Augmentation quotients for complex representation rings of dihedral groups

Abstract

Denote by D_m the dihedral group of order 2m. Let \mathit{\Re }\left(D_m\right) 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 Δⁿ(D_m)/Δⁿ⁺¹(D_m) for each positive integer n.

Keywords

dihedral group / representation / augmentation quotient

References

[1] Bachman F, Grunenfelder L. The periodicity in the graded ring associated with an integral group ring. J Pure Appl Algebra, 1974, 5: 253-264

[2] Curtis C W, Reiner I. Representation Theory of Finite Groups and Associative Algebras. Providence: AMS Chelsea Publishing, 2006

[3] Chang Shan, Tang Guoping. A basis for augmentation quotients of finite abelian groups. J Algebra, 2011, 327: 466-488

[4] Karpilovsky G. Commutative Group Algebras. Monographs and Textbooks in Pure and Applied Mathematics, Vol 78. New York: Marcel Dekker, 1983

[5] Magurn B A. An Algebraic Introduction to K-Theory. Cambridge: Cambridge University Press, 2002

[6] Serre J -P. Linear Representations of Finite Groups. Graduate Texts in Mathematics, Vol 42. New York: Springer-Verlag, 1977

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