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Cluster Geometry of Universal Schubert Polynomials I: Geometric Bases and Schubert Transitions

  • Jiarui Fei
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Abstract

We study Fulton's universal Schubert polynomials Sw(c) as regular functions on the upper unitriangular group UN. The standard triangular cluster structure associates a Schubert sf g-vector to every permutation. Their convex hull is unimodularly equivalent to Δ1×⋯×ΔN−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 Mirković--Vilonen) elements indexed by these vectors form integral bases of Fulton's standard-elementary module. We prove that Sw(c) is homogeneous under diagonal conjugation if and only if it is the corresponding canonical element, and that homogeneity of Sw(c) implies ΛQ-rigidity of Zsf gw. 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∈ S10 for which the three geometric basis elements are distinct.

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Cite

BibTeXRIS
BibTeX
@misc{fei2026cluster,
  title = {{Cluster Geometry of Universal Schubert Polynomials I: Geometric Bases and Schubert Transitions}},
  author = {Jiarui Fei},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29125},
  eprint = {2608.29125},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29125},
}
RIS
TY  - UNPB
TI  - Cluster Geometry of Universal Schubert Polynomials I: Geometric Bases and Schubert Transitions
AU  - Jiarui Fei
PY  - 2026
DA  - 2026/08/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.29125
UR  - https://arxiv.org/abs/2608.29125
AB  - We study Fulton's universal Schubert polynomials 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 Δ_1×⋯×Δ_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 Mirković--Vilonen) elements indexed by these vectors form integral bases of Fulton's standard-elementary module. We prove that S_w(c) is homogeneous under diagonal conjugation if and only if it is the corresponding canonical element, and that homogeneity of S_w(c) implies Λ_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∈ S_10 for which the three geometric basis elements are distinct.
ER  -