Coxeter-Dynkin Diagrams

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 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  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:

Coxeter-Dynkin

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:

Coxeter-Dynkin

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 letters, which has n vertices

coxeter-dynkin

  • the wreath product of the symmetric group on letters with the cyclic group of order 2, which has n+1 vertices

coxeter-dynkin

  • an order 2 subgroup of symmetric group on letters with the cyclic group of order 2, which has n+2 vertices

Coxeter-Dynkin

You can also make simplicial complexes out of Coxeter groups and I will write about this next time!

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5 thoughts on “Coxeter-Dynkin Diagrams

  1. Joseph Nebus March 22, 2015 / 9:50 pm

    I hope you do. This is a sort of notation I’ve run across but not seen explained well enough for me to understand.

    Like

  2. Yael March 28, 2015 / 3:31 am

    I love your description.

    BTW: An extra t made its way to the right hand side of stst=tstst

    Like

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