Representations of finite groups; a matrix view

Posted: February 8, 2011 in Representations of Finite Groups
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Throughout $G$ is a finite group and $\rho : G \longrightarrow \text{GL}(V)$ is a representation of $G.$

We defined an irreducible representation in here. The definition can also be given in terms of matrices.

Theorem 1. Let $\deg \rho = \dim_{\mathbb{C}} V=n.$ Then $\rho$ is reducible if and only if there exist a $\mathbb{C}$-basis $\mathcal{B}$ of $V$ and an integer $1 \leq m < n$ such that for all $g \in G$

$[\rho(g)]_{\mathcal{B}} = \begin{pmatrix} A & B \\ 0 & C \end{pmatrix},$

where $0$ here means an $m \times m$ zero matrix. The matrices $A,B$ and $C$ depend on $g.$

Proof. Well, by definition, $\rho$ is reducible if and only if $V$ is not simple as a $\mathbb{C}[G]$ – module. So $\rho$ is reducible if and only if $V$ has a non-zero $\mathbb{C}[G]$-submodule $W \neq V.$ Since $\mathbb{C}[G]$ is a semisimple ring (see the corollary in this post), $V$ is a semisimple $\mathbb{C}[G]$-module. Thus $V=W \oplus U,$ for some $\mathbb{C}[G]$-submodule $U$ of $V.$ Now, choose a $\mathbb{C}$-basis $\{v_1, \cdots , w_m \}$ for $W$ and extend it to a basis $\mathcal{B}$ for $V.$ Note that $1 \leq m < n,$ because $W \neq (0), V.$ Let $g \in G.$ Since $W$ is a $\mathbb{C}[G]$-module, we have $\rho(g)(v_i) \in W,$ for all $1 \leq i \leq m.$ So $\rho(g)(v_i)$ is a $\mathbb{C}$-linear combinations $v_1, \cdots , v_m.$ It is now clear that the matrix $[\rho(g)]_{\mathcal{B}}$ looks like the matrix given in the theorem. Conversely, if for some basis $\mathcal{B}=\{v_1, \cdots , v_n \}$ of $V,$ the matrix $[\rho(g)]_{\mathcal{B}}$ looks like the one given in the theorem, then we let $W = \text{span} \{v_1, \cdots , v_m \}.$ It is clear from the form of $[\rho(g)]_{\mathcal{B}}$ that $\rho(g)(v_i) \in W,$ for all $1 \leq i \leq m$ and hence $W$ is a $\mathbb{C}[G]$-submodule of $V.$ Thus $V$ cannot be irreducible. $\Box$

Theorem 2. There exists a basis $\mathcal{B}$ of $V$ and square matrices $A_1, \cdots , A_k$ such that for every $g \in G$

$[\rho(g)]_{\mathcal{B}} = \begin{pmatrix} A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ . & . & \cdots & . \\ . & . & \cdots & . \\ 0 & 0 & \cdots & A_k \end{pmatrix}.$

Proof. Since $V$ is a semisimple $\mathbb{C}[G]$-module, $V = \bigoplus_{i=1}^k V_i,$ for some integer $k$ and some simple $\mathbb{C}[G]$-submodules $V_i$ of $V.$ In fact, from the theory of semisimple modules, it is clear that $V_1, \cdots , V_k$ are all simple $\mathbb{C}[G]$-submodules of $V.$ Now, choose a $\mathbb{C}$– basis $\mathcal{B}_i$ for each $V_i$ and let $A_i =[\rho(g)|_{V_i}]_{\mathcal{B}_i}.$ Let $\mathcal{B} = \bigcup_{i=1}^k \mathcal{B}_i.$ Obviously $\mathcal{B}$ is a basis for $V$ and $[\rho(g)]_{\mathcal{B}}$ is the matrix given in the theorem. $\Box$