Proposition 5

The family $(\chi_{\lambda})_{\lambda\in\widehat{G}}$ is a basis of the vector space of central functions. The number of classes of simple linear representations of G is equal to the number of conjugacy classes of G.

For $a=\sum_{g\in G}a_gg\in K[G]$, write $a^{\vee}=\sum_{g\in G}a_{g^{-1}}g$; the mapping $a\mapsto a^{\vee}$ is an involutive antiautomorphism of the algebra K[G]. By formula (17), we have $e_{\lambda}=|G|^{-1}d_{\lambda}\chi^{\vee}_{\lambda}$ for every $\lambda \in \widehat{G}$. The proposition then follows from the fact that the family $(e_{\lambda})_{\lambda\in\widehat{G}}$ is a basis of the center of K[G].

Denote by $\mathscr{C}$ the set of conjugacy classes of G. Let C be an element of $\mathscr{C}$; the image $C^{-1}$ of C by the mapping $g\mapsto g^{-1}$ is a conjugacy class. The commutants of the elements of C are mutually conjugate subgroups of G; their cardinal $d(C)$ satisfies

(30) Card(G) = Card(C) $d(C)$.

In particular, we have $d(C)\cdot 1\not= 0$ in the field K.

Let $f$ be a central function on G. For any conjugacy class C, denote by $f(C)$ the common value of the $f(x)$ for $x\in C$. With this notation, the orthogonality relation for characters (VIII, p. 410, Proposition 4) can be written as

for $\lambda$ and $\mu$ in $\widehat{G}$.

Denote by A the matrix of type $\widehat{G}\times \mathscr{C}$ with entries $\chi_{\lambda}(C)$ and by B the matrix of type $\mathscr{C}\times \widehat{G}$ with entries $\chi_{\lambda}(C^{-1})d(C)^{-1}$. The sets $\widehat{G}$ and $\mathscr{C}$ have the same cardinal (Proposition 5); relation (31) expresses the fact that the matrix product AB is the matrix unit of type $\widehat{G}\times \widehat{G}$. By Proposition 11 of II, §10, No. 12, p. 360, the matrix product BA is the matrix unit of type $\mathscr{C}\times \mathscr{C}$; in other words, we have the relation

for C and $C'$ in $\mathscr{C}$ (sometimes called the “second orthogonality relation for characters”).

Let H be a subgroup of G. Let us note that the integer Card(H) divides Card(G) and that $|G|$ is not zero in K; we therefore have $|H| \not= 0$ in K.

Denote by Res$^G_H$ the linear mapping from Z(K[G]) to Z(K[H]) that sends a central function on G to its restriction to H. We have seen that if $\chi_{\pi}$ is the character of a finite-dimensional representation $\pi$ of G, then Res$^G_H(\chi_{\pi})$ is the character of the representation Res$^G_H(\pi )$ of H.