Example 5

Suppose that G is the product $G' \times G''$ of two groups. The K-linear mapping from $K[G'] \otimes_K K[G'']$ to $K[G]$ that sends $g' \otimes g''$ to $(g', g'')$ for $g' \in G'$ and $g'' \in G''$ is an algebra isomorphism. Let $(M', \pi')$ be a linear representation of $G'$ and $(M'', \pi'')$ a linear representation of $G''$. The external tensor product of $\pi'$ and $\pi''$, denoted by $\pi' \otimes \pi''$, is the representation of $G' \times G''$ in the vector space $M' \otimes M''$ defined by $(\pi' \otimes \pi'')(g', g'') = \pi'(g') \otimes \pi''(g'')$ for $(g', g'') \in G' \times G''$. If $M'$ and $M''$ are finite-dimensional free K-modules, then $M' \otimes_K M''$ is a finite-dimensional free K-module and the character of the representation $\pi' \otimes \pi''$ is given by the formula

for $g' \in G'$ and $g'' \in G''$ (Proposition 2 of III, §9, No. 2, p. 542).