Program
Cluster Duality and Canonical Basis of Grassmannian
Student No.:40
Time:Wed & Fri 08:00-09:35, Jun.1-28
Instructor:翁达平 Weng Daping  
Place:Conference Room 3, Jin Chun Yuan West Bldg.
Starting Date:2018-6-1
Ending Date:2018-6-28

Fock and Goncharov defined two cluster varieties for a given quiver, and formulated the cluster duality conjecture, which relates a canonical basis of the ring of regular functions on either one of the cluster varieties to certain tropicalization of the other. Gross, Hacking, Keel, and Kontsevich gave a sufficient condition for the duality conjecture to hold. In this course we will introduce basic definitions and theorems in cluster theory related to the duality conjecture, and use Grassmannian as our main example. At the end we will use the cluster duality to construct certain mirror dual of the Grassmannian and describe a canonical basis of highest weight representations of GL_n corresponding to multiples of fundamental weights.


Prerequisite:Basic algebraic geometry and representation theory.


Reference:

[1] Fock & Goncharov. Cluster ensembles, quantization and the dilogarithm. Ann. Sci. Ec. Norm. Super. (2009). arXiv:math/0311245.
[2] Gross, Hacking, Keel, & Kontsevich. Canonical bases for cluster algebras. J. Amer. Math. Soc. (2018). arXiv:1411.1394.
[3] Goncharov & Shen. Donaldson-Thomas transformations for moduli spaces of G-local systems. Adv. Math. (2018). arXiv:1602.06479.
[4] Weng. Donaldson-Thomas transformation of Grassmannian. arXiv:1603.00972.
[5] Shen & Weng. Cyclic sieving and cluster duality for Grassmannian. arXiv:1803.06901.