Abstract Nonsense

Dimension of the Space of Class Functions

Point of post: In this post we prove the simple result that the dimension of the space of class functions is equal to the number of conjugacy classes in $G$.

Motivation

In our last post we hinted that the dimensionality of the space $\text{Cl}(G)$ of class functions of the finite group $G$ shall be used to derive a very interestint result. As a step toward this we prove in this post that the dimension of $\text{Cl}(G)$ thought of as a subspace of the group algebra $\mathcal{A}(G)$ is the number of conjugacy classes of $G$ .

Dimension of $\text{Cl}(G)$

Let $G$ be a finite group and $\text{Cl}(G)$ the space of all class functions on $G$. Furthermore, let $C_1,\cdots,C_k$ be the  distinct conjugacy classes of $G$ then:

Theorem: $\dim_{\mathbb{C}}\text{Cl}(G)=k$.

Proof: For each $j\in[k]$ let (as usual) $\mathbf{1}_{C_j}:G\to\mathbb{C}$ denote the indicator function of $C_j$. We claim that $\left\{\mathbf{1}_{C_j}\right\}_{j\in[k]}$ forms a basis for $\text{Cl}(G)$ (note that each $\mathbf{1}_{C_j}$ is evidently a class function itself since it’s constant on the conjugacy classes of $G$). Clearly $\left\{\mathbf{1}_{C_j}\right\}_{j\in[k]}$ is linearly independent since if $c_1,\cdots,c_k\in\mathbb{C}$ were such that

$\displaystyle \sum_{j=1}^{k}c_j\mathbf{1}_{C_j}=\bold{0}$

then, choosing some representative $r_\ell\in C_\ell$ for $\ell\in[k]$ we may conclude by the pairwise-disjointness of $\left\{C_j:j\in[k]\right\}$ that

$\displaystyle 0=\sum_{j=1}^{k}c_j\mathbf{1}_{C_j}(r_\ell)=c_\ell$

and thus $c_1=\cdots=c_k=0$.

Now to see that $\displaystyle \text{span}\{\mathbf{1}_{C_j}\}_{j\in[k]}=\text{Cl}(G)$ we merely note that by previous chatracterizations of class functions for each $f\in\text{Cl}(G)$ one has that $f(C_j)$ is a singleton say $s_j$. We claim then that

$\displaystyle f=\sum_{j=1}^{k}s_j\mathbf{1}_{C_j}$

But, this is indeed obvious since given any $x\in G$ we have that $x\in C_\ell$ for precisely one $\ell\in [k]$ and

$\displaystyle f(x)=s_\ell=\sum_{j=1}^{k}s_j\mathbf{1}_{C_j}(x)$

Since $x\in G$ was arbitrary the conclusion follows. $\blacksquare$

References:

1.Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

2. Serre, Jean Pierre. Linear Representations of Finite Groups. New York: Springer-Verlag, 1977. Print