# 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 $X_w$ is nonsingular if an only if $w$ avoids the patterns $4231$ and $3412$. 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 $w\in S_n$. 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 efficient ($O(n^4)$) compared to the best known algorithms, which were all exponential in time. Furthermore, we give formulas for calculating Kazhdan-Lusztig polynomials at the maximum singular points.

This is joint work with Greg Warrington.

Speaker's Contact Info: sara(at-sign)math.mit.edu