Reduced matrix element ($A(e_i\otimes v_s)={\sum }_{\nu }{g}_{\nu }{C}_{ris}^{\nu }{v}_r$)

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.

An octet operator between baryon octet states has two reduced matrix elements, because the tensor square of the SU(3) adjoint representation contains two octets. The symmetric invariant $d_{ris}$ and antisymmetric invariant $f_{ris}$ give independent component tensors. Equivalently, represent baryons by traceless matrices $B$ and let the operator act as $F[t_i,B]+D(\{t_i,B\}-\frac{2}{3}\mathrm{tr}(t_{i}B)I)$. Normalization choices are absorbed into $F,D$; Hermitian operators permit real coefficients in a Hermitian octet basis.