Smoothness of Schubert varieties via patterns in root systems
| dc.creator | Billey, Sara | |
| dc.creator | Postnikov, Alexander | |
| dc.date | 2002-05-16 | |
| dc.date | 2004-06-01 | |
| dc.date.accessioned | 2026-07-07T04:48:33Z | |
| dc.date.available | 2026-07-07T04:48:33Z | |
| dc.description | 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. | |
| dc.description | 18 pages | |
| dc.identifier | https://arxiv.org/abs/math/0205179 | |
| dc.identifier | http://arxiv.org/abs/math/0205179 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/64087 | |
| dc.subject | Combinatorics | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14M15 | |
| dc.title | Smoothness of Schubert varieties via patterns in root systems | |
| dc.type | text |