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,*

*(*and

*tu)*^{3}*= e,**(s*This can also be written as

*u)*^{2}*= e.**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!

Joseph NebusMarch 22, 2015 / 9:50 pmI hope you do. This is a sort of notation I’ve run across but not seen explained well enough for me to understand.

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YaelMarch 28, 2015 / 3:31 amI love your description.

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

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RachaelMarch 28, 2015 / 8:20 pmThanks, will edit that out!

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