Publication

Matching fields and lattice points of simplices

Georg Peter Loho
2020
Journal Articles

Abstract

We show that the Chow covectors of a linkage matching field define a bijection of lattice points and we demonstrate how one can recover the linkage matching field from this bijection. This resolves two open questions from Sturmfels & Zelevinsky (1993) on linkage matching fields. For this, we give an explicit construction that associates a bipartite incidence graph of an ordered partition of a common set to all lattice points in a dilated simplex. Given a triangulation of a product of two simplices encoded by a set of bipartite trees, we similarly prove that the bijection from left to right degree vectors of the trees is enough to recover the triangulation. As additional results, we show a cryptomorphic description of linkage matching fields and characterise the flip graph of a linkage matching field in terms of its prodsimplicial flag complex. Finally, we relate our findings to transversal matroids through the tropical Stiefel map.

Official source

https://infoscience.epfl.ch/entities/publication/7fa97b46-1f81-4323-8eb0-e42be410ed53

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Georg Peter Loho
2020
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/7fa97b46-1f81-4323-8eb0-e42be410ed53

About this result

Related concepts (32)

Bipartite graph

In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets and , that is, every edge connects a vertex in to one in . Vertex sets and are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets and may be thought of as a coloring of the graph with two colors: if one colors all nodes in blue, and all nodes in red, each edge has endpoints of differing colors, as is required in the graph coloring problem.

Set theory

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory.

Set (mathematics)

A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics.

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