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Author ORCID Identifier

https://orcid.org/0009-0009-4695-302X

Date Available

5-6-2026

Year of Publication

2026

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy (PhD)

College

Arts and Sciences

Department/School/Program

Mathematics

Faculty

Martha Yip

Faculty

Bertrand Guillou

Abstract

The nonnegative Grassmannian admits a widely studied cell decomposition due to Alexander Postnikov, whose cells are indexed by positroids and modeled by several equivalent combinatorial objects. Subsequent work by authors including Lauren Williams, Suho Oh, and Carolina Benedetti has further developed the combinatorics and geometry of these structures. In this dissertation, we extend some of Postnikov’s combinatorial framework to the nonnegative flag variety. While cell decompositions in this setting were previously obtained, notably in work of Konstanze Rietsch, our focus is on providing new combinatorial models that make this structure more explicit and computationally tractable. We introduce flag positroid pipe dreams, a diagrammatic model analogous to Le diagrams, together with associated directed graphs and networks and use them to compute the flag bases of a flag positroid and points within the flag positroid cell of the flag variety. Using this framework, we give a constructive proof of a conjecture by Benedetti, Chávez, and Tamayo in the case of nonnegatively representable quotients, giving a complete characterization of all positroids of which a fixed positroid is such a quotient in terms of decorated permutations and diagrammatic data. Our approach also highlights the connection between flag positroids and intervals in the Bruhat order. Building on work of Jonathan Boretsky, Christopher Eur, and Williams, we show that flag positroid pipe dreams are in bijection with Bruhat intervals, where the number of ones in the diagram encodes the interval length and hence the dimension of the corresponding Richardson variety. Together, these results provide a unified combinatorial perspective on nonnegativity in flag varieties.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2026.143

Archival?

Archival

Funding Information

This research was supported by the Max Steckler Fellowship for the 2022-2023 academic year and the Cary H. Webb Fellowship for the summer of 2024.

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