Related papers: Weak Separation and Plabic Graphs
A collection $\mathcal{C}$ of $k$-element subsets of $\{1,2,\ldots,m\}$ is weakly separated if for each $I, J \in \mathcal{C}$, when the integers $1,2,\ldots,m$ are arranged around in a circle, there is a chord separating $I \backslash J$…
Plabic graphs are interesting combinatorial objects used to study the totally nonnegative Grassmannian. Faces of plabic graphs are labeled by $k$-element sets of positive integers, and a collection of such $k$-element sets are the face…
We adopt a formal and algebraic approach of Early \cite{E2} to study the positive tropical Grassmannian $\operatorname{Trop}^+ Gr_{k,n}$. Specifically, we deal with positroid subdivision of hypersimplex induced by translated blades from any…
Weakly separated collections arise in the cluster algebra derived from the Pl\"ucker coordinates on the nonnegative Grassmannian. Oh, Postnikov, and Speyer studied weakly separated collections over a general Grassmann necklace $\mathcal{I}$…
A plabic graph is a planar bicolored graph embedded in a disk, which satisfies some combinatorial conditions. Postnikov's boundary measurement map takes the space of positive edge weights of a plabic graph $G$ to a positroid cell in some…
For an odd integer $r>0$ and an integer $n>r$, we introduce a notion of weakly $r$-separated collections of subsets of $[n]=\{1,2,\ldots,n\}$. When $r=1$, this corresponds to the concept of weak separation introduced by Leclerc and…
Leclerc and Zelevinsky, motivated by the study of quasi-commuting quantum flag minors, introduced the notions of strongly separated and weakly separated collections. These notions are closely related to the theory of cluster algebras, to…
In 1998, Leclerc and Zelevinsky introduced the notion of weakly separated collections of subsets of the ordered $n$-element set $[n]$ (using this notion to give a combinatorial characterization for quasi-commuting minors of a quantum…
Studying the problem of quasicommuting quantum minors, Leclerc and Zelevinsky introduced in 1998 the notion of weakly separated sets in $[n]:=\{1,\ldots, n\}$. Moreover, they raised several conjectures on the purity for this symmetric…
We show that the partial sums of the long Pl\"ucker relations for pairs of weakly separated Pl\"ucker coordinates oscillate around $0$ on the totally nonnegative part of the Grassmannian. Our result generalizes the classical oscillating…
We discuss arrangements of equal minors of totally positive matrices. More precisely, we investigate the structure of equalities and inequalities between the minors. We show that arrangements of equal minors of largest value are in…
The Grassmannian is a disjoint union of open positroid varieties $P_v$, certain smooth irreducible subvarieties whose definition is motivated by total positivity. The coordinate ring of $P_v$ is a cluster algebra, and each reduced plabic…
For a permutation $\omega\in S_n$, Leclerc and Zelevinsky \cite{LZ} introduced a concept of $\omega$-{\em chamber weakly separated collection} of subsets of $\{1,2,...,n\}$ and conjectured that all inclusion-wise maximal collections of this…
Following the proof of the purity conjecture for weakly separated collections, recent years have revealed a variety of wider examples of purity in different settings. In this paper we consider the collection $\mathcal A_{I,J}$ of sets that…
Positroids are families of matroids introduced by Postnikov in the study of non-negative Grassmannians. In particular, positroids enumerate a CW decomposition of the totally non-negative Grassmannian. Furthermore, Postnikov has identified…
Plabic graphs are intimately connected to the positroid stratification of the positive Grassmannian. The duals to these graphs are quivers, and it is possible to associate to them cluster algebras. For the top-cell graph of $Gr_{+}(k,n)$,…
Amalgamation in the totally non-negative part of positroid varieties is equivalent to gluing copies of $Gr^{TP}(1,3)$ and $Gr^{TP}(2,3)$. Lam has proposed to represent amalgamation in positroid varieties by equivalence classes of relations…
We say that two sets $S,T\subset\{1,2,\dots,n\}$ are chord separated if there does not exist a cyclically ordered quadruple $a,b,c,d$ of integers satisfying $a,c\in S-T$ and $b,d\in T-S$. This is a weaker version of Leclerc and Zelevinsky's…
A graph polynomial $P$ is weakly distinguishing if for almost all finite graphs $G$ there is a finite graph $H$ that is not isomorphic to $G$ with $P(G)=P(H)$. It is weakly distinguishing on a graph property $\mathcal{C}$ if for almost all…
Postnikov's plabic graphs in a disk are used to parametrize totally positive Grassmannians. In recent years plabic graphs have found numerous applications in math and physics. One of the key features of the theory is the fact that if a…