Canonical-row Chern flow on Bott--Samelson towers: realizable-volume models for Schubert, Grothendieck, and Lascoux polynomials

Abstract

We construct realizable-volume models over any field for the factorially normalized homogeneous Lascoux, Lascoux-atom, and positive Grothendieck packets. Their volume minors include normalized key polynomials, Demazure atoms, Schubert polynomials, and all sign-corrected homogeneous Grothendieck components. Over $\mathbb C$, these polynomials are Lorentzian. As consequences, the supports of ordinary Grothendieck, Lascoux, and Lascoux-atom polynomials are $M^{\natural}$-convex and are exactly the lattice points of their integral generalized-polymatroid Newton polytopes. This proves, in a stronger realizable-volume form, the corresponding conjectures of Huh--Matherne--Mészáros--St.~Dizier, together with the relevant saturated-Newton-polytope and Grothendieck-support conjectures of Monical--Tokcan--Yong, Mészáros--St.~Dizier, and Mészáros--Setiabrata--St.~Dizier. The maximal-degree Grothendieck component gives the Castelnuovo--Mumford support conjecture as a special case. Our method yields further packets whose finite duals are realizable-volume polynomials. More generally, over any field, factorially normalized top-degree total-Chern polynomials of globally generated bundles are realizable-volume polynomials, and the supports of our homogeneous realizations are bases of algebraic polymatroids.