# The Cali Garmo

does Math

## Barycentric Subdivision

By Cali G , Published on Fri 10 April 2020
Category: math / simplicial complexes

We start off with some definitions.

## Simplicial Complexes

Definition: Recall that a simplicial complex $K$ is a set of simplices such that:

• every face of a simplex of $K$ is also in $K$,
• and the non-empty intersection of any two simplices $\sigma_1,\, \sigma_2 \in K$ is a face of both $\sigma_1$ and $\sigma_2$.

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 $0$-simplex, the $1$-simplex, the $2$-simplex and the $3$-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 $K$, a subdivision of $K$ is a simplicial complex $K'$ such that:

• each simplex of $K'$ is contained in a simplex of $K$
• and each simplex of $K$ is a finite union of simplicies in $K'$.

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 $\left\{v_0, v_1, \ldots, v_n\right\}$, we can let our simplex be the set of points $x$ such that So each point $x$ in the simplex can be associated to the vector $(\lambda_i)$. These $(\lambda_i)$ are known as the barycentric coordinates of $x$.

This allows us to define a subdivision based off barycenters. Let $\sigma$ be a simplex with verticese $v_0, v_1, \ldots, v_n$. The barycenter of $\sigma$ is the point: In fact, it is exactly the point in the interior of $\sigma$ whose barycenter coordinate with respect to the vertices of $\sigma$ are equal.

Let's look at an example with our $2$-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 $\sigma_i$ is the position of the $i$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 $(0,1,0)$ be our barycentric coordinate, then we associate it to the permutation $132$ since the first and third entries are equal (both $0$) and the second entry is greater than $0$.

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!