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单词 RepresentationRingVsBurnsideRing
释义

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:

[X][Vk(X)]

which can be easily extended to the function

β:Ω(G)Rk(G);
β([X])=[Vk(X)]

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.

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更新时间:2025/5/4 9:32:26