On tangent cones of Schubert varieties

TL;DR

This paper studies the tangent cones of Schubert varieties in the complete flag variety, providing conditions for when they coincide and exploring their structure using pillar entries in the rank matrix.

Contribution

It introduces a sufficient condition and a conjecture for tangent cone coincidence of Schubert varieties, and characterizes those associated with Coxeter elements.

Findings

All Schubert varieties for Coxeter elements share the same tangent cone.

A new criterion based on pillar entries determines tangent cone coincidence.

Dimension formulas for Schubert varieties are derived using pillar entries.

Abstract

We consider tangent cones of Schubert varieties in the complete flag variety, and investigate the problem when the tangent cones of two different Schubert varieties coincide. We give a sufficient condition for such coincidence, and formulate a conjecture that provides a necessary condition. In particular, we show that all Schubert varieties corresponding to the Coxeter elements of the Weyl group have the same tangent cone. Our main tool is the notion of pillar entries in the rank matrix counting the dimensions of the intersections of a given flag with the standard one. This notion is a version of Fulton's essential set. We calculate the dimension of a Schubert variety in terms of the pillar entries of the rank matrix.