Weak Separation and Plabic Graphs

TL;DR

This paper connects weak separation in quantum minors with plabic graphs, extending conjectures to positroids and proving their equivalence, which advances understanding of total positivity and combinatorial structures in the Grassmannian.

Contribution

It extends weak separation to positroids, proves conjectures relating maximal collections to plabic graphs, and establishes their combinatorial and geometric properties.

Findings

Maximal weakly separated collections correspond bijectively to plabic graphs.

The generalized conjectures of Leclerc, Zelevinsky, and Scott are proven.

The results confirm the purity and mutation connectedness of positroid stratifications.

Abstract

Leclerc and Zelevinsky described quasicommuting families of quantum minors in terms of a certain combinatorial condition, called weak separation. They conjectured that all maximal by inclusion weakly separated collections of minors have the same cardinality, and that they can be related to each other by a sequence of mutations. On the other hand, Postnikov studied total positivity on the Grassmannian. He described a stratification of the totally nonnegative Grassmannian into positroid strata, and constructed their parametrization using plabic graphs. In this paper we link the study of weak separation to plabeic graphs. We extend the notion of weak separation to positroids. We generalize the conjectures of Leclerc and Zelevinsky, and related ones of Scott, and prove them. We show that the maximal weakly separated collections in a positroid are in bijective correspondence with the plabic graphs. This correspondence allows us to use the combinatorial techniques of positroids and plabic graphs to prove the (generalized) purity and mutation connectedness conjectures.