The Exchange Graphs of Weakly Separated Collections
arXiv:1608.05723
Abstract
Weakly separated collections arise in the cluster algebra derived from the Plücker coordinates on the nonnegative Grassmannian. Oh, Postnikov, and Speyer studied weakly separated collections over a general Grassmann necklace and proved the connectivity of every exchange graph. Oh and Speyer later introduced a generalization of exchange graphs that we call -constant graphs. They characterized these graphs in the smallest two cases. We prove an isomorphism between exchange graphs and a certain class of -constant graphs. We use this to extend Oh and Speyer's characterization of these graphs to the smallest four cases, and we present a conjecture on a bound on the maximal order of these graphs. In addition, we fully characterize certain classes of these graphs in the special cases of cycles and trees.