# 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 $\{\sqrt{\lvert Z_k\rvert}f_K\}$.

## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
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