Character Theory Basics

This post illustrates some of SageMath’s character theory functionality, as well as some basic results about characters of finite groups.

Basic Definitions and Properties

Given a representation $(V,\rho)$ of a group $G$, its character is a map $ \chi: G \to \mathbb{C}$ that returns the trace of the matrices given by $\rho$:

A character $\chi$ is irreducible if the corresponding $(V,\rho)$ is irreducible.

Despite the simplicity of the definition, the (irreducible) characters of a group contain a surprising amount of information about the group. Some big theorems in group theory depend heavily on character theory.

Let’s calculate the character of the permutation representation of $D_4$. For each $g \in G$, we’ll display the pairs:

(The Sage cells in this post are linked, so things may not work if you don’t execute them in order.)

Many of the following properties of characters can be deduced from properties of the trace:

  1. The dimension of a character is the dimension of $V$ in $(V,\rho)$. Since $\rho(\text{Id})$ is always the identity matrix, the dimension of $\chi$ is $\chi(\text{Id})$.
  2. Because the trace is invariant under similarity transformations, $\chi(hgh^{-1}) = \chi(g)$ for all $g,h \in G$. So characters are constant on conjugacy classes, and are thus class functions.
  3. Let $\chi_V$ denote the character of $(V,\rho)$. Recalling the definitions of direct sums and tensor products, we see that

The Character Table

Let’s ignore the representation $\rho$ for now, and just look at the character $\chi$:

This is succinct, but we can make it even shorter. From point 2 above, $\chi$ is constant on conjugacy classes of $G$, so we don’t lose any information by just looking at the values of $\chi$ on each conjugacy class:

Even shorter, let’s just display the values of $\chi$:

This single row of numbers represents the character of one representation of $G$. If we knew all the irreducible representations of $G$ and their corresponding characters, we could form a table with one row for each character. This is called the character table of $G$.

Remember how we had to define our representations by hand, one by one? We don’t have to do that for characters, because SageMath has the character tables of small groups built-in:

This just goes to show how important the character of a group is. We can also access individual characters as a functions. Let’s say we want the last one:

Notice that the character we were playing with, $[4,2,0,0,0]$, is not in the table. This is because its representation $\rho$ is not irreducible. At the end of the post on decomposing representations, we saw that $\rho$ splits into two $1$-dimensional irreducible representations and one $2$-dimensional one. It’s not hard to see that the character of $\rho$ is the sum of rows 1,4 and 5 in our character table:

Just as we could decompose every representation of $G$ into a sum of irreducible representations, we can express any character as a sum of irreducible characters.

The next post discusses how to do this easily, by making use of the Schur orthogonality relations. These are really cool relations among the rows and columns of the character table. Apart from decomposing representations into irreducibles, we’ll also be able to prove that the character table is always square!

Edit: The promised “next post” about these topics never happened. Maybe sometime in the far future, I might come back to these topics, but no promises for now!

Written on March 20, 2015