Conormal Varieties of covexillary Schubert Varieties

TL;DR

This paper introduces an embedding of covexillary matrix Schubert varieties into Grassmannian Schubert varieties, leading to new insights into their characteristic cycles and Kazhdan-Lusztig polynomials, with implications for algebraic geometry.

Contribution

It constructs an open embedding for covexillary matrix Schubert varieties into Grassmannian Schubert varieties and characterizes their conormal varieties within cotangent bundles.

Findings

Characteristic cycles of covexillary Schubert varieties are irreducible.

Provides a new proof for Kazhdan-Lusztig polynomials of vexillary permutations.

Develops an algebraic criterion for conormal varieties as subvarieties of cotangent bundles.

Abstract

A permutation is called covexillary if it avoids the pattern $3412$. We construct an open embedding of a covexillary matrix Schubert variety into a Grassmannian Schubert variety. As applications of this embedding, we show that the characteristic cycles of covexillary Schubert varieties are irreducible, and provide a new proof of Lascoux's model computing Kazhdan-Lusztig polynomials of vexillary permutations. Combining the above embedding with earlier work of the author on the conormal varieties of Grassmannian Schubert varieties, we develop an algebraic criterion identifying the conormal varieties of covexillary Schubert and matrix Schubert varieties as subvarieties of the respective cotangent bundles.