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arXiv · 2403.16168

Quantum bumpless pipe dreams

Tuong Le·Shuge Ouyang·Leo Tao·Joseph Restivo·Angelina Zhang
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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.

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Tuong Le, Shuge Ouyang, Leo Tao, Joseph Restivo, Angelina Zhang. 2025-02-07. Quantum bumpless pipe dreams. https://doi.org/10.1017/fms.2024.112

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