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Peer-reviewedCC BY 4.0Algebraic Combinatorics

Totally nonnegative tropical flags and the totally nonnegative flag Dressian

  • Jonathan Boretsky1
  1. 1McGill University, Department of Mathematics and Statistics, 805 Sherbrooke St W, Montreal, QC H3A 3E8 (Canada)
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Abstract

We study the totally nonnegative part of the complete flag variety and of its tropicalization. We show that Lusztig’s notion of nonnegative complete flag variety coincides with the set of complete flags which have nonnegative Plücker coordinates. This mirrors the characterization of the totally nonnegative Grassmannian as the set of points in the Grassmannian with nonnegative Plücker. We then study the tropical complete flag variety and complete flag Dressian, which are two tropical versions of the complete flag variety, capturing realizable and abstract flags of tropical linear spaces, respectively. In general, the complete flag Dressian properly contains the tropical complete flag variety. However, we show that the totally nonnegative parts of these spaces coincide.

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Cite

BibTeXRIS
BibTeX
@article{boretsky2026totally,
  title = {{Totally nonnegative tropical flags and the totally nonnegative flag Dressian}},
  author = {Jonathan Boretsky},
  journal = {Algebraic Combinatorics},
  year = {2026},
  month = oct,
  doi = {10.5802/alco.511},
  url = {https://alco.centre-mersenne.org/articles/10.5802/alco.511/},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Totally nonnegative tropical flags and the totally nonnegative flag Dressian
AU  - Jonathan Boretsky
PY  - 2026
DA  - 2026/10/27
JO  - Algebraic Combinatorics
DO  - 10.5802/alco.511
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.511/
AB  - We study the totally nonnegative part of the complete flag variety and of its tropicalization. We show that Lusztig’s notion of nonnegative complete flag variety coincides with the set of complete flags which have nonnegative Plücker coordinates. This mirrors the characterization of the totally nonnegative Grassmannian as the set of points in the Grassmannian with nonnegative Plücker. We then study the tropical complete flag variety and complete flag Dressian, which are two tropical versions of the complete flag variety, capturing realizable and abstract flags of tropical linear spaces, respectively. In general, the complete flag Dressian properly contains the tropical complete flag variety. However, we show that the totally nonnegative parts of these spaces coincide.
SN  - 2589-5486
ER  -