Decompositions of Schubert varieties and small resolutions

Abstract

We construct new small resolutions of Schubert varieties X_w, provide an explicit description of X_w as a Bott–Samelson type variety when w has a (complete) BP decomposition, and simplify resolutions of Schubert varieties constructed quite generally to Gelfand–MacPherson resolutions. Our methods are applied to type A Weyl groups where we determine precisely which Schubert varieties admit small resolutions in low-rank, prove that pattern avoidance does not characterize Schubert varieties admitting small resolutions, and describe a new family of small resolutions via pattern avoidance.

KEYWORDS:

• Bott–Samelson
• BP decomposition
• fiber bundles
• flag varieties
• pattern avoidance
• Schubert varieties
• small resolutions

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 22E47
• 14M15

Acknowledgments

I thank Edward Richmond for explaining the remarkable work in Richmond–Slofstra [23] to me, and I thank Roger Zierau for many helpful conversations.