Richard P. Stanley Seminar in Combinatorics: Plabic Tangles and Cluster Promotion Maps

Inspired by the BCFW recurrence for amplituhedron tilings, we introduce the framework of plabic tangles, which uses plabic graphs to define rational maps between products of Grassmannians called promotions. Our central conjecture is that promotion maps are quasi-cluster homomorphisms, and we prove this for several classes. Promotion maps are constructed via m-vector-relation configurations on plabic graphs, which we relate to the degree (or intersection number) of the amplituhedron map on positroid varieties. We characterize all plabic trees of intersection number one and illustrate new positivity phenomena beyond cluster algebras through the example of the 4-mass box promotion. These developments suggest deep connections between promotion maps and the cluster structures of the Grassmannian and the amplituhedron, as well as — with an eye toward physics — the singularities of scattering amplitudes in planar N=4 SYM. Based on joint work with Chaim Even-Zohar, Melissa Sherman-Bennett, Ran Tessler and Lauren Williams.

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