Differential operators on Schur and Schubert polynomials

TL;DR

This paper introduces and analyzes decreasing operators on Schur and Schubert polynomials, revealing their structure, relations, and applications to identities and symmetric functions, providing new elementary proofs and extending operator classes.

Contribution

It characterizes all decreasing operators on Schur and Schubert polynomials using two fundamental operators, offering new proofs of classical identities and extending operator frameworks.

Findings

Operators $\xi$ and $ abla$ fully determine Schur function products.

Elementary proof of Giambelli and Jacobi-Trudi identities.

Extension of bosonic operators to Schubert polynomials.

Abstract

This paper deals with decreasing operators on back stable Schubert polynomials. We study two operators $\xi$ and $\nabla$ of degree $-1$, which satisfy the Leibniz rule. Furthermore, we show that all other such operators are linear combinations of $\xi$ and $\nabla$. For the case of Schur functions, these two operators fully determine the product of Schur functions, i.e., it is possible to define the Littlewood-Richardson coefficients only from $\xi$ and $\nabla$. This new point of view on Schur functions gives us an elementary proof of the Giambelli identity and of Jacobi-Trudi identities. For the case of Schubert polynomials, we construct a bigger class of decreasing operators as expressions in terms of $\xi$ and $\nabla$, which are indexed by Young diagrams. Surprisingly, these operators are related to Stanley symmetric functions. In particular, we extend bosonic operators from Schur to Schubert polynomials.