Tropical flag varieties

Brandt, Madeline; Eur, Christopher; Zhang, Leon

doi:10.1016/j.aim.2021.107695

Cited by 24 publications

(41 citation statements)

References 14 publications

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“…In fact, in general, the tropical prevariety of a collection of polynomials will properly contain the tropical prevariety of the ideal those polynomials generate. In the specific case of the complete flag variety, it is shown in [3,Example 5.2.4] that for n ≥ 6, F lDr n properly contains trF l n . Before getting to Theorem 4.11, we need to discuss one other type of Dressian.…”

Section: Tropicalizing the Complete Flag Varietymentioning

confidence: 99%

“…We consider the set of points satisfying the tropicalizations of the incidence-Plücker relations, called the complete flag Dressian, F lDr n , and the set of points satisfying the tropicalizations of all polynomials in the incidence-Plücker ideal, called the tropical complete flag variety, T rF l n . These parameterize abstract flags of tropical linear spaces and realizable flags of tropical linear spaces, respectively [3].…”

Section: Introductionmentioning

confidence: 99%

“…The tropical spaces F lDr n and T rF l n are generally different [3]. Motivated by the example of the tropical Grassmannian, we will investigate the totally non-negative (TNN) parts of these spaces.…”

Section: Introductionmentioning

confidence: 99%

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Positive Tropical Flags and the Positive Tropical Dressian

Boretsky¹

2021

Preprint

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We study the totally non-negative part of the complete flag variety and of its tropicalization. We start by showing that Lusztig's notion of non-negative complete flag variety coincides with the flags in the complete flag variety which have non-negative Plücker coordinates. This mirrors the characterization of the totally non-negative Grassmannian as those points in the Grassmannian with all non-negative Plücker coordinates. 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. The complete flag Dressian properly contains the tropical complete flag variety. However, we show that the totally non-negative parts of these spaces coincide. * We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number 557353-2021]. Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), [numéro de référence 557353-2021].

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Section: Tropicalizing the Complete Flag Varietymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Positive Tropical Flags and the Positive Tropical Dressian

Boretsky¹

2021

Preprint

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“…In Sections 5 we discuss possible generalizations of our results in terms of a possible theory of buildings and complexes for tropical flag manifolds and flag varieties, which is inspired by some recent work in this field [BEZ21], [BH20]. We also recognize that the theory of finitely generated semimodules ties up with tropical convexity, as finitely generated semimodules are also considered as tropical cones in the literature [MS09], [Jos22].…”

Section: Modules (Vector Spaces) Semimodulesmentioning

confidence: 99%

Fundamental Theorem of Projective Geometry over Semirings

Tewari¹

2021

Preprint

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We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra setup over semirings. We also provide more explicit results to understand the implications of our main theorem on maps between tropical lines in the tropical plane. Along with this we also look at geometrical connections to the rich theory of tropical geometry.

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“…Valuative functions of these polyhedra were studied in [6, 16] as invariants that behave well with respect to subdividing the associated combinatorial objects into smaller ones. In particular, matroid subdivisions have rich connection to geometry, such as compactifications of fine Schubert cells [25–27], tropical geometry [10, 36], and the

K

‐theory of Grassmannians [17, 37]. The valuativeness of many matroid invariants, such as the beta invariant and the Tutte polynomial, is a witness to such geometric connections.…”

Section: Introductionmentioning

confidence: 99%

The universal valuation of Coxeter matroids

Eur

García

Supina

2021

Bulletin of London Math Soc

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Coxeter matroids generalize matroids just as flag varieties of Lie groups generalize Grassmannians. Valuations of Coxeter matroids are functions that behave well with respect to subdivisions of a Coxeter matroid into smaller ones. We compute the universal valuative invariant of Coxeter matroids. A key ingredient is the family of Coxeter Schubert matroids, which correspond to the Bruhat cells of flag varieties. In the process, we compute the universal valuation of generalized Coxeter permutohedra, a larger family of polyhedra that model Coxeter analogues of combinatorial objects such as matroids, clusters, and posets.

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Tropical flag varieties

Cited by 24 publications

References 14 publications

Positive Tropical Flags and the Positive Tropical Dressian

Positive Tropical Flags and the Positive Tropical Dressian

Fundamental Theorem of Projective Geometry over Semirings

The universal valuation of Coxeter matroids

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