Algebraic Combinatorics· 2026Q1
Totally nonnegative tropical flags and the totally nonnegative flag Dressian
- 2citations
- Q1SCImago
- 2026year
Short summary
The totally nonnegative part of the complete flag variety is shown to be the set of complete flags with nonnegative Plücker coordinates, mirroring the Grassmannian.
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Key points
- The totally nonnegative complete flag variety equals flags with nonnegative Plücker coordinates.
- This result mirrors the characterization of the totally nonnegative Grassmannian.
- The totally nonnegative parts of the tropical complete flag variety and the complete flag Dressian coincide.
AI-generated from the title and abstract; the full text is not read.
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.
The authors' abstract, as published at the source. Algebraic Combinatorics, 2026 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics