schubmult is a python package written by Matt Samuel for computing structure constants of Schubert polynomials. It handles ordinary Schubert polynomials, double Schubert polynomials in the same set of variables (expressed positively), and double Schubert polynomials in different sets of variables. To get the latest version,
pip install schubmult
or
pip install schubmult --upgrade
if you already have it. This will install three scripts, schubmult_py (so as not to conflict with the existing software schubmult in lrcalc by Anders Buch), schubmult_double, and schubmult_yz. Since it is written in python, it is essentially platform-independent.
Try it
See the latest version on https://pypi.org/project/schubmult/ and the GitHub project on https://github.com/matthematics/schubmult.
For questions, comments, or suggestions for improvements, mail to schubmult@gmail.com.
For an explanation of what Schubert polynomials are, see Per Alexandersonn’s site at symmetricfunctions.com (Schubert polynomials).
The algorithm is based on transitioning to the elementary symmetric monomial basis (with a twist) and then using the appropriate Pieri formula. In the ordinary Schubert polynomial case the Pieri formula is due to Sottile. For double Schubert polynomials, the Pieri formula comes from my paper “A Molev-Sagan type formula for double Schubert polynomials.” For additional reference, see the slides of my talk at the Schubert Seminar on October 2nd, 2023 and the video (part 1 part 2).
--display-positive
schubmult_double has an option --display-positive to display the result positively, which applies also for –mixed-var. For double Schubert polynomials, this was proven to be possible by Gao and Xiong, Graham positivity of triple Schubert calculus (2025). The program also attempts positivity for mixed variable quantum double Schubert polynomials if this is set, which is probably also always possible, but this is not known.
Related papers
Matthew J. Samuel, A Molev-Sagan type formula for double Schubert polynomials, Journal of Pure and Applied Algebra, Volume 228, Issue 7, 2024.
Matthew J. Samuel, A Littlewood-Richardson Rule for Forest Polynomials via the Schubert Bialgebra, arXiv preprint 2606.23876
Related videos
A Molev-Sagan type formula for double Schubert polynomials (part 1)
A Molev-Sagan type formula for double Schubert polynomials (part 2)
A dual Pieri formula for double Schubert polynomials in different sets of coefficient variables
Product-coproduct duality for double Schubert polynomials
A Pieri formula for multiplying double Schubert polynomials by certain factorial Schur polynomials