Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



Construction of arbitrary Kazhdan-Lusztig polynomials in symmetric groups

Author: Patrick Polo
Journal: Represent. Theory 3 (1999), 90-104
MSC (1991): Primary 14M15; Secondary 20F55, 20G15
Published electronically: June 22, 1999
MathSciNet review: 1698201
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: To each polynomial $P$ with integral nonnegative coefficients and constant term equal to $1$, of degree $d$, we associate a certain pair of elements $(y,w)$ in the symmetric group $S_n$, where $n = 1 + d + P(1)$, such that the Kazhdan-Lusztig polynomial $P_{y,w}$ equals $P$. This pair satisfies $\ell(w) - \ell(y) = 2d + P(1) - 1$, where $\ell(w)$ denotes the number of inversions of $w$.

References [Enhancements On Off] (What's this?)

  • 1. H.H. Andersen, Schubert varieties and Demazure's character formula, Invent. Math. 79 (1985), 611-618. MR 86h:14042
  • 2. A. Beilinson, J. Bernstein and P. Deligne, Faisceaux pervers, in ``Analyse et topologie sur les espaces singuliers (I)", Astérisque 100, Soc. Math. France, 1982, pp. 3-171. MR 86g:32015
  • 3. B.D. Boe, Kazhdan-Lusztig polynomials for hermitian symmetric spaces, Trans. Amer. Math. Soc. 309 (1988), 279-294. MR 89i:22024
  • 4. M. Brion and P. Polo, Generic singularities of certain Schubert varieties, Math. Z. 231 (1999), 301-324.
  • 5. V.V. Deodhar, Local Poincaré duality and nonsingularity of Schubert varieties, Comm. Algebra 13 (1985), 1379-1388. MR 86i:14015
  • 6. V.V. Deodhar, A combinatorial setting for questions in Kazhdan-Lusztig theory, Geom. Dedicata 36 (1990), 95-119. MR 91h:20075
  • 7. F. Du Cloux, program ``Coxeter" (available at
  • 8. M. Goresky and R. MacPherson, Intersection homology II, Invent. Math. 72 (1983), 77-129. MR 84i:57012
  • 9. R. Irving, The socle filtration of a Verma module, Ann. Sci. École Norm. Sup. 21 (1988), 47-65. MR 89h:17015
  • 10. B. Iversen, ``Cohomology of sheaves", Springer-Verlag, Berlin Heidelberg 1986. MR 87m:14013
  • 11. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 81j:20066
  • 12. D. Kazhdan and G. Lusztig, Schubert varieties and Poincaré duality, in Proc. Symposia Pure Math. Vol. 36, Amer. Math. Soc., 1980, pp. 185-203. MR 84g:14054
  • 13. A. Lascoux and M.P. Schützenberger, Polynômes de Kazhdan-Lusztig pour les grassmanniennes, in ``Tableaux de Young et foncteurs de Schur en algèbre et géométrie", Astérisque 87-88, Soc. Math. France, 1981, pp. 249-266. MR 83i:14045
  • 14. S. Ramanan and A. Ramanathan, Projective normality of flag varieties and Schubert varieties, Invent. Math. 79 (1985), 217-224. MR 86j:14051
  • 15. C.S. Seshadri, Line bundles on Schubert varieties, in ``Proceedings of the Bombay colloquium on vector bundles on algebraic varieties", 1984. MR 88i:14047
  • 16. T.A. Springer, Quelques applications de la cohomologie d'intersection, Séminaire Bourbaki, Vol. 1981/1982, Astérisque, 92-93, Soc. Math. France, Paris, 1982, pp. 249-273. MR 85i:32016b
  • 17. A. Zelevinsky, Small resolutions of singularities of Schubert varieties, Funct. Anal. Appl. 17 (1982), 142-144.

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (1991): 14M15, 20F55, 20G15

Retrieve articles in all journals with MSC (1991): 14M15, 20F55, 20G15

Additional Information

Patrick Polo
Affiliation: CNRS, UMR 7539, Institut Galilée, Département de mathématiques, Université Paris-Nord, 93430 Villetaneuse, France

Received by editor(s): December 11, 1998
Received by editor(s) in revised form: April 30, 1999
Published electronically: June 22, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society