Decomposition Formula
Definition 2.5 – Decomposition Formula
Given some representation $D(G)$ (not an irrep) of some group $G$ of order $n$, the representation can be decomposed into
$$ D(G) \simeq \sum_{i} a_{i} D^{(i)}(G) $$Where $D^{(i)}(G)$ is some irrep of $G$ and $a_{i}$ is the number of times it appears in the equivalent block diagonal representation of $D(G)$.
For some element $g \in G$ belonging to the equivalence class $\mathcal{C}_{r}$, the trace is
$$ \mathrm{Tr}D(g) = \chi(\mathcal{C}_{r}) $$The coefficient $a_{i}$ can then be determined from the decomposition formula.
$$ a_{i} = \frac{1}{n} \sum_{r} p_{r} \chi^{(i)}(\mathcal{C}_{r})^{*} \chi(\mathcal{C}_{r}) $$Purpose
The main goal of representation is to figure out whether
- A given representation is irreducible or not and,
- If it is not, to decompose it into irreps.
Derivation
Consider some group $G$ of order $n$. Suppose we have a representation $D(G)$ which is not an irrep. $D(G)$ must then be equivalent to a block diagonal representation.
$$ D(G) \simeq \left(\begin{array}{c|c|c|c} D^{(a)} & \mathbb{0} & \mathbb{0} & \cdots \\ \hline \mathbb{0} & D^{(b)} & \mathbb{0} & \cdots \\ \hline \mathbb{0} & \mathbb{0} & D^{(c)} & \cdots \\ \hline \vdots & \vdots & \vdots & \ddots \end{array}\right) $$If some irrep $D^{(j)}$ appears $a_{j}$ times in the representation, then we can write the expression as a direct sum of irreps, each multiplied by $a_{j}$.
$$ D(G) \simeq a_{a} D^{(a)}(G) + a_{b}D^{(b)}(G) + a_{c}D^{(c)}(G) + \cdots = \sum_{j} a_{j} D^{(j)}(G) $$Consider now the trace of some element $g \in \cal{C}_{r}$.
$$ \chi(\mathcal{C}_{r}) = \mathrm{Tr}(D(g)) = \mathrm{Tr}\left[ \sum_{j}a_{j}D^{(j)}(g) \right] = \sum_{j} a_{j} \mathrm{Tr}\left[ D^{(j)}(g) \right] = \sum_{j} a_{j}\chi^{(j)}(\mathcal{C}_{r}) $$Now, we can apply the first orthogonality relation to this expression.
$$ \begin{align} \sum_{r} p_{r} \chi^{(i)}(\mathcal{C}_{r})^{*} \chi(\mathcal{C}_{r}) &= \sum_{r} \chi^{(i)}(\mathcal{C}_{r})^{*} \left[ \sum_{j} a_{j} \chi^{(j)}(\mathcal{C}_{r}) \right] \\ &= \sum_{j}a_{j}\left[ \sum_{r}p_{r} \chi^{(i)}(\mathcal{C}_{r})^{*} \chi^{(j)}(\mathcal{C}_{r}) \right] \\[6pt] &= \sum_{_{j}}a_{j}n\delta_{ij} = na_{i} \end{align} $$Thus, we obtain the decomposition formula.
$$ a_{i} = \frac{1}{n} \sum_{r} p_{r} \chi^{(i)}(\mathcal{C}_{r})^{*} \chi(\mathcal{C}_{r}) $$