Tropical geometry of Grassmannians and their cluster structure

Lara Bossinger (UNAM Oaxaca)

04-Dec-2020, 15:30-16:30 (3 years ago)

Abstract: The Grassmannain, or more precisely its homogeneous coordinate ring with respect to the Plücker embedding, was found to be a cluster algebra by Scott in the early years of cluster theory. Since then, this cluster structure was studied from many different perspectives by a number of mathematicians. As the whole subject of cluster algebras broadly speaking divides into two main perspectives, algebraic and geometric, so do the results regarding Grassmannian. Geometrically, the Grassmannian contains two open subschemes that are dual cluster varieties.

Interestingly, we can find tropical geometry in both directions: from the algebraic point of view, we discover relations between maximal cones in the tropicalization of the defining ideal (what Speyer and Sturmfels call the tropical Grassmannian) and seeds of the cluster algebra. From the geometric point of view, due to work of Fock--Goncharov followed by work of Gross--Hacking--Keel--Kontsevich we know that the scheme theoretic tropical points of the cluster varieties parametrize functions on the Grassmannian.

In this talk I aim to explain the interaction of tropical geometry with the cluster structure for the Grassmannian from the algebraic and the geometric point of view.

algebraic geometrycombinatorics

Audience: researchers in the topic


Tropical Geometry in Frankfurt/Zoom TGiF/Z

Series comments: Description: An afternoon seminar series on tropical geometry, known as the TGiZ ("Tropical Geometry in Zoom") or the TGiF ("Tropical Geometry in Frankfurt") seminar.

Please send an email to one of the organizers at the latest one day before a session if you wish to receive the Zoom link and password.

Videos of some past talks are available on YouTube here, while some slides can be found here.

Organizers: Andreas Gross*, Martin Ulirsch*
*contact for this listing

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