Tropical hyperplane arrangements and oriented matroids

Ardila, Federico; Develin, Mike

doi:10.1007/s00209-008-0400-z

Cited by 56 publications

(155 citation statements)

References 19 publications

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“…We write B 0 d for the set of all lattice classes [L] in V . Putting all apartments together, we see that the building B d is a geometric realization of a simplicial complex on B 0 d , namely, the flag complex of the graph on all lattice classes defined by the adjacency relation from (2). The group PG L(V ) acts in the natural way on B d and its vertex set B 0 d .…”

Section: Structure Of Mustafin Varietiesmentioning

confidence: 99%

“…Hence, the special fiber consists of two copies of P 1 k meeting transversely in one point. In the affine coordinates x = x 2 /x 1 and y = y 1 /y 2 We have the following formula for the thickness t in terms of two matrices g, h ∈ G L 2 (K ) that represent L = gL 0 and M = hL 0 relative to a reference lattice L 0 ⊂ V :…”

Section: Proposition 31 Ifmentioning

confidence: 99%

“…The tropical hypersurface T (P ) defined by this expression is an arrangement of n tropical hyperplanes (see [2]) in the (d − 1)-dimensional space A. Dual to this tropical hyperplane arrangement is a mixed subdivision [18, § 1.2] of the scaled standard simplex…”

Section: Fig 4 the Mustafin Variety Inmentioning

confidence: 99%

“…The combinatorial relationship between the mixed subdivision, the tropical hyperplane arrangement and the tropical polytope tconv( ) were introduced in [5, § 5] and further developed in [2,6]. Figure 6].…”

Section: Fig 4 the Mustafin Variety Inmentioning

confidence: 99%

“…Indeed, let L (1) and L (2) be any two lattices in V and consider = {[L (1) ], [L (2) ]}. Then, there exists an apartment A that contains , and we can represent both L (i) by vectors u (i) ∈ Z d as above.…”

Section: Hence a Polyhedral Subdivision Of D−1 × N−1 Is A Triangulatmentioning

confidence: 99%

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Mustafin varieties

Cartwright

Häbich

Sturmfels

et al. 2011

Sel. Math. New Ser.

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A Mustafin variety is a degeneration of projective space induced by a point configuration in a Bruhat-Tits building. The special fiber is reduced and CohenMacaulay, and its irreducible components form interesting combinatorial patterns. For configurations that lie in one apartment, these patterns are regular mixed subdivisions of scaled simplices, and the Mustafin variety is a twisted Veronese variety built from such a subdivision. This connects our study to tropical and toric geometry. For general configurations, the irreducible components of the special fiber are rational varieties, and any blow-up of projective space along a linear subspace arrangement can arise. A detailed study of Mustafin varieties is undertaken for configurations in the Bruhat-Tits tree of PG L(2) and in the 2-dimensional building of PG L(3). The latter yields the classification of Mustafin triangles into 38 combinatorial types. 758 D. Cartwright et al.Fig. 1 Special fibers of Mustafin varieties for d = 3 are degenerations of the projective plane. The two schemes depicted above arise from configurations of n = 4 points in B 3

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Section: Structure Of Mustafin Varietiesmentioning

confidence: 99%

Section: Proposition 31 Ifmentioning

confidence: 99%

Section: Fig 4 the Mustafin Variety Inmentioning

confidence: 99%

Section: Fig 4 the Mustafin Variety Inmentioning

confidence: 99%

Section: Hence a Polyhedral Subdivision Of D−1 × N−1 Is A Triangulatmentioning

confidence: 99%

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Mustafin varieties

Cartwright

Häbich

Sturmfels

et al. 2011

Sel. Math. New Ser.

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Tropical Oriented Matroids

Horn

2015

Springer INdAM Series

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Tropical oriented matroids were defined by Ardila and Develin in 2007. They are a tropical analogue of classical oriented matroids in the sense that they encode the properties of the types of points in an arrangement of tropical hyperplanes-in much the same way as the covectors of (classical) oriented matroids describe the types in arrangements of linear hyperplanes. Not every oriented matroid can be realised by an arrangement of linear hyperplanes though. The famous Topological Representation Theorem by Folkman and Lawrence, however, states that every oriented matroid can be represented as an arrangement of pseudohyperplanes. Ardila and Develin proved that tropical oriented matroids can be represented as mixed subdivisions of dilated simplices. In this paper I prove that this correspondence is a bijection. Moreover, I present a tropical analogue for the Topological Representation Theorem. Résumé. La notion de matroïde orienté tropical aété introduite par Ardila et Develin en 2007. Ils sont un analogue des matroïdes orientés classiques dans le sens où ils codent les propriétés des types de points dans un arrangement d'hyperplans tropicaux-d'une manière très similaireà celle dont les covecteurs des matroïdes orientés (classiques) décrivent les types de points dans les arrangements d'hyperplans linéaires. Tous les matroïdes orientés ne peuvent pasêtre représentés par un arrangement d'hyperplans linéaires. Cependant le célèbre théorème de représentation topologique de Folkman et Lawrence affirme que tout matroïde orienté peutêtre représenté par un arrangement de pseudo-hyperplans. Ardila et Develin ont prouvé que les matroïdes orientés tropicaux peuventêtre représentés par des sous-divisions mixtes de simplexes dilatés. Je prouve dans cet article que cette correspondance est une bijection. Je présente en outre, un analogue tropical du théorème de représentation topologique.

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Oriented matroids from triangulations of products of simplices

Celaya,

Loho,

Yuen

2024

Sel. Math. New Ser.

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We introduce a construction of oriented matroids from a triangulation of a product of two simplices. For this, we use the structure of such a triangulation in terms of polyhedral matching fields. The oriented matroid is composed of compatible chirotopes on the cells in a matroid subdivision of the hypersimplex, which might be of independent interest. In particular, we generalize this using the language of matroids over hyperfields, which gives a new approach to construct matroids over hyperfields. A recurring theme in our work is that various tropical constructions can be extended beyond tropicalization with new formulations and proof methods.

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Tropical hyperplane arrangements and oriented matroids

Cited by 56 publications

References 19 publications

Mustafin varieties

Mustafin varieties

Tropical Oriented Matroids

Oriented matroids from triangulations of products of simplices

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