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

TL;DR

This paper offers a new proof of Jacobi--Trudi identities using the BGG category O, connecting Kostka numbers, tensor product multiplicities, and Schur positivity in a novel algebraic framework.

Contribution

It introduces a representation-theoretic approach to classical symmetric function identities via the BGG category O for special linear Lie algebras.

Findings

Re-establishes Schur positivity of certain truncations in Jacobi--Trudi expansions.

Provides new interpretations of expansion coefficients as tensor product multiplicities.

Derives Schur positivity results for products of Schur polynomials.

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.