Affine Kazhdan-Lusztig varieties and Grobner bases

This event is partially supported by PIMS.

A Kazhdan-Lusztig variety is the intersection of a Schubert variety with an affine cell in a flag manifold. Therefore, one can obtain local equations for Schubert varieties by using coordinates on the affine cell. Building on the work of Fulton and Knutson/Miller, in finite type A, Woo and Yong gave a Gröbner basis for Kazhdan-Lusztig ideals. We will describe a generalization of their result to the affine type A flag manifold. We will define linear charts on affine flag manifolds using Bott-Samelson varieties, describe an analogue of Fulton's essential set, then use a result of Knutson on geometric vertex decompositions to show that our equations give a Gröbner basis. This is joint work with Daoji Huang.