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

Abstract

For the ordered set [n] of n elements, we consider the class B 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 B n , thus answering affirmatively a conjecture due to Leclerc and Zelevinsky. We also prove an analogous result for a hyper-simplex ∆ m n = {S ⊆ [n] : |S| = m}.for any triple i < j < k in [n] and any subset X ⊆ [n] − {i, j, k}. Throughout, for brevity we writeis called a TP-basis, or simply a basis, if the restriction map res : T P n → R B is a bijection. In other words, each TP-function is determined by its values on B, and moreover, values on B can be chosen arbitrarily. 2The first goal of this paper is to characterize B n . We give two characterizations for semi-normal bases: via a bijection to special collections of curves, that we call proper wirings, and via a bijection to certain graphical arrangements, called generalized tilings or, briefly, g-tilings (in fact, these characterizations are interrelated via planar duality). We associate to a proper wiring W (a g-tiling T ) a certain collection of subsets of [n] called its spectrum. It turns out that proper wirings and g-tilings are rigid objects, in the sense that any of these is determined by its spectrum.(More precisely, by a wiring we mean a set of n directed non-self-intersecting curves w 1 , . . . , w n in a region R of the plane homeomorphic to a circle, where each w i begins at a point s i and ends at a point s ′ i , and the points s 1 , . . . , s n , s ′ 1 , . . . , s ′ n are different and occur in this order clockwise in the boundary of R. A special wiring W that we deal with is defined by three axioms (W1)-(W3). Axiom (W1) is standard, it says that W preserves (topologically) under small deformations, i.e., no three wires have a common point, any two wires meet at a finite number of points and they cross, not touch, at each of these points. (W2) says that the common points of w i , w j follow in the opposed orders along these wires. The crucial axiom (W3) says that in the planar graph induced by W , there is a certain bijection between the faces ("chambers") whose boundary is a directed cycle and the regions ("lenses") surrounded by pieces of two wires between their neighboring common points. W is called proper if none of "cyclic" faces is a whole lens. The spectrum of W is the collection of subsets X ⊆ [n] oneto-one corresponding to the "non-cyclic" faces F , where X consists of the elements i such that F "lies on the right" from w i (according to the direction of w i ). When any two wires intersect exactly once, the dual planar graph is realized by a rhombus tiling, and vice versa (for a more general result of this sort, see [5]). In a...

Системы Разделенных Множеств И Их Геометрические Модели

“…Теперь ромбы могут перекрываться, и мы называем возникающие структуры обобщенными (ромбическими) тай-лингами. Отметим, что в работах [4], [5] изложение велось именно на языке обобщенных тайлингов.…”

“…также [4], [5]). Как и в строгом случае, основу их до-казательства составляет связь слабой разделенности с некоторыми плоскими проволочными диаграммами, называемыми далее вайрингами.…”

“…Четвертый (и самый сложный) раздел посвящен доказательству гипотез Леклерка и Зелевинского (см. также [4], [5], где даны альтернативные доказательства, использующие технику обобщенных ромбических тайлингов).…”

“…Рис. 52 Далее мы скажем кратко, опуская детали, которые можно найти в [4]. Как и раньше, камерам грассманова вайринга можно приписывать наборы цве-тов (спектры).…”

QUASI-DETERMINANTS AND q-COMMUTING MINORS

“…By now, many more commutation formulas are known for much larger collections of quantum minors (see [3,6]). The impetus for finding such results has been two-fold: (i) from the point of view of representation theory, such questions are intimately tied to the study of the canonical (or crystal) bases of Lustzig and Kashiwara [1,8,14]; (ii) from the point of view of non-commutative algebraic geometry, the study of quantum determinantal ideals provides non-commutative versions of the classical determinantal varieties [2,7,9]. Our goal is different.…”