Determinantal Formulas for SEM Expansions of Schubert Polynomials

Abstract

We show that for any permutation $w$ that avoids a certain set of 13 patterns of lengths 5 and 6, the Schubert polynomial ${\mathfrak{S}}_w$ can be expressed as the determinant of a matrix of elementary symmetric polynomials in a manner similar to the Jacobi-Trudi identity. For such $w$, this determinantal formula is equivalent to a (signed) subtraction-free expansion of ${\mathfrak{S}}_w$ in the basis of standard elementary monomials.