Direct Product (Group)

The expression of a Group as a product of Subgroups. The Characters of the representations of a direct product are equal to the products of the Characters of the representations based on the individual sets of functions. For $R_1$ and $R_2$,

$$\chi (R_1 \otimes R_2) = \chi (R_1) \chi (R_2).$$

The representation of a direct product $\Gamma_{AB}$ will contain the totally symmetric representation only if the irreducible $\Gamma_A$ equals the irreducible $\Gamma_B$.