Second Exercise Solutions

Representation Ring

By the Maschke’s Theorem, $\mathbb{Q}{C}_2$ is a Semisimple ring. Then by the Artin-Weddenburn Theorem, Simple Ring and Artin-Wedderburn Theorem.

$$1_{\mathbb{Q}{C}_2}=(1/2e+1/2g)+(1/2e-1/2g).$$

Let ${\chi }_{\pm }$ be the simple characters for each non trivial central idempotents. So that ${\chi }_{\mathbb{Q}{C}_2}=n_{+}{\chi }_{+}+n_{-}{\chi }_{-}$.

Upon to Serre’s defintion, we have Grothendieck Group

$$R_{\mathbb{Q}}(C_2)=\mathbb{Z}{\chi }_{+}\oplus \mathbb{Z}{\chi }_{-}.$$

Note that

$$g=(1/2e+1/2g)+(-1/2e+1/2g)=(1/2e+1/2g)-(1/2e-1/2g).$$

and $\chi (g^2)=\chi (g{)}^2=\chi (e)$

$$\Rightarrow n_{+}^2=n_{+}{n}_{-}^2=n_{-},n_{+}{n}_{-}=0.$$