Showing $R(G) = R(T)^W$

Question

Let $G$ be a compact connected Lie group and $T$ a maximal torus. Let $R(G)$ be the representation ring of $G$. Then restriction of reps gives a map $R(G) \to R(T)^W$, where $R(T)^W$ are the characters on $T$ that are invariant under the Weyl group $W$. How do you show that this map is a surjection? Does this involve ``inverting" the Weyl character formula?

Answer 1

Take an element of $f\in R(T)^W=\sum_\lambda c_\lambda e^\lambda$ ($\lambda$s are weights), choose a $\lambda_0$ which is maximal among those in the sum, and pass from $f$ to $f-c_{\lambda_0}\chi_{\lambda_0}$, where $\chi_{\lambda_0}$ is the irreducible character with highest weight $\lambda_0$. Repeat. After finitely many iterations you get to $0$.

Answer 2

There is a clear proof available in Fulton&Harris, Chapter 23 section 1. It is not as "slick" as the above proof but should be helpful.