We start off with some definitions.
Definition: Recall that a simplicial complex is a set of simplices such that:
- every face of a simplex of is also in ,
- and the non-empty intersection of any two simplices is a face of both and .
As basic examples, the following are simplicial complexes. In fact, they are all simplexes. A simplicial complex would be the union of some of these simplexes, possibly attached. These simplices are the -simplex, the -simplex, the -simplex and the -simplex.
We now want to take a simplicial complex and cut it up into smaller pieces. This is what is known as a subdivision.
Definition: Given a simplicial complex , a subdivision of is a simplicial complex such that:
- each simplex of is contained in a simplex of
- and each simplex of is a finite union of simplicies in .
We're going to look at a particular type of subdivision, called the barycentric subdivision. To define this I need to first talk about the barycenter.
Definition: Given a set of (indpendent) points , we can let our simplex be the set of points such that So each point in the simplex can be associated to the vector . These are known as the barycentric coordinates of .
This allows us to define a subdivision based off barycenters. Let be a simplex with verticese . The barycenter of is the point: In fact, it is exactly the point in the interior of whose barycenter coordinate with respect to the vertices of are equal.
Let's look at an example with our -simplex from above (the triangle).
It turns out you can associate a permutation naturally to faces of the barycentric subdivision of a simplex! To do this, you can give an ordering on the barycentric coordinates so that is the position of the th coordinate and by reading equal coordinates from left to right. The best way to do this is through a few examples.
If we let be our barycentric coordinate, then we associate it to the permutation since the first and third entries are equal (both ) and the second entry is greater than .
You can play this game with the barycentric subdivision from above!
If you notice, what this gives us is a way to assign to each point in our simplex a permutation!