Second Exercise for Representation Theory

Representation Ring

Prove that $R_{\mathbb{Q}}(C_2){\cong }_{RIng}\mathbb{Z}[x]/<x^2-1>$ where $R_k(G)$ is the representation ring of representations of $G$ over $k$. Equivalently¹,

$$R_k(G)=(\text{Virutal Characters},\oplus ,\otimes ).$$

Solution

Footnotes

1. Serre p. 68 and actually it is a Grothendieck ring. ↩