# The purity of set-systems related to Grassmann necklaces

1 CEMI - Central Institute of Economics and Mathematics of the RAS 2 ISA - Institute for System Analysis of the RAS

Abstract : Studying the problem of quasicommuting quantum minors, Leclerc and Zelevinsky introduced in 1998 the notion of weakly separated sets in $n:=\{1,\ldots, n\}$. Moreover, they raised several conjectures on the purity for this symmetric relation, in particular, on the Boolean cube $2^{n}$. In 0909.1423math.CO we proved these purity conjectures for the Boolean cube $2^{n}$, the discrete Grassmanian $\binom{n}{r}$, and some other set-systems. Oh, Postnikov, and Speyer in arxiv:1109.4434 proved the purity for weakly separated collections inside a positroid which contain a Grassmann necklace $\mathcal {N}$ defining the positroid. We denote such set-systems as $\mathcal{Int}\mathcal {N}$. In this paper we give an alternative and shorter proof of the purity of $\mathcal{Int}\mathcal {N}$ and present a stronger result. More precisely, we introduce a set-system $\mathcal{Out}\mathcal {N}$ complementary to $\mathcal{Int}\mathcal {N }$, in a sense, and establish its purity. Moreover, we prove Theorem~3 that these two set-systems are weakly separated from each other. As a consequence of Theorem~3, we obtain the purity of set-systems related to pairs of weakly separated necklaces Proposition 4 and Corollaries 1 and 2. Finally, we raise a conjecture on the purity of both the interior and exterior of a generalized necklace.

Keywords : set systems Grassmann necklaces

Author: Vladimir Danilov - Alexander Karzanov - Gleb Koshevoy -

Source: https://hal.archives-ouvertes.fr/