Green-to-red sequences for positroids

Abstract

L-diagrams are combinatorial objects that parametrize cells of the totally nonnegative Grassmannian, called positroid cells, and each L-diagram gives rise to a cluster algebra which is believed to be isomorphic to the coordinate ring of the corresponding positroid variety. We study quivers arising from these diagrams and show that they can be constructed from the well-behaved quivers associated to Grassmannians by deleting and merging certain vertices. Then, we prove that quivers coming from arbitrary L-diagrams, and more generally reduced plabic graphs, admit a particular sequence of mutations called a green-to-red sequence.

Positroid varieties and cluster algebras

“…the introduction of [SSBW19]. For instance, the existence of a green-to-red sequence [FS18], together with the constructions of [GHKK18] endow C[ curious Lefschetz property, which has implications for extension groups of certain Verma modules that we aim to explore in future work.…”

Universal quivers

“…Remarkably, the analogue of Theorem 3.6 fails for reduced plabic graphs, for the following reason. As shown by N. Ford and K. Serhiyenko [10,Theorem 1.2], the quiver associated with any reduced plabic graph has a reddening sequence. This property is preserved by quiver mutations and by passing to a full subquiver, see G. Muller [22,Theorem 17,Corollary 19].…”

Building maximal green sequences via component preserving mutations

“…Additionally there are branches of the exchange graph, in which no amount of mutations can lead to a maximal green sequence; meaning random computer generated mutations are extremely unlikely to produce maximal green sequences for these quivers. In addition to finite mutation type quivers, headway has been made on specific families of quivers such as minimal mutation-infinite quivers [32] and quivers which are associated to reduced plabic graphs [17]. This gives us many quivers for which we know reddening or maximal green sequences for.…”