Characters of the Feigin–Stoyanovsky Subspaces and Brion's Theorem

We give an alternative proof of the main result of [B. Feigin, M. Jimbo, S. Loktev, T. Miwa, E. Mukhin, The Ramanujan J., 7:3 (2003), 519–530]; the proof relies on Brion's theorem about convex polyhedra. The result itself can be viewed as a formula for the character of the Feigin–Stoyanovsky subspace of an integrable irreducible representation of the affine Lie algebra $\widehat{\mathfrak{sl}_n}(\mathbb{C})$. Our approach is to assign integer points of a certain polytope to vectors comprising a monomial basis of the subspace and then compute the character by using (a variation of) Brion's theorem.