representation ring vs burnside ring

Let G be a finite groupMathworldPlanetmath and let k be any field. If X is a G-set, then we may consider the vector space Vk(X) over k which has X as a basis. In this manner Vk(X) becomes a representation of G via action induced from X and linearly extended to Vk(X). It can be shown that Vk(X) only depends on the isomorphism class of X, so we have a well-defined mapping:


which can be easily extended to the function


where on the left side we have the Burnside ring and on the right side the representation ringMathworldPlanetmath. It can be shown, that β is actually a ring homomorphismMathworldPlanetmath, but in most cases it neither injectivePlanetmathPlanetmath nor surjectivePlanetmathPlanetmath. But the following theoremMathworldPlanetmath due to Segal gives us some properties of β:

Theorem (Segal). Let β:Ω(G)R(G) be defined as above with rationals as the underlying field. If G is a p-group for some prime number p, then β is surjective. Furthermore β is an isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if and only if G is cyclic.

Title representation ring vs burnside ring
Canonical name RepresentationRingVsBurnsideRing
Date of creation 2013-03-22 19:19:05
Last modified on 2013-03-22 19:19:05
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Theorem
Classification msc 20C99