 Reduced matrix element

A tensor operator transforming in an irreducible representation $R$ between states in $V$ defines an intertwining operator $R\otimes V\to V$. In a basis of these intertwining maps, its scalar coefficients $g_{\nu }$ are reduced matrix elements, independent of the component indices. Schur lemma makes the number of independent coefficients the multiplicity of $V$ in $R\otimes V$. The component dependence is fixed by the associated Clebsch-Gordan coefficients.