Affine Patches on Positroid Varieties and Affine Pipe Dreams (Thesis)

TL;DR

This thesis explores affine patches on positroid varieties in the Grassmannian, connecting them to Kazhdan-Lusztig varieties, and introduces affine pipe dreams as a generalization of existing combinatorial tools.

Contribution

It establishes a new correspondence between affine patches on positroid varieties and Kazhdan-Lusztig varieties, and extends pipe dreams to the affine setting.

Findings

Affine patches correspond to Kazhdan-Lusztig varieties in the affine Grassmannian.

Introduces a new term order related to subword complexes and Stanley-Reisner ideals.

Defines affine pipe dreams as generalizations of Cauchon and Le diagrams.

Abstract

The objects of interest in this thesis are positroid varieties in the Grassmannian, which are indexed by juggling patterns. In particular, we study affine patches on these positroid varieties. Our main result corresponds these affine patches to Kazhdan-Lusztig varieties in the affine Grassmannian. We develop a new term order and study how these spaces are related to subword complexes and Stanley-Reisner ideals. We define an extension of pipe dreams to the affine case and conclude by showing how our affine pipe dreams are generalizations of Cauchon and Le diagrams.