Quantum bumpless pipe dreams
Abstract
Schubert polynomials are polynomial representatives of Schubert classes in the cohomology of the complete flag variety and have a combinatorial formulation in terms of bumpless pipe dreams. Quantum double Schubert polynomials are polynomial representatives of Schubert classes in the torus-equivariant quantum cohomology of the complete flag variety, but no analogous combinatorial formulation had been discovered. We introduce a generalization of the bumpless pipe dreams called quantum bumpless pipe dreams, giving a novel combinatorial formula for quantum double Schubert polynomials as a sum of binomial weights of quantum bumpless pipe dreams. We give a bijective proof for this formula by showing that the sum of binomial weights satisfies a defining transition equation.
Cite
@article{arxiv.2403.16168,
title = {Quantum bumpless pipe dreams},
author = {Tuong Le and Shuge Ouyang and Leo Tao and Joseph Restivo and Angelina Zhang},
journal= {arXiv preprint arXiv:2403.16168},
year = {2025}
}
Comments
22 pages, 20 figures