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Quantum bumpless pipe dreams

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

TL;DR

This work provides a combinatorial QBPD framework that evaluates quantum double Schubert polynomials by summing binomial weights over QBPDs associated to each permutation. It establishes the main identity by a transition-equation-based bijective proof, using four key bijections and a stability analysis to align QBPD weights with the quantum recursive structure. While the QBPD formula captures the correct monomial and binomial contributions, cancellations arise and increase with size, highlighting both the naturalness and limits of the combinatorial model. The results bridge classical Schubert calculus with torus-equivariant quantum cohomology and offer a structured path toward deeper combinatorial understanding of quantum deformations, with clear directions for refining or extending the combinatorial framework.

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.

Quantum bumpless pipe dreams

TL;DR

This work provides a combinatorial QBPD framework that evaluates quantum double Schubert polynomials by summing binomial weights over QBPDs associated to each permutation. It establishes the main identity by a transition-equation-based bijective proof, using four key bijections and a stability analysis to align QBPD weights with the quantum recursive structure. While the QBPD formula captures the correct monomial and binomial contributions, cancellations arise and increase with size, highlighting both the naturalness and limits of the combinatorial model. The results bridge classical Schubert calculus with torus-equivariant quantum cohomology and offer a structured path toward deeper combinatorial understanding of quantum deformations, with clear directions for refining or extending the combinatorial framework.

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.
Paper Structure (11 sections, 20 theorems, 35 equations, 20 figures, 2 tables)

This paper contains 11 sections, 20 theorems, 35 equations, 20 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Rothe Diagrams
  4. Double Schubert polynomials
  5. Quantum double Schubert polynomials
  6. Quantum bumpless pipe dreams
  7. Droop moves and lift moves
  8. Stability
  9. Proof of Theorem \ref{['thm:qdform']}
  10. Cancellation Analysis
  11. Future Directions

Key Result

Theorem 2.1

For any $k$ and any permutation $w$, where $q_{cd}:= q_{c}q_{c+1}\dots q_{d-1}$.

Figures (20)

  • Figure 1: Rothe diagram for $4213$
  • Figure 2: A non-example of a QBPD
  • Figure 3: QBPDs for $4213$
  • Figure 4: A droop move (light color indicates possibilities)
  • Figure 5: A lift move (light color indicates possibilities)
  • ...and 15 more figures

Theorems & Definitions (48)

  • Theorem 2.1: Monk's rule for quantum double Schubert polynomials qbsm
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • Theorem 3.4
  • Example 3.5
  • Corollary 3.6
  • Definition 3.7
  • Lemma 3.8
  • proof
  • ...and 38 more