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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Sara C. Billey and Gregory S. Warrington
Journal: Trans. Amer. Math. Soc. 355 (2003), 3915-3945
MSC (2000): Primary 14M15; Secondary 05E15
Published electronically: June 24, 2003
MathSciNet review: 1990570
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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 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.


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Additional Information

Sara C. Billey
Affiliation: Department of Mathematics, 2-363c, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350
Email: billey@math.mit.edu, billey@math.washington.edu

Gregory S. Warrington
Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003
Address at time of publication: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Email: warrington@math.umass.edu, gwar@alumni.princeton.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03019-8
PII: S 0002-9947(03)03019-8
Received by editor(s): March 19, 2001
Received by editor(s) in revised form: January 28, 2002
Published electronically: June 24, 2003
Additional Notes: Work supported by NSF grant DMS-9983797
Article copyright: © Copyright 2003 American Mathematical Society