| Pl\"ucker environments, wiring and tiling diagrams, and weakly separated set-systems (2009) | |||||||||
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.. Comment: 42 pages. This revision differs from the original version by minor stilistic improvements, typos corrections and reference updates | |||||||||
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