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When do Schubert polynomial products stabilize?

Andrew Hardt, David Wallach

TL;DR

This work resolves a central question about when products of Schubert polynomials stabilize under row/column shifts by deriving a closed formula for the back-stabilization number: $\mathrm{BS}(u,v)=\max_{i\ge0}(\lambda_i(u)+\lambda_i(v)-i)$. The authors develop a rich combinatorial framework using colored words and a colored shuffle algebra that realizes back-stable (quasi)symmetric functions and back-stable Schubert/slide polynomials, enabling a constructive comparison with ordinary structure constants. They prove the Back-Stabilization Theorem, establish forward-stabilization via dualities with $w_0$-conjugation, and provide a parallel forward-stabilization formula, $\mathrm{FS}(u,v)=\max_{i\le 1+\max(\mathrm{FS}(u),\mathrm{FS}(v))}(\Lambda_i(u)+\Lambda_i(v)+i-1)$, clarifying when Schubert products fully realize in a given flag variety. The paper also develops the algebraic underpinnings (DC-triviality, increasing suffix theory) and introduces back-stable key polynomials to extend the stability analysis to broader bases, with implications for Schubert calculus and related polynomial families.

Abstract

The "back-stabilization number" for products of Schubert polynomials is the distance the corresponding permutations must be shifted before the structure constants stabilize. We give an explicit formula for this number and thereby prove a conjecture of N. Li in a strengthened form. This leads to an additional result: a formula for the smallest $n$ such that a given Schubert product expands completely over $S_n$. Our method is to explore back-stable fundamental slide polynomials and their products combinatorially, in the context of their associated words. We use three main tools: (i) an algebra consisting of "colored words", with a modified shuffle product, and which contains the rings of back (quasi)symmetric functions as subquotients; (ii) the combinatorics of increasing suffixes of reduced words; and (iii) the lift of differential operators to the space of colored words.

When do Schubert polynomial products stabilize?

TL;DR

This work resolves a central question about when products of Schubert polynomials stabilize under row/column shifts by deriving a closed formula for the back-stabilization number: . The authors develop a rich combinatorial framework using colored words and a colored shuffle algebra that realizes back-stable (quasi)symmetric functions and back-stable Schubert/slide polynomials, enabling a constructive comparison with ordinary structure constants. They prove the Back-Stabilization Theorem, establish forward-stabilization via dualities with -conjugation, and provide a parallel forward-stabilization formula, , clarifying when Schubert products fully realize in a given flag variety. The paper also develops the algebraic underpinnings (DC-triviality, increasing suffix theory) and introduces back-stable key polynomials to extend the stability analysis to broader bases, with implications for Schubert calculus and related polynomial families.

Abstract

The "back-stabilization number" for products of Schubert polynomials is the distance the corresponding permutations must be shifted before the structure constants stabilize. We give an explicit formula for this number and thereby prove a conjecture of N. Li in a strengthened form. This leads to an additional result: a formula for the smallest such that a given Schubert product expands completely over . Our method is to explore back-stable fundamental slide polynomials and their products combinatorially, in the context of their associated words. We use three main tools: (i) an algebra consisting of "colored words", with a modified shuffle product, and which contains the rings of back (quasi)symmetric functions as subquotients; (ii) the combinatorics of increasing suffixes of reduced words; and (iii) the lift of differential operators to the space of colored words.

Paper Structure

This paper contains 24 sections, 55 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Words and permutations
  3. Words
  4. Permutations
  5. Lehmer code
  6. Colored words and slide polynomials
  7. Colored words and the colored shuffle product
  8. Compatible sequences
  9. Back-symmetric and back-quasisymmetric functions
  10. Back-stable Schubert and slide polynomials
  11. The algebra of words
  12. Homomorphism between $\mathcal{Q}$ and rings of polynomials
  13. Weight-preserving maps on elements of $\mathcal{Q}$
  14. Differential operators on Schubert and slide polynomials
  15. Proof of the back-stabilization conjecture
  16. ...and 9 more sections

Key Result

Theorem 1.3

Conjecture conj:li-back-stable is true. Moreover, for all $u,v\in S_{\mathbb{Z}_+}$,

Theorems & Definitions (128)

  • Definition 1.1
  • Conjecture 1.2: Li-back-stable-conjecture
  • Theorem 1.3: Back-Stabilization Theorem
  • Corollary 1.4
  • Example 1.5
  • Theorem 1.6: Forward-Stability Theorem
  • Corollary 1.7
  • Example 1.8
  • Remark 2.1
  • Definition 2.2
  • ...and 118 more