Grassmannians, plabic graphs, and mirror symmetry for cluster varieties

Tim Magee (King's College London)

Wednesday 22nd March, 2023 16:00-17:00 Maths 311B


The open positroid variety U in the Grassmannian Y has two flavors of cluster structure, an "A cluster structure" and an "X cluster structure".  I'll review cluster ensemble maps from A to X and discuss how such a map induces another cluster ensemble map from the mirror of X to the mirror of A.  Using this pair of maps, we will see that we can identify two Landau-Ginzburg mirror constructions for the pair (U,Y)-- one due to Marsh-Rietsch, and the other based on work of Gross-Hacking-Keel-Kontsevich.  When one of these maps identifies the inputs (U,Y) of the two constructions, the other identifies the LG mirror outputs.  As a corollary, we will also relate the polytopes and toric degenerations of Rietsch-Williams with those in the Gross-Hacking-Keel-Kontsevich framework.  Based on ongoing joint work with Lara Bossinger, Mandy Cheung, and Alfredo Nájera Chávez.

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