter groups are:
- the symmetric group on n letters, which has n vertices
- the wreath product of the symmetric group on n letters with the cyclic group of order 2, which has n+1 vertices
- an order 2 subgroup of symmetric group on n letters with the cyclic group of order 2, which has n+2 vertices
You can also make simplicial complexes out of Coxeter groups and I will write about this next time!