Forum of Mathematics, Sigma (Jan 2025)

Quantum bumpless pipe dreams

  • Tuong Le,
  • Shuge Ouyang,
  • Leo Tao,
  • Joseph Restivo,
  • Angelina Zhang

DOI
https://doi.org/10.1017/fms.2024.112
Journal volume & issue
Vol. 13

Abstract

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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.

Keywords