Dual of a rational Schur module ($D_{\lambda }^{\ast }\cong D_{(-{\lambda }_m,\dots ,-{\lambda }_1)}$)

The dual of $D_{\lambda }$ has dominant label $-w_0\lambda =(-{\lambda }_m,\dots ,-{\lambda }_1)$. Substitution $x_i\mapsto x_i^{-1}$ in the alternant character formula for the general linear group and reversal of determinant columns gives that character. The numerator and denominator reversal signs and common monomial factors cancel.