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The Tropical Geometry of Flag Positroids.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	The Tropical Geometry of Flag Positroids./
Author:	Boretsky, Jonathan.
Description:	1 online resource (156 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 85-12, Section: B.
Contained By:	Dissertations Abstracts International85-12B.
Subject:	Mathematics. -
Online resource:	click for full text (PQDT)
ISBN:	9798382775623

The Tropical Geometry of Flag Positroids.
Boretsky, Jonathan.

The Tropical Geometry of Flag Positroids. - 1 online resource (156 pages)

Source: Dissertations Abstracts International, Volume: 85-12, Section: B.

Thesis (Ph.D.)--Harvard University, 2024.

Includes bibliographical references

Recently, there has been interest in the tropicalization of the Grassmannian, known as the tropical Grassmannian. It has been shown that the tropical Grassmannian, and a related tropical space called the Dressian, can be used to construct subdivisions of matroid polytopes into matroid polytopes. There has also been interest "nonnegative" and "flag" versions of the tropical Grassmannian and the Dressian, which can be used to construct subdivisions of "nonnegative" and "flag" versions of matroid polytopes. This thesis combines these perspectives, exploring "nonnegative flag" versions of these objects. We show that, for flags whose ranks consist of consecutive integers, many nice results that hold in the previously studied settings have "nonnegative flag" analogues.The nonnegative flag variety Fl≥0 r;n of rank r=(r1,..., rk), defined by Lusztig, can be described as the set of rank r flags of linear subspaces in Rn which can be represented by a matrix whose minors are all positive. We show that, for flag varieties of consecutive rank, this equals the subset of the flag variety with nonnegative Plucker coordinates. This generalizes the well-established fact, proven independently by many authors including Rietsch, Talaska and Williams, Lam, and Lusztig, that the nonnegative Grassmannian equals the subset of the Grassmannian with nonnegative Plucker coordinates, and yields a straightforward condition to determine whether a flag of consecutive rank lies in the nonnegative flag variety. A flag F in any nonnegative flag variety determines a combinatorial object called a flag positroid, defined in terms of the set of nonzero Plucker coordinates of F. We characterize flag positroids of consecutive ranks as oriented flag matroids whose constituents are positively oriented matroids.We also explore flag varieties using tropical geometry, which is the study of algebraic geometry over the (min,+) semifield. The tropical flag variety and the flag Dressian are tropical spaces parameterizing realizable and abstract flags of tropical linear spaces, respectively. In general, the flag Dressian strictly contains the tropical flag variety. However, we show that the nonnegative tropical flag variety and the nonnegative flag Dressian, which parameterize positively realizable and abstract flags of positive tropical linear spaces, respectively, are equal for flags of consecutive ranks. This generalizes the equality of the nonnegative Dressian and the nonnegative tropical Grassmannian, proven by Speyer and Williams.The flag Dressian is closely related to coherent subdivisions of flag matroid polytopes: it is a polyhedral complex whose cones parameterize coherent subdivisions of flag matroid polytopes into smaller flag matroid polytopes. We specialize this relationship to the nonnegative flag Dressian. We prove that the restriction of this parameterization to the cones of the nonnegative flag Dressian instead parameterizes coherent subdivisions of flag positroid polytopes into smaller flag positroid polytopes. In the special case of the positive (non-flag) Dressian, this recovers the description of coherent subdivisions of hypersimplices into positroid polytopes by Lukowski, Parisi, and Williams. In the special case of the nonnegative complete flag Dressian, this describes coherent subdivisions of Bruhat interval polytopes into Bruhat interval polytopes.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2024

Mode of access: World Wide Web

ISBN: 9798382775623Subjects--Topical Terms:

527692
Mathematics.
Subjects--Index Terms:

Algebraic geometryIndex Terms--Genre/Form:

554714
Electronic books.

The Tropical Geometry of Flag Positroids.
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500 $a Source: Dissertations Abstracts International, Volume: 85-12, Section: B.
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504 $a Includes bibliographical references
520 $a Recently, there has been interest in the tropicalization of the Grassmannian, known as the tropical Grassmannian. It has been shown that the tropical Grassmannian, and a related tropical space called the Dressian, can be used to construct subdivisions of matroid polytopes into matroid polytopes. There has also been interest "nonnegative" and "flag" versions of the tropical Grassmannian and the Dressian, which can be used to construct subdivisions of "nonnegative" and "flag" versions of matroid polytopes. This thesis combines these perspectives, exploring "nonnegative flag" versions of these objects. We show that, for flags whose ranks consist of consecutive integers, many nice results that hold in the previously studied settings have "nonnegative flag" analogues.The nonnegative flag variety Fl≥0 r;n of rank r=(r1,..., rk), defined by Lusztig, can be described as the set of rank r flags of linear subspaces in Rn which can be represented by a matrix whose minors are all positive. We show that, for flag varieties of consecutive rank, this equals the subset of the flag variety with nonnegative Plucker coordinates. This generalizes the well-established fact, proven independently by many authors including Rietsch, Talaska and Williams, Lam, and Lusztig, that the nonnegative Grassmannian equals the subset of the Grassmannian with nonnegative Plucker coordinates, and yields a straightforward condition to determine whether a flag of consecutive rank lies in the nonnegative flag variety. A flag F in any nonnegative flag variety determines a combinatorial object called a flag positroid, defined in terms of the set of nonzero Plucker coordinates of F. We characterize flag positroids of consecutive ranks as oriented flag matroids whose constituents are positively oriented matroids.We also explore flag varieties using tropical geometry, which is the study of algebraic geometry over the (min,+) semifield. The tropical flag variety and the flag Dressian are tropical spaces parameterizing realizable and abstract flags of tropical linear spaces, respectively. In general, the flag Dressian strictly contains the tropical flag variety. However, we show that the nonnegative tropical flag variety and the nonnegative flag Dressian, which parameterize positively realizable and abstract flags of positive tropical linear spaces, respectively, are equal for flags of consecutive ranks. This generalizes the equality of the nonnegative Dressian and the nonnegative tropical Grassmannian, proven by Speyer and Williams.The flag Dressian is closely related to coherent subdivisions of flag matroid polytopes: it is a polyhedral complex whose cones parameterize coherent subdivisions of flag matroid polytopes into smaller flag matroid polytopes. We specialize this relationship to the nonnegative flag Dressian. We prove that the restriction of this parameterization to the cones of the nonnegative flag Dressian instead parameterizes coherent subdivisions of flag positroid polytopes into smaller flag positroid polytopes. In the special case of the positive (non-flag) Dressian, this recovers the description of coherent subdivisions of hypersimplices into positroid polytopes by Lukowski, Parisi, and Williams. In the special case of the nonnegative complete flag Dressian, this describes coherent subdivisions of Bruhat interval polytopes into Bruhat interval polytopes.
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2024
538 $a Mode of access: World Wide Web
650 4 $a Mathematics. $3 527692
650 4 $a Theoretical mathematics. $3 1180455
653 $a Algebraic geometry
653 $a Combinatorics
653 $a Flag varieties
653 $a Positroids
653 $a Tropical geometry
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710 2 $a Harvard University. $b Mathematics. $3 1193029
773 0 $t Dissertations Abstracts International $g 85-12B.
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