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We study the structure of the representation ring of the symmetric group in dimension six. The Kronecker coefficients and all power formulas of irreducible representations of $S_6$ are computed using the character theory of finite groups. In addition, by direct sum decomposition of tensor products of different irreducible representations of $S_6$, we characterise generators of the representation ring $\mathcal{R}(S_6)$, show that its unit group $U(\mathcal{R}(S_6))$ is a Klein four-group and related results on the structure of primitive idempotents. Furthermore, we introduce Harrison center theory to study the representation ring and show that the Harrison center of the cubic form induced by the generating relations of $\mathcal{R}(S_6)$ is isomorphic to itself. Finally, we conclude with some open problems for future consideration., Comment: 21 pages; revised version