# Twisted Schubert polynomials

## TL;DR

This paper introduces twisted Schubert polynomials, proves their positivity, and provides combinatorial formulas for their coefficients, extending previous results and connecting to the Pieri rule.

## Contribution

It defines twisted Schubert polynomials, proves their monomial positivity, and offers combinatorial formulas, extending earlier positivity results and linking to classical Schubert calculus.

## Key findings

- Twisted Schubert polynomials are monomial positive.
- A combinatorial formula for their coefficients is provided.
- Connections to the Pieri rule and positivity of difference operators are established.

## Abstract

We prove that twisted versions of Schubert polynomials defined by $\widetilde{\mathfrak S}_{w_0} = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}$ and $\widetilde{\mathfrak S}_{ws_i} = (s_i+\partial_i)\widetilde{\mathfrak S}_w$ are monomial positive and give a combinatorial formula for their coefficients. In doing so, we reprove and extend a previous result about positivity of skew divided difference operators and show how it implies the Pieri rule for Schubert polynomials. We also give positive formulas for double versions of the $\widetilde{\mathfrak S}_w$ as well as their localizations.