Recall the definition of the [representation ring](http://ncatlab.org/nlab/show/representation+ring) $R(G)$ of a compact Lie group $G$. I'd like a reference that gives me basic ring-theoretic properties that $R(G)$ always has, or enough info that I can check for various properties myself.



What sort of ring-theoretic properties does the representation ring of a compact Lie group possess?
Assume that $G$ is connected. Let $T$ be a maximal torus of $G$. Restriction induces a map $R(G) \to R(T)$. Note that $R(T)$ is a Laurent polynomial ring in $r$ variables where $r$ is the rank. Because the conjugates of $T$ fill up $G$, Peter-Weyl implies that this map is injective. Moreover, if $W$ is the Weyl group, then the image of this map lies in $R(T)^W$, and I think this map is always an isomorphism. So $R(G)$ is about as nice as possible: in particular, it is a finitely generated integral domain, so Noetherian of finite Krull dimension, etc. A lot is known about invariants of Weyl groups so you can probably get even more explicit information than this.


*Example.* Let $G = \text{U}(n)$. Then $R(T)$ is a Laurent polynomial ring in $n$ variables, the Weyl group is $W \cong S\_n$ acting by permutation, and the invariant subring is $\mathbb{Z}[e\_1, ..., e\_n, e\_n^{-1}]$ where $e\_i$ is the $i^{th}$ elementary symmetric polynomial, which corresponds to $\Lambda^i(\mathbb{C}^n)$. 


