Monday, August 16, 2021

3:00 p.m. (sharp)

PhD Candidate: Roger Bai

Supervisor: Joel Kamnitzer

Thesis title: Cluster Structure for Mirkovic-Vilonen Cycles and Polytopes

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We look at the Mirkovic-Vilonen (MV) basis for semisimple Lie algebras and compare this to the associated cluster algebra to investigate the question of whether or not the cluster variables are in the MV basis.

We begin with finding analogues of the cluster structure among MV cycles and MV polytopes. In particular, we show the exchange relations correspond to an equation involving MV polytopes.

We extend a result of Baumann-Kamnitzer in relating valuations of an MV cycle and the dimension of homomorphism spaces of its associated preprojective algebra module. In doing so, we are able to give a partial result for an exchange relation involving MV cycles in low dimensions.

In joint work with Dranowski and Kamnitzer, we present a way to calculate the fusion product of MV cycles in type A through a generalization of the Mirkovic-Vybornov isomorphism.

Finally, we finish with examining the A_3 example and show directly that the cluster variables in this case are in the MV basis.

A copy of the thesis can be found here: Cluster_Structure_for_MV_Cycles_and_PolytopesFinal

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