nLab
Artin-Lam induction exponent

Contents

Context

Group Theory

Representation theory

Contents

Definition

Definition

For GG a finite group, its Artin-Lam induction exponent is the minimum e(G)e(G) among positive natural numbers such that the e(G)e(G)-dimensional trivial representation is in the ideal of the rational representation ring of GG which is generated by GG-representations that are induced by cyclic subgroups of GG.

This definition according to the abstract of Madsen, Thomas & Wall 1983. Other authors take the smallest e(G)e(G) such that e(G)χe(G) \cdot \chi is in this ideal for all rational representations χ\chi (e.g. Yamauchi 70), in which case e(G)e(G) is the exponent of the additive group underlying the quotient by the ideal, whence the terminology.

References

The original article:

Further discussion:

  • Kenichi Yamauchi, On the 2-part of Artin Exponent of Finite Groups, Science Reports of the Tokyo Kyoiku Daigaku, Section A Vol. 10, No. 263/274 (1970), 234-240 (jstor:43698746)

  • K. K. Nwabueze, Some definition of the Artin exponent of finite groups (arXiv:math/9611212)

  • S. Jafari and H. Sharifi, On the Artin exponent of some rational groups, Communications in Algebra, 46:4, 1519-152 (doi:10.1080/00927872.2017.1347665)

Textbook account:

  • Charles Curtis, Irving Reiner, Methods of representation theory – With applications to finite groups and orders – Vol. I, Wiley 1981 (pdf)

Application to discussion of free group actions on n-spheres:

Last revised on October 27, 2021 at 14:30:34. See the history of this page for a list of all contributions to it.