Abstract
A collection đ of k-element subsets of {1,2,...,m} is weakly separated if for each I,Jâđ, when the integers 1,2,...,m are arranged around a circle, there is a chord separating IâJ from JâI. Oh, Postnikov and Speyer constructed a correspondence between weakly separated collections which are maximal by inclusion and reduced plabic graphs, a class of networks defined by Postnikov which give coordinate charts on the Grassmannian of k-planes in m-space. As a corollary, they proved Scottâs Purity Conjecture, which states that a weakly separated collection is maximal by inclusion if and only if it is maximal by size. In this note, we describe maximal weakly separated collections corresponding to symmetric plabic graphs, which give coordinate charts on the Lagrangian Grassmannian, and prove a symmetric version of the Purity Conjecture.
Highlights
Two k-element subsets I and J of [m] := {1, 2, . . . , m} are weakly separated if, when the integers 1, 2, . . . , m are arranged around a circle, there is a chord separating I\J from J\I
Scott showed that a collection of k-element subsets of [m] that are pairwise weakly separated has order at most k(m â k) + 1, and made the following Purity Conjecture: a weakly separated collection of k-element subsets of [m] is maximal by inclusion if and only if it is maximal by size [8]
We have shown that the face PlĂŒcker map F restricts to an isomorphism from a totally nonnegative cell in Î(2n) to the space of positive, symmetric face weightings of a symmetric plabic graph
Summary
Two k-element subsets I and J of [m] := {1, 2, . . . , m} are weakly separated if, when the integers 1, 2, . . . , m are arranged around a circle, there is a chord separating I\J from J\I. The face labels of a plabic graph form a weakly separated collection which is maximal by size [9]. That a weakly separated collection is maximal by inclusion if and only if it is the set of face labels of a plabic graph for Gr>0(k, m). (2) C is the set of face labels of a symmetric plabic graph for Î>0(2n) It follows that a symmetric weakly separated collection is maximal by inclusion if and only if it is maximal by size. The totally nonnegative part of Î(2n) has an analogous stratification, with Î>0(2n) being the unique top-dimensional cell, and our results apply to these lower-dimensional cells, as well as to Î>0(2n)
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