Coxeter Complex

In my last post

https://picturethismaths.wordpress.com/2015/03/21/coxeter-dynkin-diagrams/

I talked about Coxeter groups and their corresponding Coxeter-Dynkin Diagrams. Here is a little recall of the definition of  a Coxeter group:

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. 

Now I’m going to describe a simplicial complex called the Coxeter complex. One of these can be constructed out of any Coxeter group. Let’s bullet point the steps of the construction and after each follow the example of the group with two commuting generators, given by the Coxeter-Dynkin diagram below

Screen Shot 2015-04-06 at 09.06.08

and the relations s 2= e, t 2= e, and st=ts.

  • Firstly take all the parabolic subgroups of the Coxeter group. these are groups generated by subsets of the vertices.

For our example the parabolic subgroups are

<e>={e}, <s>={e,s}, <t>={e,t}, <s,t>={e,s, t, st=ts}

  • Secondly form a list of all the cosets of these subgroups. Cosets are things you get from premultiplying your subgroup by a chosen element in the main group.

For our example the cosets for each of the parabolic subgroups are

  <e>={e}  has cosets<e>={e}, s<e>={s}, t<e>={t}, and st<e>{st}                                                   <s>={e,s}  has cosets <s>={e,s}, s<s>={e,s}=<s>, t<s>={t,ts=st}, and st<s>={t, st}                        <t>={e,t}  has cosets <e>={e,t}, s<t>={s, st}, t<t>={e, t}, and st<t>={s, st }                                     <s,t>={e,s,t,st}  has cosets <s,t>=s<s,t>=t<s,t>=st<s,t>={e,s,t,st}                  

We see that some of these are the same so we end up with cosets:

Cosets with one element: <e>, s<e>, t<e>, st<e>

Cosets with two elements: <s>, t<s>, <t>, s<t>

Cosets with four elements: <s,t>

  • Create your complex by setting the vertices to be the cosets, and creating an edge (1-simplex) between two cosets whenever the elements of one are a subset of the elements of another. Then create a triangle (2 simplex) whenever you have one coset inside another which is in turn inside another. Continue like this with triangular pyramids from 4 cosets (3 simplex) etc.

Finally we get to draw some pictures! Here are the vertices:

Cosets

And here are the edges (there is an edge between every other coset and <s,t> since <s,t> is the whole group, so we have missed these out):

Edges

We can rearrange the edges, then putting in the triangles (and colouring them pink) gives us:

Coxeter complex

Many mathematicians like to miss out the middle coset, which represents the whole group. Then our complex would look like this:

Coxeter Complex

Notice that and t act on the Coxeter complex. Multiplying all the cosets by s reflects in the middle vertical line and multiplying all the cosets by t reflects in the middle horizontal line:

Group actionThis is true for any group and their coxeter complex. As another example, here is a photo of a model for the coxeter complex of the symmetric group on 4 elements, or

Coxeter-dynkin diagram

which was made by someone in my department.

Coxeter complex

We see hexagons and squares. The middle vertex acts as a pair with the left vertex on half of the hexagons and with the right vertex on the other half. They are hexagons because the vertices have a line between them, and thus have braid relation of order 3. The two outer vertices (which commute) act on the square, like our square in the previous example! The shapes actually fit together and tesselate, as shown in this next picture.

Tesselating

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One thought on “Coxeter Complex

  1. emilydumont55 April 12, 2015 / 2:52 am

    I’ve played with those square/hexagon structures before.. good post! x

    Like

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