Weak separation and plabic graphs

By Suho Oh, Alexander Postnikov, David E. Speyer. 2011

127 citations recorded by the source corpus.

Leclerc and Zelevinsky described quasicommuting families of quantum minors in terms of a certain combinatorial condition, called weak separation. They conjectured that all inclusion‐maximal weakly separated collections of minors have the same cardinality, and that they can be related to each other by a sequence of mutations.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f3283bfe535e2b58cb02e85f91631aaf3408cb70

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Cluster algebras (2014, unknown): “Bicolored triangulations and subdivisions are special cases of plabic tilings [46].”
Positroids and non-crossing partitions (2013, methodology): “ be uniquely constructed by choosing a non-crossing partition (S 1;:::;S t) of [n], and then putting the structure of a connected positroid on each block S i. The rst statement was also discovered in [OPS], where it is stated without proof, and in [For13]. We also give an alternative description of ”
Infinitely many Lagrangian fillings (2020, background): “It should be noted that the article [64] succeeded in constructing finitely many Lagrangian fillings of the maximal-tb Legendrian (n,m)-torus link, as many as maximal pairwise weakly separated n-element subsets [50, 58] of [1, n+m], a finite number which is bounded above by nm+ 1.”
Dimer models and cluster categories of Grassmannians (2013, background): “By a more recent result of Oh–Postnikov–Speyer [25], every cluster consisting of minors arises in this way (see also related results in [6]).”
The positive Dressian equals the positive tropical Grassmannian (2020, background): “nts K(i;j) precisely correspond to the Young diagrams whose complementary Young diagram is a rectangle. It is not hard to see that the collection fK(i;j)gis a maximal weakly separated set collection [OPS15], and hence form a cluster for the cluster algebra structure [Sco06]. Example 3.17. Figure 2de”
A categorification of Grassmannian cluster algebras (2013, background): “As conjectured by Scott [24] and proved by Oh–Postnikov–Speyer [22], any maximal set S of non-crossing subsets has k(n− k) + 1 elements.”