# 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 -

1. [...] dealing with characters, there’s something we can do to rework our expression for the inner product on the space of class [...]

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2. [...] course an irreducible character — like all characters — is a class function. We can describe it by giving its values on each conjugacy class. And so we lay out the [...]

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3. [...] Inner Products in the Character Table As we try to fill in the character table, it will help us to note another slight variation of our inner product formula: [...]

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4. [...] of the subspace it spans. Of course whatever subspace this is, it has to fit within the space of class functions, and so it can’t have any more basis elements than the dimension of this larger space. That [...]

Pingback by Consequences of Orthogonality « The Unapologetic Mathematician | October 25, 2010 | Reply

5. [...] Now let be any irreducible representation of , with character . We know that the multiplicity of in is given by the inner product . This, we can calculate: [...]

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6. [...] a quick corollary, we find that the irreducible characters span a subspace of the space of class functions with dimension equal to the number of conjugacy classes in . Since this is the dimension of the [...]

Pingback by The Character Table is Square « The Unapologetic Mathematician | November 19, 2010 | Reply

7. [...] table is square, we know that irreducible characters form an orthonormal basis of the space of class functions. And we also know another orthonormal basis of this space, indexed by the conjugacy classes [...]

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