Cluster Geometry of Universal Schubert Polynomials I: Geometric Bases and Schubert Transitions
We study Fulton's universal Schubert polynomials $\mathfrak S_w(c)$ as regular functions on the upper unitriangular group $U_N$. The standard triangular cluster structure associates a Schubert $\sf g$-vector to every permutation. Their convex hull is unimodularly equivalent to $\Delta_1\times\cdots\times\Delta_{N-1}$, and their root-degree fibers are parabolic Bruhat intervals realized by the strata of staircase quiver Grassmannians. The geometric (i.e., generic, canonical, and Mirkovi\'c--Vilonen) elements indexed by these vectors form integral bases of Fulton's standard-elementary module. We prove that $\mathfrak S_w(c)$ is homogeneous under diagonal conjugation if and only if it is the corresponding canonical element, and that homogeneity of $\mathfrak S_w(c)$ implies $\Lambda_Q$-rigidity of $Z_{\sf g_w}$. We also classify simultaneously the unit columns of the geometric-to-Schubert transitions, determine support components of the PBW-to-geometric and code-to-Schubert transitions, and exhibit a permutation $w\in S_{10}$ for which the three geometric basis elements are distinct.