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- W4293489690 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 0909.1423[math.CO] we proved these purity conjectures for the Boolean cube $2^{[n]}$, the discrete Grassmanian ${[n]choose r}$, 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 $mathcal{I}nt(mathcal N )$. In this paper we give an alternative (and shorter) proof of the purity of $mathcal{I}nt(mathcal N )$ and present a stronger result. More precisely, we introduce a set-system $mathcal{O}ut(mathcal N )$ complementary to $mathcal{I}nt(mathcal N )$, 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." @default.
- W4293489690 created "2022-08-29" @default.
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- W4293489690 date "2013-12-11" @default.
- W4293489690 modified "2023-09-27" @default.
- W4293489690 title "The purity of set-systems related to Grassmann necklaces" @default.
- W4293489690 doi "https://doi.org/10.48550/arxiv.1312.3121" @default.
- W4293489690 hasPublicationYear "2013" @default.
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