Isaacs, $\textit{Character Theory of Finite Groups}$, Lemma(2.10)

If $g\in G$ and $g\neq1$, then $\rho(g)=0.$ Also $\rho(1)=|G|$.

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Pf: Obviously.

Isaacs, $\textit{Character Theory of Finite Groups}$, Lemma(2.11)

$$\rho=\sum_{i=1}^k\chi_i(1)\chi_i.$$

Pf:

  • $\mathbb{C}[G]=\bigoplus_{M_i\in\mathcal{M}(\mathbb{C}[G])}M_i(\mathbb{C}[G])$.
  • $n_{M_i}(\mathbb{C}[G])=dim(M_i)=\chi_i(1)$

 

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