Normalized skew Schur polynomials are Lorentzian

Abstract

We prove the conjecture of Huh, Matherne, M\'esz\'aros, and St.~Dizier that the normalization of every skew Schur polynomial in finitely many variables is Lorentzian. We first realize every nonzero skew Schur polynomial in finitely many variables as a specialization of a Schubert polynomial and prove that it is dually Lorentzian. The dual Jacobi--Trudi identity then identifies its normalization with the finite dual of a skew Schur polynomial obtained by rectangular complementation. As a consequence, skew Kostka numbers satisfy log-concavity inequalities along the root directions.