# Each irreducible representation of $G$ is of the form $\rho_1\otimes\rho_2$, for some irreducible representations of $G1$ and $G2$

Each irreducible representation of $$G_1\times G_2$$ is isomorphic to a representation $$\rho_1\otimes\rho_2$$, where $$\rho_i$$ is an irreducible representation of $$G_i$$ for $$i=1,2.$$

So, this is the theorem $$10$$ in Jean-Pierre Serre book "Linear Representations of Finite Groups".

I understood half of the proof, but there is a line which he says: "it suffices to show that each class function $$f$$ on $$G_1\times G_2$$, which is orthogonal to the characters on the form of $$\chi_1(s_1)\chi_2(s_2)$$, is zero".

I don't get why showing that each class function is zero proves that each irreducible representation of $$G$$ is of the form $$\rho_1\otimes\rho_2$$, my guess is that this idea has some relation with the theorem 6 of the book that says: "The characters $$\chi_1,...,\chi_h$$ form an orthonormal basis of $$H$$", but i can't fully comprehend this.

• If the given characters did not form a basis, then there would be some non-zero element in the orthogonal complement of their span. He shows that there is not. Mar 30 '20 at 17:08
• And why this proves that each irreducible representation is of the form i cited above? Mar 30 '20 at 17:15
• Because the characters mentioned there are precisely the characters of the representations claimed to form a basis. Mar 30 '20 at 17:17

Here you have and alternative (and I think more understandable and full detailed) proof of the result you want to prove.

$$\textbf{Proposition}$$ Let $$\phi^1:G_1\to \text{GL}(V)$$ and $$\phi^2:G\to \text{GL}(V)$$ two group representations of $$G_1$$ and $$G_2$$ with dimensions $$d_1$$ and $$d_2$$, respectively, over the same vectorial space $$V$$. Denoting as $$D^1(g^1_i)$$ y $$D^2(g_j^2)$$ the matrices of the representations $$\phi^1$$ and $$\phi^2$$, then $$D^1(g^1_i) \otimes D^2(g_j^2)$$ is a representation with dimension $$d_1d_2$$ of $$G_1\times G_2$$.

$$\textit{Proof}.$$ By the properties of the Kronecker product, it followes that $$\begin{equation*} \begin{split} [D^1(g_i^1)\otimes D^2(g_j^2)][D^1(g_k^1)\otimes D^2(g_l^2)] & =[D^1(g_i^1)D^1(g_k^1)]\otimes [D^2(g_j^2)D^2(g_l^2)]\\ &=D^1(g_i^1g_k^1)\otimes D^2(g_j^2g_l^2). \end{split} \end{equation*}$$

$$\textbf{Theorem}$$ With the same notation, if the representations $$\phi^1$$ and $$\phi^2$$ are irreducible then $$D^1(g^1_i) \otimes D^2(g_j^2)$$ of hte group $$G_1\times G_2$$ is also irreducible. Moreover, all the irreducible representations of $$G_1\times G_2$$ are the direct product of an irreducible representation of $$G_1$$ times other of $$G_2$$.

$$\textit{Proof}$$ It is known that the irreducibility of $$\phi^1$$ and $$\phi^2$$ is equivalent to the condition $$\begin{equation*} \sum_{g\in G_1} \chi^1(g)^*\chi^1(g)=|G_1| \quad \text{y} \quad \sum_{h\in G_2} \chi^2(h)^*\chi^2(h)=|G_2|, \end{equation*}$$ where $$\chi^1$$ and $$\chi^2$$ are the characters of $$\phi^1$$ y $$\phi^2$$, respectively. Then, $$\begin{equation*} \begin{split} |G_1\times G_2|& =|G_1||G_2|=\left(\sum_{g\in G_1} \chi^1(g)^*\chi^1(g)\right)\left(\sum_{h\in G_2} \chi^2(h)^*\chi^2(h)\right)\\ &=\sum_{g\in G_1}\sum_{h\in G_2} \left(\chi^1(g)\chi^2(h)\right)\left(\chi^1(g)^*\chi^2(h)^*\right). \end{split} \end{equation*}$$

On the other hand, using that the characters of the Kronecker product of the two representations is the product of the characters of the two representations, then $$\begin{equation*} |G_1\times G_2|=\sum_{g'\in G_1\times G_2} \chi(g')\chi(g')^*, \end{equation*}$$ where $$\chi$$ are the characters of the representation on the direct product $$G_1\times G_2$$. Thus, we infer that the representation $$D^1(g^1_i) \otimes D^2(g_j^2)$$ of the group $$G_1\times G_2$$ is also irreducible.

For the second part, let $$m_1$$ and $$m_2$$ the number of irreducible representations of $$G_1$$ and $$G_2$$, respectively, whose dimensions are denoted as $$d_{i,1}$$ and $$d_{j,2}$$. Hence, $$\begin{equation*} |G_1|=\sum_{i=1}^{m_1} d_{i,1}^2 \quad \text{y} \quad |G_2|=\sum_{j=1}^{m_2} d_{j,2}^2. \end{equation*}$$ The irreducible representations of $$G_1\times G_2$$ that are obtained as the Kronecker product of irreducible representations of $$G_1$$ and $$G_2$$ will have dimension $$d_{i,1}d_{j,2}$$. Therefore, the sum of the squares of the dimensions $$d_k$$ of the irreducible representations $$G_1\times G_2$$ is $$\begin{equation*} \sum_{k=1}^{m_1m_2}d_k^2= \sum_{i=1}^{m_1} \sum_{j=1}^{m_2} d_{i,1}^2d_{j,2}^2 = |G_1||G_2|=|G_1\times G_2|. \end{equation*}$$ We conclude that the Kronecker product of the irreducible representations of $$G_1$$ and $$G_2$$ determine all the irreducible representacions of $$G_1\times G_2$$.