每日定理12

Isaacs, $\textit{Character Theory of Finite Groups}$, Theorem(2.4)

Let $\mathcal{K}_1,~\mathcal{K}_2,\cdots,~\mathcal{K}_r$ be the conjugacy classes of a group $G$. Let $K_i=\sum_{x\in\mathcal{K}_i}x\in\mathbb{C}[G]$. Then the $K_i$ form a basis for $\mathbf{Z}(\mathbb{C}[G])$ and if $K_iK_j=\sum a_{ijv}K_v$, then the multiplication constants $a_{ijv}$ are nonnegative integers.

Pf:

• If $z=\sum a_gg\in\mathbf{Z}(\mathbb{C}[G])$ and $h\in G$, we have $z=h^{-1}zh=\sum a_gg^h=\sum a_gg$.
• Pick $g\in\mathcal{K}_v$. Then $a_{ijv}$ is the coefficient of $g$ in $K_iK_j$. That is $|\{(x,y)|x\in\mathcal{K}_i,y\in\mathcal{K}_j,xy=g\}|$.

每日定理12

原文：https://www.cnblogs.com/zhengtao1992/p/10803528.html