Equivalence, Irreducibility and Characters
For any finite group, there is a specific number of distinct representations. Furthermore, their most important properties can be determined systematically.
Equivalence
Definition 2.2 – Equivalent Representations
Consider the representations $D(G)$ and $D'(G)$. If there exists a similarity transformation such that
$$ D'(g_{a}) = S^{-1} D(g_{a})S $$$$ \begin{align} D'(g_{a})D'(g_{b}) &= S^{-1}D(g_{a})S S^{-1} D(g_{b}) S = S^{-1} D(g_{a})D(g_{b}) S \\[6pt] &= S^{-1} D(g_{a}g_{b}) S = D'(g_{a}g_{b}) \end{align} $$Then the representations $D(G)$ and $D'(G)$ are called equivalent. Otherwise, the representations are non-equivalent.
When identifying the distinct representations of a group, we consider any mutually equivalent representations as one. Our interest lies in non-equivalent representations of a group.
Direct Sum Representations
Consider at least two representations of a group: $D^{(1)}(G)$ and $D^{(2)}(G)$. We can construct a third representation by taking the direct sum of them.
$$ D'(G) = D^{(1)}(G) \oplus D^{(2)}(G) = \left( \begin{array}{c|c} D^{(1)}(G)_{n_{1} \times n_{1}} & \mathbb{0}_{n_{1} \times n_{2}} \\ \hline \mathbb{0}_{n_{2} \times n_{1}} & D^{(2)}(G)_{n_{2} \times n_{2}} \end{array} \right) $$Note that there is nothing new about this representation. Everything we know about $D^{(1)}(G)$ and $D^{(2)}(G)$, we know about $D'(G)$.
Irreducibility
Definition 2.3 – Reducible Representations
A representation is reducible if it is diagonal by blocs, such as $D'(G)$, the direct sum representation above.
If no similarity transformation exists to render a representation block diagonal, that representation is said to be irreducible.
When identifying the distinct representations of a group, we are looking for all non-equivalent and irreducible representations. Any other representation can be constructed by similarity transformations and direct sums of these fundamental representations.
The Trivial Representation
Every group has a trivial representation, $D^{(1)}(G)$ where each element maps to the scalar one.
$$ D^{(1)}(g) = 1 \quad \forall \space g \in G $$Character
Definition 2.4 – Representation and Element Characters
The character of a representation element is the trace of that element’s matrix.
$$ \chi(g) = \mathrm{Tr}D(g) $$The character of a representation is the set of all distinct characters of its elements.
$$ \chi(G) = \{ \chi(e), \chi(g_{2}), \chi(g_{3}), \dots, \chi(g_{n}) \} $$The character of equivalent representations is identical.
$$ \mathrm{Tr}D' = D'_{\alpha\alpha} = (S^{-1} D S)_{\alpha \alpha} = S^{-1}_{\alpha\beta} D_{\beta\gamma} S_{\gamma\alpha} = D_{\beta\gamma}S_{\gamma\alpha}S^{-1}_{\alpha\beta} = D_{\beta\gamma} \delta_{\gamma\beta} = D_{\beta\beta} = \mathrm{Tr}D $$So, the characters of representations will play a key role in classifying the non-equivalent and irreducible representations of a group.