Finest positroid subdivisions from maximal weakly separated collections

TL;DR

This work analyzes the positive tropical Grassmannian $\mathrm{Trop}^{+}Gr_{k,n}$ through blade arrangements, showing how translated blades from maximal weakly separated collections induce positroid subdivisions of the hypersimplex $\Delta_{k,n}$. It introduces weighted blade arrangements $\mathcal{Z}_{k,n}$ and a boundary-map framework that identifies when such arrangements realize $\mathrm{Trop}^{+}Gr_{k,n}$ via the quotient fan $\overline{\mathcal{Z}_{k,n}}$, and proves a key necessary-and-sufficient condition (Theorem 1) for when a maximal $w$-collection yields a simplicial cone; in the special case $k=2$, every maximal $w$-collection of $2$-element sets yields a simplicial cone, making all resulting subdivisions finest. A second theorem, proved using reduced plabic graphs, shows that boundary maps preserve maximality of weakly separated collections, implying that translated blades from any maximal $w$-collection produce the finest positroid subdivision and that flips between maximal collections correspond to adjacent maximal cones in $\overline{\mathcal{Z}_{k,n}}$. The results firmly connect combinatorial weak separation, plabic graph techniques, and tropical geometry, illuminating the structure of finest positroid subdivisions and their adjacency relations in the positive tropical Grassmannian.

Abstract

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 maximal weakly separated collection. One of our main results gives a necessary and sufficient condition on a maximal weakly separated collection to form a positroid subdivision of a hypersimplex corresponding to a simplicial cone in $\rm Trop^+Gr_{k,n}$. For k = 2 our condition says that any weakly separated collection of two-elements sets gives such a simplicial cone, and all cones are of such a form. We also show that the maximality of any weakly separated collection is preserved under the boundary map, which armatively answers a question by Early in \cite{E1}. Plabic graphs, invented by Postnikov \cite{P}, are of use in proving this result. As a corollary, we get that all those positroid subdivisions are the finest. Thus, the flip of two maximal weakly separatedcollections corresponds to a pair of adjacent maximal cones in positive tropical Grassmannian.

Figures (21)

• Figure 1: The expansions of $\pi_{l_pl_q}^{(L)}$ and $\sum\limits_{\{l_p,l_q\} \in A}\pi_{l_pl_q}^{(L)}$
• Figure 2: Two non-simplicial bipyramid maximal cones in $\overline{\mathcal{Z}_{3,6}}$.
• Figure 3: The weighted blade $-\beta_{135} + \beta_{145} + \beta_{136} + \beta_{235}$
• Figure 4:
• Figure 5:

Theorems & Definitions (76)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6