Corollary 2

a) The family of characters of the irreducible representations of $G$ is a basis of $\Theta (G^{\sharp})$;

b) The mapping $u: R(G)\otimes_{\mathbf{Z}}\mathbf{C}\rightarrow \Theta (G^{\sharp})$ such that $u(\pi \otimes 1) =\chi_{\pi}$ for every $\pi \in \widehat{G}$ is an isomorphism of algebras over $\mathbf{C}$.

The first assertion results from Proposition 9 and Corollary 1.

Since the classes of the representations $\pi \in \widehat{G}$ form a basis of the free $\mathbf{Z}$-module $R(G)$, the mapping $u$ is well-defined. It is a morphism of $\mathbf{C}$-algebras, and it is an isomorphism by a).