Combinatorial characterization of Schubert polynomial structure constants

Characterize the structure constants c^{\gamma}_{\alpha,\beta} in products of Schubert polynomials \mathfrak{S}_{\alpha}\mathfrak{S}_{\beta}, including a combinatorial description that determines their values for triples of permutations.

Background

Schubert polynomials are indexed by permutations and have connections to algebraic geometry, Lie theory, and representation theory. Their products expand in the Schubert basis with integer structure constants c{\gamma}_{\alpha,\beta}. The paper presents understanding these constants as an important open problem and supplies datasets of permutation triples labeled by the corresponding coefficients.

References

An important open problem in the study of Schubert polynomials is understanding their {\em structure constants}.

— Machine Learning meets Algebraic Combinatorics: A Suite of Datasets Capturing Research-level Conjecturing Ability in Pure Mathematics  (2503.06366 - Chau et al., 9 Mar 2025) in Section 3, subsection “Schubert Polynomial Structure Constants (Open Problem)”

We conjecture that the double support is actually determined by the single support.

— Non-vanishing of Single, Double, and Triple Schubert Structure Constants  (2608.17378 - Chen et al., 18 Aug 2026) in Conjecture 1.1, Section 1 (Introduction)

The following Conjecture \ref{conjb} captures this asymmetry.

For any $u,v\in S_\infty$,

{}2(u,v)={}3(u,v)\cap{}3(v,u).

— Non-vanishing of Single, Double, and Triple Schubert Structure Constants  (2608.17378 - Chen et al., 18 Aug 2026) in Conjecture 1.2, Section 1 (Introduction), immediately following equation (1.2)