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

Abstract

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.