 Multiplicity of an irreducible representation

In a completely reducible group representation $V={⨁}_{\lambda }{m}_{\lambda }{V}_{\lambda }$, the nonnegative integer $m_{\lambda }$ counts the copies of the corresponding irreducible representation. For complex representations of a finite group, character orthogonality gives $m_{\lambda }=⟨{\chi }_{\lambda },{\chi }_V⟩$.