 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 $t\to 0$ gives the Weyl dimension formula $\dim L(\lambda )={\prod }_{\alpha >0}(\lambda +\rho ,\alpha )/(\rho ,\alpha )$. Factors converting roots to coroots cancel.