Graham positivity of triple Schubert calculus

TL;DR

This paper proves conjectures related to Graham positivity in Schubert calculus, establishing new positivity results for expansion coefficients and skew divided difference operators in the context of Schubert polynomials.

Contribution

It introduces a refined Graham positivity theorem and confirms Kirillov's conjecture on positivity of skew divided difference operators applied to Schubert polynomials.

Findings

Proved Samuel's Graham positivity conjecture for double Schubert polynomials.

Established positivity of skew divided difference operators on Schubert polynomials.

Provided a refined version of Graham's positivity theorem.

Abstract

We prove Samuel's conjecture on certain Graham positivity of the expansion coefficient of two double Schubert polynomials in three sets of variables by establishing a refined version of Graham's positivity theorem. As a corollary, we prove Kirillov's conjecture on the positivity of skew divided difference operators applied to Schubert polynomials.