Maximal singular loci of Schubert varieties in $SL(n)/B$

Abstract

Schubert varieties in the flag manifold SL(n)/B play a key role in our understanding of projective varieties. One important problem is to determine the locus of singular points in a variety. In 1990, Lakshmibai and Sandhya showed that the Schubert variety Xw is nonsingular if and only if w avoids the patterns 4231 and 3412. They also gave a conjectural description of the singular locus of Xw. In 1999, Gasharov proved one direction of their conjecture. In this paper we give an explicit combinatorial description of the irreducible components of the singular locus of the Schubert variety Xw for any element w ∈ G-fraktur signn. In doing so, we prove both directions of the Lakshmibai-Sandhya conjecture. These irreducible components are indexed by permutations which differ from w by a cycle depending naturally on a 4231 or 3412 pattern in w. Our description of the irreducible components is computationally more efficient (O(n6)) than the previously best known algorithms, which were all exponential in time. Furthermore, we give simple formulas for calculating the Kazhdan-Lusztig polynomials at the maximum singular points.