AG : Oliver Daisey (Durham) : Cluster structures for type E_n homogeneous varieties
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AG : Oliver Daisey (Durham) : Cluster structures for type E_n homogeneous varieties
10 décembre / 13:30 - 17:00
This presentation is divided into two parts.
First talk: Cluster structures for type E_n homogeneous varieties
Abstract: We will present a gentle overview of cluster algebras of geometric type, and introduce one of their generalisations due to Lam and Pylyavskyy. This will lead to an elegant geometric description of seeds and their mutations via coverings of algebraic varieties by torus charts, which we will explore explicitly for the case of Grassmannians. Finally, we will introduce a sequence of mathematical objects parametrised by the Dynkin diagrams E_4, E_5, E_6, E_7, and tease at a finite type cluster structure we can induce on each of them.
Second talk: A Laurent Phenomenon for the Cayley Plane
Abstract: We describe a Laurent phenomenon for the Cayley plane, which is the homogeneous variety associated to the cominuscule representation of E_6. The corresponding Laurent phenomenon algebra has finite type and appears in a natural sequence of LPAs indexed by the E_n Dynkin diagrams for \(n\leq6\). We conjecture the existence of a further finite type LPA, associated to the Freudenthal variety of type E_7.
AG : Oliver Daisey (Durham) : Cluster structures for type E_n homogeneous varieties