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

الأصل · EN

For the ordered set [n] of n elements, we consider the class ₙ of bases B of tropical Plücker 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 ₙ, thus answering affirmatively a conjecture due to Leclerc and Zelevinsky. We also prove an analogous result for a hyper-simplex Δₙᵐ={S[n] |S|=m}.