Positroid varieties and q,t-Catalan numbers
HARVARD-MIT ALGEBRAIC GEOMETRY
Thomas Lam - University of Michigan
Positroid varieties are subvarieties of the Grassmannian obtained by intersecting cyclic rotations of Schubert varieties. 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). Unlike the Grassmannian, the cohomology of open positroid varieties is not pure.
The q,t-rational Catalan numbers satisfy remarkable symmetry and unimodality properties, and these arise from the Koszul duality phenomenon in the derived category of the flag variety, and from the curious Lefschetz phenomenon for cluster varieties. Our work is also related to knot homology and to the cohomology of compactified Jacobians.
This talk is based on joint work with Pavel Galashin.