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

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 $X_w$ is nonsingular if and only if $w$ avoids the patterns $4231$ and $3412$. They also gave a conjectural description of the singular locus of $X_w$. 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 $X_w$ for any element $w\in \mathfrak{S}_n$. 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 ($O(n^6)$) 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.