Smoothness of Schubert varieties via patterns in root systems

Abstract

The aim of this article is to present a smoothness criterion for Schubert varieties in generalized flag manifolds $G/B$ in terms of patterns in root systems. We generalize Lakshmibai-Sandhya's well-known result that says that a Schubert variety in $SL(n)/B$ is smooth if and only if the corresponding permutation avoids the patterns 3412 and 4231. Our criterion is formulated uniformly in general Lie theoretic terms. We define a notion of pattern in Weyl group elements and show that a Schubert variety is smooth (or rationally smooth) if and only if the corresponding element of the Weyl group avoids a certain finite list of patterns. These forbidden patterns live only in root subsystems with star-shaped Dynkin diagrams. In the simply-laced case the list of forbidden patterns is especially simple: besides two patterns of type $A_3$ that appear in Lakshmibai-Sandhya's criterion we only need one additional forbidden pattern of type $D_4$. Remarkably, several other important classes of elements in Weyl groups can also be described in terms of forbidden patterns. For example, the fully commutative elements in Weyl groups have such a characterization. In order to prove our criterion we used several known results for the classical types. For the exceptional types, our proof is based on computer verifications. In order to conduct such a verification for the computationally challenging type $E_8$, we derived several general results on Poincaré polynomials of cohomology rings of Schubert varieties based on parabolic decomposition, which have an independent interest.