Differential operators on Schur and Schubert polynomials

Gleb Nenashev

Published 2020-05-17Version 1

This paper is about decreasing operators on back stable Schubert polynomials. We study two operators $\xi$ and $\nabla$ of degree $-1$, which satisfy 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 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 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.