# The Unapologetic Mathematician

## Class Functions

Our first observation about characters takes our work from last time and spins it in a new direction.

Let’s say $g$ and $h$ are conjugate elements of the group $G$. That is, there is some $k\in G$ so that $h=kgk^{-1}$. I say that for any $G$-module $V$ with character $\chi$, the character takes the same value on both $g$ and $h$. Indeed, we find that

\displaystyle\begin{aligned}\chi(h)&=\chi\left(kgk^{-1}\right)\\&=\mathrm{Tr}\left(\rho\left(kgk^{-1}\right)\right)\\&=\mathrm{Tr}\left(\rho(k)\rho(g)\rho(k)^{-1}\right)\\&=\mathrm{Tr}\left(\rho(g)\right)\\&=\chi(g)\end{aligned}

We see that $\chi$ is not so much a function on the group $G$ as it is a function on the set of conjugacy classes $K\subseteq G$, since it takes the same value for any two elements in the same conjugacy class. We call such a complex-valued function on a group a “class function”. Clearly they form a vector space, and this vector space comes with a very nice basis: given a conjugacy class $K$ we define $f_K:G\to\mathbb{C}$ to be the function that takes the value $1$ for every element of $K$ and the value $0$ otherwise. Any class function is a linear combination of these $f_K$, and so we conclude that the dimension of the space of class functions in $G$ is equal to the number of conjugacy classes in $G$.

The space of class functions also has a nice inner product. Of course, we could just declare the basis $\{f_K\}$ to be orthonormal, but that’s not quite what we’re going to do. Instead, we’ll define

$\displaystyle\langle\chi,\psi\rangle=\frac{1}{\lvert G\rvert}\sum\limits_{g\in G}\overline{\chi(g)}\psi(g)$

The basis $\{f_K\}$ isn’t orthonormal, but it is orthogonal. However, we can compute:

\displaystyle\begin{aligned}\langle f_K,f_K\rangle&=\frac{1}{\lvert G\rvert}\sum\limits_{g\in G}\overline{f_K(g)}f_K(g)\\&=\frac{1}{\lvert G\rvert}\sum\limits_{k\in K}\overline{f_K(k)}f_K(k)\\&=\frac{1}{\lvert G\rvert}\sum\limits_{k\in K}1\\&=\frac{\lvert K\rvert}{\lvert G\rvert}\end{aligned}

Incidentally, this is the reciprocal of the size of the centralizer $Z_k$ of any $k\in K$. Thus if we pick a $k$ in each $K$ we can write down the orthonormal basis $\{\sqrt{\lvert Z_k\rvert}f_K\}$.

October 15, 2010