= Weyl integration formula for U(n)
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{title2=$\frac1{n!}\prod_{i<j}|z_i-z_j|^2$}
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 $\det(z_i^{\lambda_j+n-j})/\det(z_i^{n-j})$.
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