Counting irreducible representations via the adjoint representation

Let $G$ be a finite group and $(\mathbb{C}[G], \rho)$ be the (left) regular representation of $G$. By Maschke's theorem, the representation is semisimple and hence can be expressed as $$\mathbb{C}[G] \cong V_1^{\dim V_1}\oplus\cdots\oplus V_m^{\dim V_m}.$$ What makes $(\mathbb{C}[G],\rho)$ a "fundamental" representation is that (1) $\dim\text{Hom}_G(\mathbb{C}[G],V) = \dim V$ and hence, (2) any irreducible representation of $G$ is included in the above decomposition.

Indeed, suppose $W$ is an irreducible representation of $G$ and is not isomorphic to any of the $V_i$'s. Then $$\dim\text{Hom}_G(\mathbb{C}[G], W) = \sum_{i=1}^m \dim V_i \dim\text{Hom}_G(V_i,W) = 0$$ by Schur's lemma, which contradicts $\dim W\ge 1$.

Now to a fundamental result:

Sketch of Proof. Step 1. Construct a linear isomorphism $\Phi:\mathbb{C}[G]\xrightarrow{\sim} \text{End}(V_1)\oplus\cdots\oplus\text{End}(V_m)$ by extending each $\rho_i: G\to \text{GL}(V_i)$ linearly to $\rho_i:\mathbb{C}[G]\to\text{End}(V_i)$ given by $$\rho_i\left(\sum_{g\in G}a_g g\right) = \sum_{g\in G} a_g\rho_i(g)\in \text{End}(V_i),$$ then putting together the extended $\rho_i$'s: $\Phi = (\rho_1,\dots,\rho_m)$.

Step 2. But $\Phi$ is not $G$-linear with respect to the left regular representation $(\mathbb{C}[G],\rho)$. To fix this, we observe that $$\Phi(ghg^{-1}) = \bigoplus_{i=1}^m \rho_{\text{End}(V_i)}(g)(\Phi(h)),$$ hence motivating the adjoint representation $(\mathbb{C}[G],\rho_{ad})$ given by $$\rho_{ad}(g)\left(\sum_{h\in G} a_h h\right) = \sum_{h\in G} a_h ghg^{-1}.$$ This makes $\Phi$ $G$-linear as a map $$\mathbb{C}[G]\xrightarrow{\sim}\bigoplus_{i=1}^m \text{End}(V_i).$$

Step 3. Now take $G$-invariants on both sides: the RHS gives $m$ by Schur's lemma, and the LHS should be the number of conjugacy classes in $G$. Indeed, \begin{align*} f = \sum_{h\in G} a_h h\in (\mathbb{C}[G],\rho_{ad})^G &\iff f = \sum_{h\in G} a_h ghg^{-1} = \sum_{h\in G} a_{g^{-1}hg}h, \ \forall g\in G \\ &\iff a_h = a_{g^{-1}hg}, \ \forall g,h\in G. \end{align*} So $f$ has the same coefficients for $h$'s in the same conjugacy class of $G$. Hence $\left(\sum_{g\in\mathcal{C}_i} g\right)_{i=1}^{m'}$ for the conjugacy classes $\mathcal{C}_1,\dots,\mathcal{C}_{m'}$ of $G$ is a basis for $(\mathbb{C}[G], \rho_{ad})^G$. $\square$

References

1. G. James and M. Liebeck, Representations and characters of groups, Cambridge University Press, 2001.
2. T. Schedler, Group representation theory, Lecture Notes, Imperial College London, 2021.