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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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  • 1. A. Beilinson and J. Bernstein, Localization of $\mathfrak{g}$-modules, C. R. Acad. Sci. Paris Ser. I Math 292 (1981), 15-18. MR 82k:14015
  • 2. S. Billey, Pattern avoidance and rational smoothness of Schubert varieties, Adv. Math. 139 (1998), no. 1, 141-156. MR 99i:14058
  • 3. S. Billey and V. Lakshmibai, Singular Loci of Schubert Varieties, Progress in Mathematics, no. 182, Birkhäuser, 2000. MR 2001j:14065
  • 4. S. Billey and T. K. Lam, Vexillary elements in the hyperoctahedral group, J. Alg. Combin. 8 (1998), no. 2, 139-152. MR 2000d:05124
  • 5. S. Billey and G. Warrington, Kazhdan-Lusztig polynomials for $321$-hexagon-avoiding permutations, J. Alg. Comb. 13 (2001), 111-136.
  • 6. F. Brenti, A combinatorial formula for Kazhdan-Lusztig polynomials, Invent. Math. 118 (1994), no. 2, 371-394. MR 96c:20074
  • 7. -, Combinatorial expansions of Kazhdan-Lusztig polynomials, J. London Math. Soc. 55 (1997), no. 2, 448-472. MR 99a:05143
  • 8. -, Kazhdan-Lusztig polynomials and $R$-polynomials from combinatorial point of view, Discrete Math 193 (1998), no. 1-3, 93-116. MR 2000c:05154
  • 9. F. Brenti and R. Simion, Explicit formulae for some Kazhdan-Lusztig polynomials, J. Alg. Comb. 11 (2000), no. 3, 187-196. MR 2001e:05137
  • 10. M. Brion and P. Polo, Generic singularities of certain Schubert varieties, Math. Z. 231 (1999), no. 2, 301-324. MR 2000f:14078
  • 11. J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjectures and holonomic systems, Invent. Math. 64 (1981), 387-410. MR 83e:22020
  • 12. J. B. Carrell and J. Kuttler, On the smooth points of T-stable varieties in $G/B$ and the Peterson map, Invent. Math. 151 (2003), no. 2, 353-379.
  • 13. C. Chevalley, Sur les décompositions cellulaires des espaces $G/B$, Proceedings of Symposia in Pure Mathematics 56 (1994), no. Part I, 1-25. MR 95e:14041
  • 14. A. Cortez, Singularites generiques et quasi-resolutions des variétés de Schubert pour le groupe lineaire, C. R. Acad. Sci. Paris Ser. I Math. 33 (2001), 561-566.
  • 15. V. Deodhar, Local Poincaré duality and non-singularity of Schubert varieties, Comm. Algebra 13 (1985), 1379-1388. MR 86i:14015
  • 16. W. Fulton, Young tableaux; with applications to representation theory and geometry, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, New York, 1997. MR 99f:05119
  • 17. V. Gasharov, Sufficiency of Lakshmibai-Sandhya singularity conditions for Schubert varieties, Compositio Math. 126 (2001), no. 1, 47-56. MR 2002d:14078
  • 18. J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, 1990. MR 92h:20002
  • 19. C. Kassel, A. Lascoux, and C. Reutenauer, The singular locus of a Schubert variety, IRMA Prepublication (2001); to appear in J. Algebra.
  • 20. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 81j:20066
  • 21. V. Lakshmibai and B. Sandhya, Criterion for smoothness of Schubert varieties in $SL(n)/B$, Proc. Indian Acad. Sci. (Math Sci.) 100 (1990), no. 1, 45-52. MR 91c:14061
  • 22. V. Lakshmibai and C. S. Seshadri, Singular locus of a Schubert variety, Bull. Amer. Math. Soc. 11 (1984), no. 2, 363-366. MR 85j:14095
  • 23. A. Lascoux, Polynômes de Kazhdan-Lusztig pour les variétés de Schubert vexillaires, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 6, 667-670. MR 96g:05144
  • 24. A. Lascoux and M.-P. Schützenberger, Polynômes de Kazhdan & Lusztig pour les grassmanniennes, Young tableaux and Schur functors in algebra and geometry (Torun, 1980), Soc. Math. France, Paris, 1981, pp. 249-266. MR 83i:14045
  • 25. A. Lascoux and M.-P. Schützenberger, Schubert polynomials and the Littlewood-Richardson rule, Letters in Math. Physics 10 (1985), 111-124. MR 87g:20021
  • 26. L. Manivel, Generic singularities of Schubert varieties, arXiv:math.AG/0105239 (2001).
  • 27. L. Manivel, Le lieu singulier des variétés de Schubert, Internat. Math. Res. Notices (2001), no. 16, 849-871.
  • 28. P. Polo, Construction of arbitrary Kazhdan-Lusztig polynomials in symmetric groups, Represent. Theory 3 (1999), 90-104, (electronic). MR 2000j:14079
  • 29. B. Shapiro, M. Shapiro, and A. Vainshtein, Kazhdan-Lusztig polynomials for certain varieties of incomplete flags, Discrete Math. 180 (1998), 345-355. MR 99b:14054
  • 30. J. Stembridge, On the fully commutative elements of Coxeter groups, J. Alg. Combin. 5 (1996), no. 4, 353-385. MR 97g:20046
  • 31. A. V. Zelevinskii, Small resolutions of singularities of Schubert varieties, Functional Anal. Appl. 17 (1983), no. 2, 142-144. MR 85b:14069

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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

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

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