Related papers: Tropical Quiver Grassmannians
Based on the notion of vectors and linear subspaces for a matroid, we develop a theory of flats and hyperplane arrangements for T-matroids, where T is a tract. This leads to several cryptomorphic descriptions of T-matroids: in terms of its…
In this paper, we develop a tropical analog of the classical flag variety that we call the flag Dressian. We find relations, which we call "tropical incidence relations", for when one tropical linear space is contained in another, and show…
We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide two algorithms for finding them in affine spaces of complementary dimension to the zero set. We use these…
Given $d$ row vectors of $n$ tropical numbers, $d<n$, the tropical Stiefel map constructs a version of their row space, whose Pl\"ucker coordinates are tropical determinants. We explicitly describe the fibers of this map. From the viewpoint…
Degenerations of linear series on smooth projective varieties approaching multicomponent varieties $X$ give rise to certain quiver representations in the category of linear series over $X$, which yield rational maps from $X$ to the…
We study Dressians of matroids using the initial matroids of Dress and Wenzel. These correspond to cells in regular matroid subdivisions of matroid polytopes. An efficient algorithm for computing Dressians is presented, and its…
We develop the rudiments of a finite-dimensional representation theory of groups over idempotent semifields by considering linear actions on tropical linear spaces. This can be considered a tropical representation theory, a characteristic…
We introduce combinatorial objects which are parameterized by the positive part of the tropical Grassmannian $Gr(k,n)$. Our method is to relate the Grassmannian to configuration spaces of flags. By work of the first author, and of Goncharov…
We study linear degenerations of flag varieties from the point of view of tropical geometry. We define the linear degenerate flag Dressian and prove a correspondence between: $(a)$ points in the linear degenerate flag Dressian, $(b)$ linear…
Tropicalisation (with trivial coefficients) is a process that turns a polynomial equation into a combinatorial predicate on subsets of the set of variables. We show that for each minuscule representation of a simple reductive group, there…
We study the moduli space of $d$-dimensional linear subspaces contained in a fixed $(d+1)$-dimensional linear variety $X$, and its tropicalization. We prove that these moduli spaces are linear subspaces themselves, and thus their…
We explore several facets of tropical subrepresentations of a linear representation of a group over the tropical semifield $\mathbb{T}$. A key role in the study of tropical subrepresentations is played by two types of modules over a…
In this paper we address two of the major foundational questions in the theory of matroids over rings. First, we provide a cryptomorphic axiomatisation, by introducing an analogue of the base polytope for matroids. Second, we describe a…
We show that any quiver Grassmannian associated with a rigid representation of a quiver is a rational variety using torus localization techniques.
When filtering a topological space by a single parameter, the theory of quiver representations provides a complete framework for decomposing the resulting persistence module to obtain its barcode. This is achieved by interpreting the…
Recent developments in particle physics have revealed deep connections between scattering amplitudes and tropical geometry. From the heart of this relationship emerged the chirotropical Grassmannian $\text{Trop}^\chi \text{G}(k,n)$ and the…
We tropicalize the rational map that takes triples of points in the projective plane to the plane of quadrics passing through these points. The image of its tropicalization is contained in the tropicalization of its image. We identify these…
Following up on our previous work, we study single-element extensions of transversal valuated matroids. We show that tropical presentations of valuated matroids with a minimal set of finite entries enjoy counterparts of the properties…
In this text, we characterize the representation type of an acyclic quiver by the properties of its associated quiver Grassmannians. This characterization utilizes and extends known results about singular quiver Grassmannians and cell…
In tropical algebraic geometry, the solution sets of polynomial equations are piecewise-linear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral subcomplex of the Grobner fan. The tropical…