Representation ring

Remarks

By Maschke's theorem every representation is a direct sum of irreducible representations, uniquely up to order. So $R(G)$ is free abelian with basis the classes of the irreducible representations, and its rank is the number of conjugacy classes of $G$. The character $\chi_V$ satisfies $\chi_{V \oplus W} = \chi_V + \chi_W$ and $\chi_{V \otimes W} = \chi_V \chi_W$, so it gives an injective ring homomorphism from $R(G)$ to the ring of class functions $G \to \C$. Example: for $G = \Z/3\Z$, the irreducible representations are $1, \rho, \rho^2$, where $\rho$ sends a generator to $e^{2\pi i/3}$, and $R(G) \cong \Z[\rho]/(\rho^3 - 1)$. Non-example: the element $[1] - [\rho]$ of $R(\Z/3\Z)$ is not the class of a representation, since its multiplicity of $\rho$ is negative. The same construction works over other fields and for compact groups with continuous finite-dimensional representations.