In this paper, we introduce a method for computing the primitive decomposition of idempotents in any semisimple finite group algebra, utilizing its matrix representations and Wedderburn decomposition. Particularly, we use this method to calculate the examples of the dihedral group algebras $\mathbb{C}[D_{2n}]$ and generalized quaternion group algebras $\mathbb{C}[Q_{4m}]$. Inspired by the orthogonality relations of the character tables of these two families of groups, we obtain two sets of trigonometric identities. Furthermore, a group algebra isomorphism between $\mathbb{C}[D_8]$ and $\mathbb{C}[Q_8]$ is described, under which the two complete sets of primitive orthogonal idempotents of these group algebras correspond bijectively.

The complex representation rings of finite groups are the fundamental class of fusion rings, categorified by the corresponding fusion categories of complex representations. The category of ${\mathbb{Z}}_{+}$-modules of finite rank over such a representation ring is also semisimple. In this paper, we classify the irreducible based modules of rank up to 5 over the complex representation ring $r(S_4)$ of the symmetric group $S_4$. In total, 16 inequivalent irreducible based modules were obtained. In this process, the MATLAB program was used in order to obtain some representation matrices. Based on such a classification result, we further discuss the categorification of based modules over $r(S_4)$ by module categories over the complex representation category $\mathrm{R}\mathrm{e}\mathrm{p}(S_4)$ of $S_4$ arisen from projective representations of certain subgroups of $S_4$.