Exercise 1, § 7

Take G to be the group $\mathbf{U}(n,\mathbf{C})$, and T to be the subgroup consisting of the diagonal matrices; for $t=$ diag($t_1, . . . , t_n$) in T and $1\leq i\leq n$, put $\varepsilon_i(t) =t_i$. Denote by $\sigma$ the identity representation of G on $\mathbf{C}^n$.

a) The group X(T) has basis $\varepsilon_1, . . . , \varepsilon_n$; show that every element of $\mathbf{Z}[X(T)]^W$ is of the form $e^{k(\varepsilon_1+\cdot \cdot \cdot+\varepsilon_n)}P(e^{\varepsilon_1}, . . . , e^{\varepsilon_n})$, where $k$ is a relative integer and P a symmetric polynomial in $n$ variables with integer coefficients.

b) Show that the representations $\wedge \wedge^r\sigma (1\leq r\leq n)$ are irreducible.

c) Show that the homomorphism $u:\mathbf{Z}[X_1,X_2, . . . ,X_n][X^{-1}_n]\rightarrow R(G)$ such that $u(X_i) = [\wedge \wedge^i\sigma ]$ is an isomorphism.