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Given a ring R and a group $G$, I can consider the group ring $R[G]$ and then take the algebraic $K$-theory $K(R[G])$. This the $K$-theory of the category $\operatorname{Rep}_R(G)$. As a variant, one ...

According to Carters Lower K-theory of finite groups for a finite group $G$ we have $$ K_{-1} (\mathbb Z G) = \mathbb Z^r \oplus \mathbb Z_2^s $$ where $s$ is the sum over all irreducible ...