Example 6

Let $(V, \pi)$ be a linear representation of G such that V is a finite-dimensional free K-module. We define a representation $\rho$ of $G \times G$ in $\operatorname{End}_K(V)$ by the formula

The K-module isomorphism $\theta_V : V^* \otimes_K V \to \operatorname{End}_K(V)$ of II, §4, No. 2, p. 271 is an isomorphism of representations from the external tensor product $\pi^\vee \otimes \pi$ to $\rho$. The mapping $g \mapsto \rho(1, g)$ (resp. $g \mapsto \rho(g, 1)$) is a representation of G isomorphic to $\pi^{\dim_K V}$ (resp. $(\pi^\vee)^{\dim_K V}$).

Let L be a commutative K-algebra, and let $(M, \pi)$ be a linear representation of the group G. The group homomorphism $\pi_{(L)} : G \to \mathbf{GL}(M_{(L)})$ defined by $g \mapsto \operatorname{Id}_L \otimes \pi(g)$ is a linear representation of G in the L-module $M_{(L)}$, called the linear representation of G deduced from the representation $\pi$ by extension of the ring of scalars K to L.

Suppose that K is a field and that L is a nonzero commutative K algebra. Let $(M, \pi)$ and $(M', \pi')$ be linear representations of G. The representations $\pi$ and $\pi'$ are isomorphic if and only if $\pi_{(L)}$ and $\pi'_{(L)}$ are (VIII, p. 37, Theorem 3).

Suppose, moreover, that the algebra L is an extension of K. Consider the rings $R_K(G)$ and $R_L(G)$ defined in Example 1 of VIII, p. 198. Extension of scalars defines a ring homomorphism

This homomorphism is injective, and an element $\xi \in R_K(G)$ is effective if and only if $u(\xi )$ is (VIII, p. 195, Theorem 1).