Toric degenerations of Gr(2,n) and Gr(3,6) via plabic graphs

TL;DR

This paper explores the connection between toric degenerations of Grassmannians and combinatorial objects called plabic graphs, establishing a bijection for Gr(2,n) and highlighting differences for Gr(3,6).

Contribution

It provides an explicit bijection between toric degenerations from tropical Grassmannians and plabic graphs for Gr(2,n), and shows this does not extend to Gr(3,6).

Findings

Bijection established for Gr(2,n) toric degenerations and plabic graphs.

The bijection does not hold for Gr(3,6).

Highlights differences in combinatorial structures between Gr(2,n) and Gr(3,6).

Abstract

We establish an explicit bijection between the toric degenerations of the Grassmannian $\textrm{Gr}(2,n)$ arising from maximal cones in tropical Grassmannians and the ones coming from plabic graphs corresponding to $\textrm{Gr}(2,n)$. We show that a similar statement does not hold for $\textrm{Gr}(3,6)$.