Which Schubert varieties are local complete intersections?

TL;DR

This paper characterizes which Schubert varieties in GL_n are local complete intersections using pattern avoidance, providing explicit equations and exploring combinatorics, with applications to Kostant polynomials and cohomology rings.

Contribution

It offers a pattern avoidance characterization of lci Schubert varieties and explicit equations for their neighborhoods, advancing understanding of their geometric and combinatorial structure.

Findings

Pattern avoidance characterizes lci Schubert varieties.

Explicit minimal equations for neighborhoods at the identity.

Applications to Kostant polynomials and cohomology ring presentations.

Abstract

We characterize by pattern avoidance the Schubert varieties for GL_n which are local complete intersections (lci). For those Schubert varieties which are local complete intersections, we give an explicit minimal set of equations cutting out their neighborhoods at the identity. Although the statement of our characterization only requires ordinary pattern avoidance, showing that the Schubert varieties not satisfying our conditions are not lci appears to require working with more general notions of pattern avoidance. The Schubert varieties defined by inclusions, originally introduced by Gasharov and Reiner, turn out to be an important subclass, and we further develop some of their combinatorics. Applications include formulas for Kostant polynomials and presentations of cohomology rings for lci Schubert varieties.