Quantum Schubert polynomials and quantum Schur functions

TL;DR

This paper introduces new quantum symmetric functions and establishes their connections to quantum Schubert polynomials, providing formulas and conjectures that extend classical combinatorial and algebraic results into the quantum setting.

Contribution

It defines quantum multi-Schur functions, proves their equivalence to quantum double Schubert polynomials for certain permutations, and extends classical formulas to the quantum case.

Findings

Quantum multi-Schur functions coincide with quantum double Schubert polynomials for restricted vexillary permutations.

Quantum analogs of classical formulas like Nagelsbach-Kostka and Jacobi-Trudi are established for Grassmannian permutations.

Conjectures are proposed for the structure of quantum polynomials for 321-avoiding permutations.

Abstract

We introduce the quantum multi-Schur functions, quantum factorial Schur functions and quantum Macdonald polynomials. We prove that for restricted vexillary permutations the quantum double Schubert polynomial coincides with some quantum multi-Schur function and prove a quantum analog of the Nagelsbach-Kostka and Jacobi-Trudi formulae for the quantum double Schubert polynomials in the case of Grassmannian permutations. We prove, also, an analog of the Billey-Jockusch-Stanley formula for quantum Schubert polynomials. Finally we formulate two conjectures about the structure of quantum double and quantum Schubert polynomials for 321-avoiding permutations.