Non-Induced Representations of Finite Cyclic Groups

Abstract

Let $K$ be an algebraically closed field of characteristic $0$ and let $G$ be a finite cyclic group of order $n$. In this note we prove, using induction on the number of prime divisors of $n$, that $R_K(G)/I \cong \mathbb{Z}[X]/\langle \Phi_n(X) \rangle$ where $R_K(G)$ denotes the ring of $K$-representations of $G$ and $I$ is the sum of ideals $\mathrm{Ind}_H^G(R_K(H))$ of $R_K(G)$ as $H$ varies over all proper subgroups of $G$. This gives us an idea of how many representations of $G$ are not induced from representations of a proper subgroup.