Leading coefficient of an exponential alternant ($\det (e^{t{\ell }_i{c}_j})=t^{\left(\genfrac{}{}{0ex}{}{m}{2}\right)}\Delta (\ell )\Delta (c)/{\prod }_{k=0}^{m-1}k!+O(t^{\left(\genfrac{}{}{0ex}{}{m}{2}\right)+1})$)

For distinct ${\ell }_i$ and $c_j$, expand the exponential entries in powers of $t$. Nonzero determinant terms first occur at the distinct powers $0,1,\dots ,m-1$. The coefficient is the product of their two Vandermonde determinants divided by the factorial product. Ratios of such alternants yield the Weyl dimension formula by taking a torus character to the identity.