An explicit formula for the Hilbert series of a partial flag variety

For a semisimple, simply-connected linear algebraic group, $G$, and parabolic subgroup, $P\subseteq G$, we use the fact that the Hilbert polynomial of the equivariant embedding of $G\mathrm{/}P$ is equal to the Hilbert function to compute an explicit formula for the Hilbert series of $G\mathrm{/}P$ in terms of the dimensions of finitely many irreducible representations of $G$. As an example, we compute the Hilbert series of the adjoint variety of $SL(n+1,\mathbb{C})$. We conclude by computing the linear term of the numerator of the Hilbert series of any fundamental representation in type $A$.