 Weyl integration formula for U(n)

For the unitary group, the diagonal maximal torus has coordinates $z_i=e^{i{\theta }_i}$ and Weyl group $S_n$. The Weyl integration formula weights independent uniform angles by $n{!}^{-1}{\prod }_{i<j}|z_i-z_j{|}^2$. The Vandermonde determinant $\det (z_i^{n-j})$ has squared torus integral $n!$, by Fourier orthogonality. Multiplying an irreducible character of highest weight $\lambda$ by this determinant gives an alternating Laurent polynomial. Its leading alternant has coefficient one; character orthogonality for compact groups forces every other alternant coefficient to vanish. This yields the quotient $\det (z_i^{{\lambda }_j+n-j})/\det (z_i^{n-j})$.