Decompositions of Schubert varieties and small resolutions

Abstract

We construct new small resolutions of Schubert varieties Xw , provide an explicit description of Xw 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.