Polynomiality of Stretched Schubert Structure Constants and Key Coefficients

Abstract

We prove that monomial coefficients in affine families of key and Schubert polynomials are eventually polynomial. The proof combines Demazure operators with vector partition functions and, in the Schubert case, P.~Magyar's orthodontic formula. These coefficient results extend to finite products. Using M.~Watanabe's Schubert duality, we deduce that stretched Schubert structure constants are eventually polynomial, proving a conjecture of I.~Pak and Z.~Slonim. For key polynomials, this resolves the polynomiality part of a conjecture of P.~Alexandersson and E.~Alhajjar.

Disclosure

“formula in the three partition parameters [Ras04]. Acknowledgments We thank Igor Pak for asking about polynomiality of stretched Schubert coefficients; that question was the starting point for this paper. OpenAI’s Codex was used during the preparation of this paper for proof exploration and editorial assistance. References [AA19] Per Alexandersson and Elie Alhajjar. Ehrhart positivity and Demazure characters.”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T18:59:53.791848Z

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