Weak Separation, Pure Domains and Cluster Distance

International audience Following the proof of the purity conjecture for weakly separated sets, recent years have revealed a variety of wider classes of pure domains in different settings. In this paper we show the purity for domains consisting of sets that are weakly separated from a pair of “generic” sets I and J. Our proof also gives a simple formula for the rank of these domains in terms of I and J. This is a new instance of the purity phenomenon which essentially differs from all previously known pure domains. We apply our result to calculate the cluster distance and to give lower bounds on the mutation distance between cluster variables in the cluster algebra structure on the coordinate ring of the Grassmannian. Using a linear projection that relates weak separation to the octahedron recurrence, we also find the exact mutation distances and cluster distances for a family of cluster variables.

Arrangements of equal minors in the positive Grassmannian

International audience We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with <i>sorted sets</i>, which earlier appeared in the context of <i>alcoved polytopes</i> and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the <i>Eulerian number</i>. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the <i>weakly separated sets</i>. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the \textitpositive Grassmannian and the associated <i>cluster algebra</i>.

The purity of set-systems related to Grassmann necklaces

International audience 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 $\binom{[n]}{r}$, 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.

Combinatorics of Positroids

International audience Recently Postnikov gave a combinatorial description of the cells in a totally-nonnegative Grassmannian. These cells correspond to a special class of matroids called positroids. There are many interesting combinatorial objects associated to a positroid. We introduce some recent results, including the generalization and proof of the purity conjecture by Leclerc and Zelevinsky on weakly separated sets.