Smoothness of Schubert varieties via patterns in root subsystems

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 A3 that appear in Lakshmibai-Sandhya's criterion we only need one additional forbidden pattern of type D4. In terms of these patterns, the only difference between smoothness and rational smoothness is a single pattern in type B2. 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 E8, we derived several general results on Poincaré polynomials of cohomology rings of Schubert varieties based on parabolic decomposition, which have an independent interest.