= Lowest-degree alternant proof of the Weyl dimension formula
For a <root system> with $N$ positive <roots>, put $A_\nu(H)=\sum_{w\in W}\det(w)e^{(w\nu,H)}$. Its degree-$m$ Taylor coefficient is alternating separately in $\nu$ and $H$, and hence divisible in each set of variables by the product of positive-root linear forms. Thus it vanishes for $m<N$ and its degree-$N$ term is $C\prod_{\alpha>0}(\nu,\alpha)\prod_{\alpha>0}(H,\alpha)$. The <Weyl denominator formula> at $\nu=\rho$ determines the nonzero constant $C$. Taking the ratio $A_{\lambda+\rho}(tH)/A_\rho(tH)$ for regular $H$ and letting $t\to0$ gives the <Weyl dimension formula> $\dim L(\lambda)=\prod_{\alpha>0}(\lambda+\rho,\alpha)/(\rho,\alpha)$. Factors converting roots to <coroots> cancel.
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