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Let $\Phi$ be a root system with Weyl group $\operatorname{Weyl}(\Phi)$, let $\Phi^+$ be a set of positive roots for $\Phi$ and $\rho$ be the half sum of the elements of $\Phi^+$. Then the Weyl's denominator formula can be stated in the following form: $$\sum_{w\in\operatorname{Weyl}(\Phi)}{sgn(w)e^{w(\rho)}}=\prod_{\alpha\in\Phi^+}{(e^{\alpha/2}-e^{-\alpha/2})}.$$ I heard that taking $\Phi=A_{n-1}$, the above formula is equivalent to the Vandermonde identity: $$\sum_{\sigma\in\frak{S}_n}{\varepsilon(\sigma)X_{\sigma(1)}^0\cdots X_{\sigma(n)}^{n-1}}=\prod_{i<j}{(X_j-X_i)}.$$ Can anyone give me a reference where I can find the above calculation?

Many thanks!

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    $\begingroup$ Wikipdia (Weyl character formula) says Exercise 9 in Chapter 10 of Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer. $\endgroup$
    – Friedrich Knop
    Commented Apr 21, 2017 at 11:32

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