Plücker environments, wiring and tiling diagrams, and weakly separated set-systems

Details Plücker environments, wiring and tiling diagrams, and weakly separated set-systems Plücker environments, wiring and tiling diagrams, and weakly separated set-systems

2009-02-19

arxiv.org/abs/0902.3362v3

Advances in Mathematics, 224, 2010, 1-44

Mathematics

Combinatorics

47 pages. In this revision we add an Appendix containing results on weakly separated set-systems in a hyper-simplex and relate

Scientific paper

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 $\Delta_n^m=\{S\subseteq[n]\colon |S|=m\}$.

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