Maximal singular loci of Schubert varieties in $SL(n)/B$
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- by Sara C. Billey and Gregory S. Warrington PDF
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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.References
- Alexandre Beĭlinson and Joseph Bernstein, Localisation de $g$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15–18 (French, with English summary). MR 610137
- Sara C. Billey, Pattern avoidance and rational smoothness of Schubert varieties, Adv. Math. 139 (1998), no. 1, 141–156. MR 1652522, DOI 10.1006/aima.1998.1744
- Sara Billey and V. Lakshmibai, Singular loci of Schubert varieties, Progress in Mathematics, vol. 182, Birkhäuser Boston, Inc., Boston, MA, 2000. MR 1782635, DOI 10.1007/978-1-4612-1324-6
- Sara Billey and Tao Kai Lam, Vexillary elements in the hyperoctahedral group, J. Algebraic Combin. 8 (1998), no. 2, 139–152. MR 1648468, DOI 10.1023/A:1008633710118
- S. Billey and G. Warrington, Kazhdan-Lusztig polynomials for $321$-hexagon-avoiding permutations, J. Alg. Comb. 13 (2001), 111–136.
- Francesco Brenti, A combinatorial formula for Kazhdan-Lusztig polynomials, Invent. Math. 118 (1994), no. 2, 371–394. MR 1292116, DOI 10.1007/BF01231537
- Francesco Brenti, Combinatorial expansions of Kazhdan-Lusztig polynomials, J. London Math. Soc. (2) 55 (1997), no. 3, 448–472. MR 1452258, DOI 10.1112/S0024610797004948
- Francesco Brenti, Kazhdan-Lusztig and $R$-polynomials from a combinatorial point of view, Discrete Math. 193 (1998), no. 1-3, 93–116. Selected papers in honor of Adriano Garsia (Taormina, 1994). MR 1661365, DOI 10.1016/S0012-365X(98)00137-X
- Francesco Brenti and Rodica Simion, Explicit formulae for some Kazhdan-Lusztig polynomials, J. Algebraic Combin. 11 (2000), no. 3, 187–196. MR 1771610, DOI 10.1023/A:1008741113381
- Michel Brion and Patrick Polo, Generic singularities of certain Schubert varieties, Math. Z. 231 (1999), no. 2, 301–324. MR 1703350, DOI 10.1007/PL00004729
- J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), no. 3, 387–410. MR 632980, DOI 10.1007/BF01389272
- 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.
- C. Chevalley, Sur les décompositions cellulaires des espaces $G/B$, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991) Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 1–23 (French). With a foreword by Armand Borel. MR 1278698
- 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.
- Vinay V. Deodhar, Local Poincaré duality and nonsingularity of Schubert varieties, Comm. Algebra 13 (1985), no. 6, 1379–1388. MR 788771, DOI 10.1080/00927878508823227
- William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
- Vesselin Gasharov, Sufficiency of Lakshmibai-Sandhya singularity conditions for Schubert varieties, Compositio Math. 126 (2001), no. 1, 47–56. MR 1827861, DOI 10.1023/A:1017585921369
- James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
- C. Kassel, A. Lascoux, and C. Reutenauer, The singular locus of a Schubert variety, IRMA Prepublication (2001); to appear in J. Algebra.
- David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
- V. Lakshmibai and B. Sandhya, Criterion for smoothness of Schubert varieties in $\textrm {Sl}(n)/B$, Proc. Indian Acad. Sci. Math. Sci. 100 (1990), no. 1, 45–52. MR 1051089, DOI 10.1007/BF02881113
- V. Lakshmibai and C. S. Seshadri, Singular locus of a Schubert variety, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 2, 363–366. MR 752799, DOI 10.1090/S0273-0979-1984-15309-6
- Alain 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 (French, with English and French summaries). MR 1354702
- Alain Lascoux and Marcel-Paul Schützenberger, Polynômes de Kazhdan & Lusztig pour les grassmanniennes, Young tableaux and Schur functors in algebra and geometry (Toruń, 1980), Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 249–266 (French). MR 646823
- Alain Lascoux and Marcel-Paul Schützenberger, Schubert polynomials and the Littlewood-Richardson rule, Lett. Math. Phys. 10 (1985), no. 2-3, 111–124. MR 815233, DOI 10.1007/BF00398147
- L. Manivel, Generic singularities of Schubert varieties, arXiv:math.AG/0105239 (2001).
- L. Manivel, Le lieu singulier des variétés de Schubert, Internat. Math. Res. Notices (2001), no. 16, 849–871.
- Patrick Polo, Construction of arbitrary Kazhdan-Lusztig polynomials in symmetric groups, Represent. Theory 3 (1999), 90–104. MR 1698201, DOI 10.1090/S1088-4165-99-00074-6
- B. Shapiro, M. Shapiro, and A. Vainshtein, Kazhdan-Lusztig polynomials for certain varieties of incomplete flags, Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995), 1998, pp. 345–355. MR 1603684, DOI 10.1016/S0012-365X(97)00124-6
- John R. Stembridge, On the fully commutative elements of Coxeter groups, J. Algebraic Combin. 5 (1996), no. 4, 353–385. MR 1406459, DOI 10.1023/A:1022452717148
- A. V. Zelevinskiĭ, Small resolutions of singularities of Schubert varieties, Funktsional. Anal. i Prilozhen. 17 (1983), no. 2, 75–77 (Russian). MR 705051
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
- MR Author ID: 341999
- 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
- MR Author ID: 677560
- Email: warrington@math.umass.edu, gwar@alumni.princeton.edu
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 3915-3945
- MSC (2000): Primary 14M15; Secondary 05E15
- DOI: https://doi.org/10.1090/S0002-9947-03-03019-8
- MathSciNet review: 1990570