Grassmannian twists categorified

Abstract: The Grassmannian of k-dimensional subspaces of an n-dimensional space carries a birational automorphism called the twist (or sometimes the Donaldson–Thomas transformation), defined by Berenstein–Fomin–Zelevinsky and Marsh–Scott. This automorphism respects the cluster algebra structure on the coordinate ring, being a quasi-cluster automorphism in the sense of Fraser. By work of Muller–Speyer, similar results hold for positroid strata in the Grassmannian. The cluster algebras in this picture have been categorified, by Jensen–King–Su in the case of the full Grassmannian, and by myself for more general (connected) positroid varieties. In this talk I will report on joint work with İlke Çanakçı and Alastair King, in which we describe the twist in terms of these categorifications. The key ingredient is provided by perfect matching modules, certain combinatorially defined representations for a quiver 'with faces', and I will also explain this construction.

mathematical physicscommutative algebraalgebraic geometryrings and algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

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Longitudinal Algebra and Geometry Open ONline Seminar (LAGOON)

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