Augmentation quotients for real representation rings of cyclic groups

Abstract

Denote by $C_m$ the cyclic group of order m. Let ${\mathcal {R}}(C_m)$ be its real representation ring, and $\Delta (C_m)$ its augmentation ideal. In this paper, we give an explicit ${\mathbb {Z}}$-basis for the n-th power $\Delta ^{n}(C_m)$ and determine the isomorphism class of the n-th augmentation quotient $\Delta ^n(C_m)/\Delta ^{n+1}(C_m)$ for each positive integer n.