Original title: Canonical-row Chern flow on Bott--Samelson towers: realizable-volume models for Schubert, Grothendieck, and Lascoux polynomials
Khai-Hoan Nguyen-Dang, Zhenpeng Wang · September 2026
Mathematicians study complex algebraic formulas called polynomials that help count geometric shapes. For a long time, researchers struggled to prove which powers of variables can appear in certain formulas, called Grothendieck and Lascoux polynomials. The authors solved this by building realizable-volume models over any field. These models represent algebraic formulas using geometric volumes. Their approach covers Lascoux, Lascoux-atom, and positive Grothendieck packets. Their minors also include Schubert and key polynomials. Over the complex numbers, these polynomials are LorentzianA property of polynomials related to their signs and combinatorial convexity., meaning they have strong concavity properties. The authors showed that the powers of variables in these polynomials form discrete shapes called M-natural convex sets. This matches the lattice points of their Newton polytopesGeometric shapes defined by the exponents of terms in a polynomial., proving several open conjectures about these structures.
Imagine you have a set of books with different spine widths. You want to arrange them on a shelf so that the pattern of thick and thin books forms a perfect rectangle when viewed from above. Normally, it's unclear if such an arrangement is possible. In this paper, for certain polynomials, the set of ways to combine terms (like book widths) always fits perfectly into a specific geometric shape called a polytope.
This work settles long-standing mathematical conjectures about the shape of polynomials used in algebra and geometry. It gives mathematicians a clearer understanding that can influence future research in combinatorics and related fields.
The Lorentzian property is only established over the complex numbers, not necessarily for other fields. The results are specific to homogeneous polynomials and certain packets, so they may not generalize to all polynomials. The work is theoretical, with no immediate practical applications mentioned.
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Canonical-row Chern flow on Bott--Samelson towers: realizable-volume models for Schubert, Grothendieck, and Lascoux polynomials — Khai-Hoan Nguyen-Dang, Zhenpeng Wang
Abstract licence: CC0
Explanation generated by Synthesis from the abstract — it is not the original paper. This is a preprint, so it hasn't necessarily been peer reviewed.