University of Birmingham > Talks@bham > Geometry and Mathematical Physics seminar > Compactifications of cluster varieties and convexity

Compactifications of cluster varieties and convexity

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  • UserAlfredo Nájera Chávez, UNAM Oaxaca
  • ClockTuesday 11 February 2020, 16:00-17:00
  • HousePhysics West 115.

If you have a question about this talk, please contact Timothy Magee.

In 2014, Gross, Hacking, Keel and Kontsevich (GHKK) introduced the so-called theta basis for a cluster algebra. The elements of the theta basis theta functions are a very special class of functions on a cluster variety. In order to describe theta functions one considers sophisticated objects called scattering diagrams and broken lines inside them. A key insight of the work of GHKK is that theta functions on a cluster varieties play a similar role to the role played by the characters of an algebraic torus in toric geometry. In particular, they showed that one can consider “positive subsets” inside the scattering diagram to compactify a cluster variety. Various projective varieties arising in representation theory (such as flag varieties, Grassmannians, certain Schubert varieties, etc.) fit this framework. The definition of a positive set is of algebraic nature and is a notion directly linked to the positivity property of cluster algebras. The purpose of this talk is to present a geometric/combinatorial interpretation of positive sets. The main result I will talk about is the following: a set is positive if and only if it is “broken line convex”.

This is joint work with Man-Wai Cheung and Timothy Magee

This talk is part of the Geometry and Mathematical Physics seminar series.

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