Grassmannian twists categorified

Matthew Pressland (University of Glasgow)

Wednesday 23rd February, 2022 16:00-17:00 Maths 110

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 Ä°lke 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.

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