Pl\"ucker environments, wiring and tiling diagrams, and weakly separated set-systems

TL;DR

This paper introduces a new class of bases for tropical Pl"ucker functions represented by wiring diagrams and tilings, proving they encompass all maximum weakly separated set-systems, thus confirming a conjecture.

Contribution

It defines the class _n of bases, shows they are representable via wiring diagrams and tilings, and proves they include all maximum weakly separated set-systems, confirming a conjecture.

Findings

Bases in _n are representable by wiring diagrams and tilings.

Maximum weakly separated set-systems are contained in _n.

The results confirm a conjecture by Leclerc and Zelevinsky.

Abstract

For the ordered set $[n]$ of $n$ elements, we consider the class $\Bscr_n$ of bases $B$ of tropical Pl\"ucker functions on $2^{[n]}$ such that $B$ can be obtained by a series of mutations (flips) from the basis formed by the intervals in $[n]$. We show that these bases are representable by special wiring diagrams and by certain arrangements generalizing rhombus tilings on the $n$-zonogon. Based on the generalized tiling representation, we then prove that each weakly separated set-system in $2^{[n]}$ having maximum possible size belongs to $\Bscr_n$, thus answering affirmatively a conjecture due to Leclerc and Zelevinsky. We also prove an analogous result for a hyper-simplex $\Delta_n^m=\{S\subseteq[n]\colon |S|=m\}$.