Revisiting Jacobi-Trudi identities via the BGG category $\mathcal{O}$

By interpreting Kostka numbers as tensor product multiplicities in the BGG category O for the special linear Lie algebras, we provide a new proof of the classical Jacobi–Trudi identities for skew Schur polynomials, derived from the celebrated Weyl character formula. We re-establish the Schur positivity of certain truncations in the Jacobi–Trudi expansion of skew Schur polynomials and obtain Schur positivity results for similar truncations in the Jacobi–Trudi-type expansion of the product of two Schur polynomials. Furthermore, we interpret the coefficients in the Schur polynomial expansions of these Jacobi–Trudi truncations as tensor product multiplicities in the BGG category O.