We start off with some definitions.
Simplicial Complexes
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.
Subdivision
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).
Permutations
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!