= Alternant orthogonality on a unitary torus
{title2=$\int_T A_\kappa\overline{A_\eta}\,dt=n!\,\delta_{\kappa\eta}$}
For strictly decreasing integer tuples $\kappa,\eta$, let $A_\kappa(z)=\det(z_i^{\kappa_j})$ on the diagonal <maximal torus> of the <unitary group>. Expanding both <determinants>, <Fourier orthogonality> makes the integral zero unless the exponent tuples have the same permutation orbit. Strict decreasing order makes that equivalent to $\kappa=\eta$; the $n!$ matching permutation terms then each contribute one. An alternating <Laurent polynomial> is a finite linear combination of these alternants. For an <irreducible character> $\chi_\lambda$, its highest monomial gives coefficient one for $A_{\lambda+\delta}$ in $\chi_\lambda A_\delta$, with $\delta=(n-1,\ldots,0)$. The <Weyl integration formula> and <Schur orthogonality relations> force the sum of squared coefficients to be one, so all other coefficients vanish. This proves the unitary <Weyl character formula> directly.
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