= Weyl denominator
{c}
{title2=$\Delta=e^\rho\prod_{\alpha>0}(1-e^{-\alpha})$}
The Weyl denominator is the alternating expression in the <Weyl denominator formula>. Its product form uses the <Weyl vector> $\rho$, half the positive-root sum. On a compact <maximal torus>, the factor $e^\rho$ has unit modulus; if it is only defined on a covering torus, the squared modulus still descends. Thus $|\Delta(t)|^2=\prod_{\alpha>0}|1-\alpha(t)|^2$ is a globally defined density in the <Weyl integration formula>. For the <unitary group> an integer central shift gives the <Vandermonde determinant> $\det(z_i^{n-j})$, whose modulus is the same.
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