Sara Billey

Letwbe an element of the Weyl groupS_n, and letX_wbe the Schubert variety associated towin the flag manifoldSL_n(C)=B.Lakshmibai and Sandhya showed thatX_wis smooth if andonly ifwavoids the patterns 4231 and3412.Using twotestsfor rationalsmoothness dueto Carrell and Peterson, we show that rational smoothnessofX_wischaracterized bypatternavoidancefortypesB,CandDaswell.Akeystepin the proof ofthis resultis asequence of rules for factoring the Poincare polynomialsforthecohomologyringofX_w,generalizingtheworkof Gasharov.Thepatternscharacterizingrationalsmoothnessarethen extended to afull characterization of smoothnessof Schubert varieties for the classical groups.