# representation ring vs burnside ring

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

 $[X]\mapsto[V_{k}(X)]$

which can be easily extended to the function

 $\beta:\Omega(G)\to R_{k}(G);$
 $\beta([X])=[V_{k}(X)]$

where on the left side we have the Burnside ring and on the right side the representation ring. It can be shown, that $\beta$ is actually a ring homomorphism, but in most cases it neither injective nor surjective. But the following theorem due to Segal gives us some properties of $\beta$:

Theorem (Segal). Let $\beta:\Omega(G)\to R_{\mathbb{Q}}(G)$ be defined as above with rationals as the underlying field. If $G$ is a $p$-group for some prime number $p$, then $\beta$ is surjective. Furthermore $\beta$ is an isomorphism if and only if $G$ is cyclic.

Title representation ring vs burnside ring RepresentationRingVsBurnsideRing 2013-03-22 19:19:05 2013-03-22 19:19:05 joking (16130) joking (16130) 4 joking (16130) Theorem msc 20C99
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