자료유형  

 학위논문

Control Number  

0017162132

International Standard Book Number  

9798382775623

Dewey Decimal Classification Number  

510

Main Entry-Personal Name  

Boretsky, Jonathan.

Publication, Distribution, etc. (Imprint  

[S.l.] : Harvard University., 2024

Publication, Distribution, etc. (Imprint  

Ann Arbor : ProQuest Dissertations & Theses, 2024

Physical Description  

156 p.

General Note  

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

General Note  

Advisor: Williams, Lauren.

Dissertation Note  

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

Summary, Etc.  

요약Recently, 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.

Subject Added Entry-Topical Term  

Mathematics.

Subject Added Entry-Topical Term  

Theoretical mathematics.

Index Term-Uncontrolled  

Algebraic geometry

Index Term-Uncontrolled  

Combinatorics

Index Term-Uncontrolled  

Flag varieties

Index Term-Uncontrolled  

Positroids

Index Term-Uncontrolled  

Tropical geometry

Added Entry-Corporate Name  

Harvard University Mathematics

Host Item Entry  

Dissertations Abstracts International. 85-12B.

Electronic Location and Access  

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Control Number  

joongbu:658560