The Tropical Geometry of Flag Positroids
Notice bibliographique
Résumé
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 of rank r=(r_1,..., r_k), defined by Lusztig, can be described as the set of rank r flags of linear subspaces in R^n 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 Plücker 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 Plücker 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 Plücker 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.
Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.
Comment cette classification a été obtenuedéplier
Prédiction machine sur la base complète
Imitation des enseignantsNi prévalence calibrée, ni vérité terrain. Validation humaine à venir. Le volet Gemma est une étiquette directe du modèle pour chaque travail de la base, lue sur la notice réduite au titre. Le volet Codex est un classifieur appris des 10 348 étiquettes directes de Codex et calibré sur les taux pondérés de l'échantillon; les champs sans appui suffisant ne portent aucun appel Codex. Le mode candidate est l'union des deux volets; le consensus est leur intersection. Ces sorties portent le statut machine_predicted_unvalidated et ne sont pas des étiquettes humaines.
Scores du classifieur distillé par catégorie (deux têtes)
| Catégorie | Codex | Gemma |
|---|---|---|
| Métarecherche | 0,000 | 0,001 |
| Méta-épidémiologie (sens strict) | 0,000 | 0,000 |
| Méta-épidémiologie (sens large) | 0,000 | 0,000 |
| Bibliométrie | 0,001 | 0,001 |
| Études des sciences et des technologies | 0,001 | 0,002 |
| Communication savante | 0,002 | 0,002 |
| Science ouverte | 0,000 | 0,001 |
| Intégrité de la recherche | 0,000 | 0,001 |
| Charge utile insuffisante (le modèle a refusé de juger) | 0,004 | 0,000 |
Scores machine (provisoires)
Les deux têtes enseignantes du modèle étudiant, lues sur ce travail. Un score ordonne la base pour la relecture; il n'affirme jamais une catégorie, et le statut de validation accompagne chaque rangée tel quel.
Scores de référence d'un modèle non mature (critères de maturité non atteints, 7 itérations). Un score ordonne; il n'affirme jamais une catégorie.
score_only:v0-immature-baseline · tel quel depuis la passe de notation : score_only signifie que le nombre peut ordonner les travaux, et qu'aucune étiquette de catégorie n'en découleClassification
machine, non validéePrédiction automatique; un appel candidat d’une seule source (Gemma direct ou Codex distillé), pas un consensus.
Le détail, modèle par modèle et score par score, se trouve en fin de page sous « Comment cette classification a été obtenue ».