Positroid varieties and $q,t$ -Catalan numbers

Abstract: Positroid varieties are subvarieties of the Grassmannian defined as intersections of rotations of Schubert varieties in my work with Knutson and Speyer. They also appear in the work of Shende-Treumann-Williams-Zaslow as moduli spaces of constructible sheaves with microsupport in a Legendrian link. $\newline$ We show that the "top open positroid variety" has mixed Hodge polynomial given by the $q,t$-rational Catalan numbers (up to a simple factor). The $q,t$-rational Catalan numbers satisfy remarkable symmetry and unimodality properties, and we show that these follow from the curious Lefschetz phenomenon for cluster varieties. The cohomologies of open positroid varieties are shown to be related to Khovanov-Rosanzky knot homology.$\newline$ This talk is based on joint work with Pavel Galashin.

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