 Alternant character formula for the general linear group

For a dominant integer tuple $\lambda$ and $\delta =(m-1,\dots ,0)$, the rational Schur character is $\det (x_i^{{\lambda }_j+m-j})/\det (x_i^{m-j})$. For partitions it is a symmetric polynomial; for general tuples it is Laurent on the diagonal torus. The formula follows by comparing the trace of a permuted tensor power with the Frobenius alternant character formula and using independence of symmetric-group characters. Apparent poles at coincident nonzero eigenvalues are removable.