Augmentation quotients for complex representation rings of generalized quaternion groups

Abstract

Denote by Q _m the generalized quaternion group of order 4m. Let R(Q _m) be its complex representation ring, and Δ(Q _m) its augmentation ideal. In this paper, the author gives an explicit Z-basis for the Δ ⁿ(Q _m) and determines the isomorphism class of the n-th augmentation quotient $\frac{{{\Delta ^n}\left( {{Q_m}} \right)}}{{{\Delta ^{n + 1}}\left( {{Q_m}} \right)}}$ for each positive integer n.

Keywords

Generalized quaternion groups / Representation ring / Augmentation quotients

References

[1] Fulton W., Harris J.. Representation Theory: A First Course, 1991, New York: Springer-Verlag

[2] Curtis C. W., Reiner I.. Representation Theory of Finite Groups and Associative Algebras, 2006, Providence: American Mathematical Society

[3] Chang S., Chen H., Tang G. P.. Augmentation quotients for complex representation rings of dihedral groups. Front. Math. China, 2012, 7: 1-18

[4] Chang S.. Augmentation quotients for complex representation rings of point groups. J. Anhui University (Natural Science Edition), 2014, 38: 13-19

[5] Chang, S., Augmentation quotients for real representation rings of cyclic groups, Front. Math. China, preprint.

[6] Karpilovsky G.. Commutative Group Algebras, 1983, New York: Marcel Dekker

[7] Chang S., Tang G. P.. A basis for augmentation quotients of finite abelian groups. J. Algebra, 2011, 327: 466-488