# ALGEBRAIC COMBINATORICS

The Purity Conjecture in type $C$
Algebraic Combinatorics, Volume 3 (2020) no. 6, pp. 1401-1416.

A collection $𝒞$ of $k$-element subsets of $\left\{1,2,...,m\right\}$ is weakly separated if for each $I,J\in 𝒞$, when the integers $1,2,...,m$ are arranged around a circle, there is a chord separating $I\setminus J$ from $J\setminus I$. 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 Scott’s 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.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.145
Classification: 05C70,  05C75,  05E99
Keywords: plabic graphs, weakly separated collections, plabic tilings, symmetric plabic graphs, total positivity, Lagrangian Grassmannian.
Karpman, Rachel 1

1 Otterbein University Department of Mathematics and Actuarial Science 1 S. Grove Street Westerville OH 43081, USA
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Karpman, Rachel. The Purity Conjecture in type $C$. Algebraic Combinatorics, Volume 3 (2020) no. 6, pp. 1401-1416. doi : 10.5802/alco.145. https://alco.centre-mersenne.org/articles/10.5802/alco.145/

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