相关论文: Positive Tropical Flags and the Positive Tropical …

We study the totally nonnegative part of the complete flag variety and of its tropicalization. We show that Lusztig's notion of nonnegative complete flag variety coincides with the flags in the complete flag variety which have nonnegative…

We study positivity notions for the tropicalization of type A flag varieties and the flag Dressian. We focus on the hollow case, where we have one constituent of rank 1 and another of corank 1. We characterize the three different notions…

For Grassmannians, Lusztig's notion of total positivity coincides with positivity of the Plucker coordinates. This coincidence underpins the rich interaction between matroid theory, tropical geometry, and the theory of total positivity.…

Flag matroids are combinatorial abstractions of flags of linear subspaces, just as matroids are of linear subspaces. We introduce the flag Dressian as a tropical analogue of the partial flag variety, and prove a correspondence between: (a)…

Given integers $1 \le k_1 < \cdots < k_l \le n-1$, let $\text{Fl}_{k_1,\dots,k_l;n}$ denote the type $A$ partial flag variety consisting of all chains of subspaces $(V_{k_1}\subset\cdots\subset V_{k_l})$ inside $\mathbb{R}^n$, where each…

Tropical algebraic geometry is the geometry of the tropical semiring (R, min, +). The theory of total positivity is a natural generalization of the study of matrices with all minors positive. In this paper we introduce the totally positive…

We study the relation between the integer tropical points of a cluster variety (satisfying the full Fock-Goncharov conjecture) and the totally positive part of the tropicalization of an ideal presenting the corresponding cluster algebra.…

The Dressian and the tropical Grassmannian parameterize abstract and realizable tropical linear spaces; but in general, the Dressian is much larger than the tropical Grassmannian. There are natural positive notions of both of these spaces…

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…

In this paper we study tropicalization of Grassmannian and linear varieties. In particular, we study the tropical linear spaces cor- responding to the phylogenetic trees. We prove that corresponding to each subtree of the phylogenetic tree…

We investigate the tropical analogues of totally positive and totally nonnegative matrices. These arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued…

Let $G$ be a Kac-Moody group, split over $\mathbb R$. The totally nonnegative part of $G$ and its (ordinary) flag variety $G/B^+$ was introduced by Lusztig. It is known that the totally nonnegative parts of $G$ and $G/B^+$ have remarkable…

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…

The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative…

A complete flag in $\mathbb{R}^n$ is a sequence of nested subspaces $V_1 \subset \cdots \subset V_{n-1}$ such that each $V_k$ has dimension $k$. It is called totally nonnegative if all its Pl\"ucker coordinates are nonnegative. We may view…

While the positive Grassmannian is deeply understood through the rich combinatorics of plabic graphs and positroid cells, its tropical counterpart, the positive tropical Grassmannian Trop$_{>0}G(k,n)$, has lacked a comparable structural…

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…

In this article, we present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of finite symmetries. We compute the tropical Grassmannian TGr$_0(3,8)$, and show that it refines the…

The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) $J$-total positivity on the full flag variety of an arbitrary…

The nonnegative Grassmannian is a cell complex with rich geometric, algebraic, and combinatorial structures. Its study involves interesting combinatorial objects, such as positroids and plabic graphs. Remarkably, the same combinatorial…