Cluster Structure for Mirkovic-Vilonen Cycles and Polytopes

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 A3 example and show directly that the cluster variables in this case are in the MV basis.

Department dc:contributor.department

Mathematics

Year dc:date.issued

2022

Author dc:creator

• Bai, Yuguang

Advisor dc:contributor.advisor

• Kamnitzer, Joel

• cluster algebras
• Mirkovic-Vilonen

Statement dc:rights

• Attribution 4.0 International

Licence dc:rights.uri

• http://creativecommons.org/licenses/by/4.0/

Handle dc:identifier.uri

http://hdl.handle.net/1807/110728

OAI identifier oai:identifier

oai:utoronto.scholaris.ca:1807/110728