Consider a finite group $G$ and assume it decomposes as
$$G\cong\displaystyle\bigoplus_{k=1}^n G_k.$$
Say that I know the character table for all of $G_k$. Can I construct the character table for $G$?
We consider complex irreducible representations, i.e. group homomorphisms $\pi : G\to\operatorname{End}(\mathbb C^n).$
In particular, the case $n=2$ is of interest, since I have a situation like this.
Best Answer
Yes. The complex irreducible representations of $G_1\times G_2$ are tensor products $(V_1\otimes V_2,\rho_1\otimes\rho_2)$ of complex irreducible representations $(V_1,\rho_1)$ of $G_1$ and $(V_2,\rho_2)$ of $G_2$. The character associated to a tensor product is just a product of characters, $\chi_{\small V_1\otimes V_2}(g_1,g_2)=\chi_{V_1}(g_1)\chi_{V_2}(g_2)$. The conjugacy classes of a direct product $G_1\times G_2$ are Cartesian products $K_1\times K_2$ of conjugacy classes $K_1$ of $G_1$ and $K_2$ of $G_2$. Therefore, the character table of $G_1\times G_2$ is just the "Kronecker product" of the tables for the groups $G_1$ and $G_2$ (viewing the tables as matrices).