Remark

We identify the algebra End$_K(V_{\lambda})$ with the matrix algebra $\mathbf{M}_{d_{\lambda}}(K)$ via the basis $(e_{\lambda ,j})$ of $V_{\lambda}$. The mapping $\mathscr{F}^{-1}$ is an isomorphism from the algebra $\prod_{\lambda}\mathbf{M}_{d_{\lambda}}(K)$ to the algebra K[G]. For $\mu\in \widehat{G}$, denote by $E^\mu_{ij}$ the element of $\prod_{\lambda}\mathbf{M}_{d_{\lambda}}(K)$ whose component of index $\mu$ is the matrix unit $E_{ij}$ of $\mathbf{M}_{d_\mu}(K)$ (II, §10, No. 3, p. 341) and whose other components are zero; set $u^\mu_{ij}=\mathscr{F}^{-1}(E^\mu_{ij})$. The family of elements $u^{\lambda}_{ij}$, for $\lambda \in \widehat{G}, 1\leqslant i\leqslant d_{\lambda}$, $1\leqslant j\leqslant d_{\lambda}$, is a basis of the algebra K[G]; the multiplication table is

Moreover, by formula (15), we have