Augmentation quotients for complex representation rings of dihedral groups

Abstract

Denote by D m the dihedral group of order 2m. Let R(D m ) be its complex representation ring, and let Δ(D m ) be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quotient Δ n (D m )/Δ n+1 (D m ) for each positive integer n.

Stability in graded rings associated with commutative augmented rings

“…There is a significant fact that all of known results show that the isomorphism class of Q n does not depend on n when n is large enough. In particular, Bachman and Grunenfelder showed the sequence {Q n (ZG)} becomes stationary up to isomorphism for any finite abelian group G in [4], the author, Chen and Tang obtained similar results in [2] for complex representation rings of dihedral groups and all finite abelian groups, Wu and Tang proved in [19] that, for any finite abelian group G, there exists a positive integer n 0 such that Q n (Ω G ) ∼ = Q n+1 (Ω G ) for any n n 0 . Motivated by this, we raise the following theorem.…”

“…Their augmentation maps are induced by the degree of representations and the cardinality of fixed points of finite H-sets, respectively. The details are presented in [1], [2] and [3].…”

“…for any positive integer n. This problem has been well studied in [4]- [15] for integral group rings, in [2], [16]- [18] for representation rings, and in [19]- [24] for Burnside rings.…”

Augmentation Quotients for Burnside Rings of some Finite $p$-Groups

“…Two related problems of recent interest have been to investigate the augmentation ideals and their consecutive quotients for integral group rings and representation rings of finite groups. These problems have been well studied in [6], [7], [8], [9], [10], [11], [12], [13] and [14].…”