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Recall the definition of the 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.

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  • $\begingroup$ What about structure? $R(G)$ is always a $\lambda$-ring, although it has even more structure than this, e.g. it's acted on by all Schur functors. $\endgroup$
    – Qiaochu Yuan
    Commented Feb 9, 2014 at 23:10
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    $\begingroup$ I'm thinking things like Noetherian, Henselian, finite (whatever) dimension, etc. The sorts of things that come up in vanilla commutative algebra. $\endgroup$
    – David Roberts ♦
    Commented Feb 9, 2014 at 23:24
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    $\begingroup$ As the answer you've accepted indicates, $R(G)$ is pretty boring as a ring; I think the proper point of view on it is as a ring-with-basis. (And then it's very interesting.) $\endgroup$
    – Allen Knutson
    Commented Feb 11, 2014 at 4:49

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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 $\Lambda^i(\mathbb{C}^n)$.

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    $\begingroup$ Note that a good reference for the structure of representation rings is section VI.2 of Springer GTM 98 Representations of Compact Lie Groups by Brocker and tom Dieck. In particular, the ring is finitely generated (hence noetherian) and has more precise properties if $G$ is simply connected, etc. $\endgroup$
    – Jim Humphreys
    Commented Feb 10, 2014 at 0:15
  • $\begingroup$ @Jim: thanks for the reference! My claim above is Proposition 2.1 in that section, and the more precise information when $G$ is simply connected is Corollary 2.11. In this case $R(G)$ is actually the polynomial ring on $r$ variables where $r$ is the rank. $\endgroup$
    – Qiaochu Yuan
    Commented Feb 10, 2014 at 0:56
  • $\begingroup$ In fact, I just found that Segal proved $R(G)$ is finitely generated for any compact $G$, by using finite generation over $R(U(n))$. $\endgroup$
    – David Roberts ♦
    Commented Feb 10, 2014 at 4:23
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    $\begingroup$ @David: I should have added an online link to Segal's 1968 paper: numdam.org/numdam-bin/fitem?id=PMIHES_1968__34__113_0 (which Serre discusses in the finite group case, in his textbook on finite group representations). $\endgroup$
    – Jim Humphreys
    Commented Feb 10, 2014 at 13:11

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