Determinantal Formulas for SEM Expansions of Schubert Polynomials

TL;DR

This paper establishes determinantal formulas for certain Schubert polynomials associated with pattern-avoiding permutations, enabling explicit subtraction-free expansions in elementary symmetric polynomials.

Contribution

It introduces a new determinantal representation for Schubert polynomials of pattern-avoiding permutations, extending classical identities.

Findings

Determinantal formulas exist for permutations avoiding specific patterns.

These formulas lead to explicit subtraction-free expansions.

The approach generalizes classical Jacobi-Trudi identities.

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.