Maximal Singular Loci of Schubert Varieties in $SL(n)/B$Sara BilleyMIT
April 20,

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 KazhdanLusztig polynomials at the maximum singular points. This is joint work with Greg Warrington. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

