Criterion for smoothness of Schubert varieties in Sl(n)/B

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Abstract

LetG=Sl(n) andB, the Borel subgroup ofG consisting of upper triangular matrices. LetwS n andX(w)=BwB(modB), the associated Schubert variety inG/B. In this paper, we give a geometric criterion for the smoothness ofX(w). This criterion admits a neat combinatorial description in terms of the permutationw.