Non-Induced Representations of Finite Cyclic Groups

TL;DR

This paper characterizes the quotient of the ring of representations of a finite cyclic group over an algebraically closed field of characteristic zero, by the ideal generated by induced representations from proper subgroups, revealing the structure of non-induced representations.

Contribution

It provides an explicit isomorphism describing the structure of non-induced representations of finite cyclic groups over algebraically closed fields of characteristic zero.

Findings

The quotient ring is isomorphic to a polynomial ring modulo a cyclotomic polynomial.

The structure of non-induced representations is explicitly characterized.

Induction on prime divisors of the group order is used in the proof.

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.