Examples of cluster varieties from plabic graphs I

Abstract: Cluster varieties were introduced by Fock and Goncharov as geometric counterparts of Fomin and Zelevinsky’s cluster algebras. Simply speaking, cluster varieties are algebraic varieties with an atlas of torus charts, whose transition maps are captured by certain combinatorial process called cluster mutations. Many interesting geometric objects turn out to be examples of cluster varieties, and one can then use cluster theoretical techniques to study these geometric objects. In this lecture series, we will discuss various examples of cluster varieties whose combinatorics can be captured by plabic graphs, including Grassmannians and double Bruhat/Bott-Samelson cells of $SL_n$. This lecture series will be complementary to Linhui Shen’s lecture series on cluster theory.

Lecture 1: $Gr(2,n)$ and $M(0,n)$ $\newline$ We discuss the cluster structures on Grassmannian $Gr(2,n)$ and on the moduli space of $n$ points in $\mathbb{P}^1$. These are examples of cluster varieties of Dynkin $A_{n-3}$ mutation type and their combinatorics are captured by triangulations of an $n$-gon.

algebraic geometrycombinatoricsdifferential geometrygeometric topologyquantum algebrarepresentation theorysymplectic geometry

Audience: researchers in the topic

Legendrians, Cluster algebras, and Mirror symmetry

Series comments: Schedule
School: January 4–8, 2021
Conference: January 11–15, 2021