  Using the BGG resolution, Akin presented a realization of the Jacobi-Trudi
expansion of a Schur polynomial as the Euler-Poincar\'e characteristic of an
exact sequence in the category of polynomial representations of the general
linear groups. Zelevinskii gave an independent treatment for a general skew
Schur polynomial. Based on their construction of resolutions, one can obtain
the Schur positivity of certain truncations of the Jacobi-Trudi expansion of a
skew Schur polynomial. By interpreting Kostka numbers as tensor product
multiplicities in the BGG category $\mathcal{O}$ of the special linear Lie
algebras, we establish a new relationship between the Schur positivity of these
truncations of Jacobi-Trudi expansion and the BGG resolution. Specifically, we
show that coefficients in the combination of the Schur polynomial of these
truncations of Jacobi-Trudi expansion also appear as tensor product
multiplicities in the BGG category $\mathcal{O}$. This point of view also
enables us to derive the Jacobi-Trudi identity directly from the celebrated
Weyl character formula.
