A Gr\"obner basis for Kazhdan-Lusztig ideals of the flag variety of affine type A

TL;DR

This paper provides explicit equations forming a Gr"obner basis for Kazhdan-Lusztig ideals in affine type A flag varieties, generalizing previous results and offering a linear parametrization of Schubert cells.

Contribution

It introduces a linear parametrization of Schubert cells in affine type A and explicitly constructs a Gr"obner basis for Kazhdan-Lusztig ideals in this setting.

Findings

Explicit equations generating Kazhdan-Lusztig ideals

The equations form a Gr"obner basis in affine type A

Generalization of Woo-Yong's type A result

Abstract

A Kazhdan-Lusztig variety is the intersection of a locally-closed Schubert cell with an opposite Schubert variety in a flag variety. We present a linear parametrization of the Schubert cells in the affine type A flag variety via Bott-Samelson maps, and give explicit equations that generate the Kazhdan-Lusztig ideals in these coordinates. Furthermore, our equations form a Gr\"obner basis for the Kazhdan-Lusztig ideals. Our result generalizes a result of Woo-Yong that gave a Gr\"obner basis for Kazhdan-Lusztig ideals in the type A flag variety.