Let $G$ be a compact Lie group. I am trying to characterize the algebraic properties of the representation ring $R(G)$ of $G$. In the case of the $n$-torus, the representation ring $R(T)$ is isomorphic to the ring $\mathbb{Z}[x_1,x_2,...,x_n,({x_1x_2...x_n})^{-1}]$, whose dimension is seen to be $n+1$. An internet search suggests that the dimension of the representation ring of a compact Lie group $G$ is equal to $\text{rank}(G)+1$ and cites Segal's paper titled 'The representation-ring of a compact Lie group' as a reference. In the paper, Segal gives a detailed description of the prime ideals of $R(G)$, but does not give an explicit computation of the dimension of the representation ring. Can someone please redirect me to a source or provide a proof for $\text{dim}R(G)=\text{rank}(G)+1$?
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9$\begingroup$ If $G$ is connected, $R(G)=R(T)^W$, where $T$ is a maximal torus and $W$ the Weyl group. Then $\dim R(G)=\dim R(T)^W=\dim R(T)= \operatorname{rk}G+1 $. $\endgroup$– abxCommented Jun 30, 2023 at 10:06
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$\begingroup$ Note also that when $G$ is not connected, the various connected components of $\mathrm{Spec}(R(G))$ have different dimensions in general. See for example the case of even orthogonal groups in H. Minami, "THE REPRESENTATION RINGS OF ORTHOGONAL GROUPS" Osaka J. Math. (8) 1971, 243–250. $\endgroup$– Stefan DawydiakCommented Jul 3, 2023 at 16:57
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