The purity of set-systems related to Grassmann necklaces

Studying the problem of quasi-commuting quantum minors, Leclerc and Zelevinsky [3] introduced the notion of weakly separated sets in [n] := {1, . . . , n}. They raised conjectures on the purity, with respect to this weakly separation relation, of certain set-systems. In particular, a conjecture on the purity of the whole Boolean cube 2; equivalently: on the max-clique purity of the graph on 2 whose edges are generated by this relation. (Here a finite graph G is called pure if all (inclusion-wise) maximal cliques in it have the same cardinality.) Answering those conjectures, [1] proved the purity of 2 and some other set-systems, including the discrete Grassmannian ( [n] r ) . In [5] the purity was proved for a certain weakly separated set-system inside the so-called positroid defined by a Grassmann necklace N . We denote this set-system as Int(N ). It is a special collection in the discrete Grassmannian, and the whole discrete Grassmannian is obtained by considering the ‘largest’ necklace. In this paper we give an alternative (and shorter) proof of the purity of Int(N ) and present a stronger result. More precisely, we introduce a set-system Out(N ) complementary to Int(N ), in a sense, and establish its purity. This is a consequence of the result (Theorem 3) that the set-systems Int(N ) and Out(N ) are weakly separated from each other, which means that any element of the former is weakly separated from any element of the latter. To prove this, we use a technique of plabic tilings from [5]. As one more consequence of Theorem 3, we also demonstrate the purity of set-systems related to pairs of weakly separated necklaces (Proposition 4 and Corollaries 1 and 2). Finally, we introduce a notion of generalized necklaces and claim the purity of the corresponding interior and exterior set-systems related to them; this extends the statement of Theorem 3. (This claim and a further generalization were recently shown in [2]. Note that results in [2] and [6] give rise to a cluster algebra structure on the interior and exterior of a generalized necklace. However, the natural questions on a positroid concerning the purity and the existence of a cluster algebra structure are still open.)