The Unapologetic Mathematician

Mathematics for the interested outsider

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 Posted by | Algebra, Group theory, Representation Theory | 7 Comments

   

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