Leading coefficients of Kazhdan-Lusztig polynomials and fully commutative elements
R.M. Green
Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO 80309-0395, USA
DOI: 10.1007/s10801-008-0156-x
Abstract
Let W be a Coxeter group of type [( A)\tilde] n -1 \widetilde{A}_{n-1} . We show that the leading coefficient, μ ( x, w), of the Kazhdan-Lusztig polynomial P x, w is always equal to 0 or 1 if x is fully commutative (and w is arbitrary).
Pages: 165–171
Keywords: keywords Kazhdan-Lusztig polynomials; affine Weyl groups; fully commutative elements; 0-1 conjecture
Full Text: PDF
References
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2. Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)
3. Jones, B.C.: Leading coefficients of Kazhdan-Lusztig polynomials for Deodhar elements. J. Algebr. Comb. (in press).
4. Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53, 165-184 (1979)
5. Kazhdan, D., Lusztig, G.: Schubert varieties and Poincaré duality. Proc. Symp. Pure Math. 36, 185- 203 (1980)
6. Lascoux, A.: Polynômes de Kazhdan-Lusztig pour les variétés de Schubert vexillaires. C. R. Acad. Sci. Paris 321, 667-670 (1995)
7. Lascoux, A., Schützenberger, M.-P.: Polynômes de Kazhdan & Lusztig pour les grassmanniennes. In: Young Tableaux and Schur Functors in Algebra and Geometry, Toruń,
1980. Asterisque, vol. 87-88, pp. 249-266. Soc. Math. France, Paris (1981)
8. Lusztig, G.: Some problems in the representation theory of finite Chevalley groups. Proc. Symp. Pure Math. 37, 313-317 (1980)
9. Lusztig, G.: Some examples of square integrable representations of semisimple p-adic groups. Trans. Am. Math. Soc. 277, 623-653 (1983)
10. Lusztig, G.: Cells in affine Weyl groups. In: Algebraic Groups and Related Topics. Adv. Studies Pure Math., vol. 6, pp. 255-287. North-Holland and Kinokuniya, Tokyo and Amsterdam (1985)
11. McLarnan, T.J., Warrington, G.S.: Counterexamples to the 0-1 conjecture. Represent. Theory 7, 181- 195 (2003)
12. Shi, J.Y.: Fully commutative elements in the Weyl and affine Weyl groups. J. Algebra 284, 13-36 (2005)
13. Shi, J.Y.: Fully commutative elements and Kazhdan-Lusztig cells in the finite and affine Coxeter groups, II. Proc. Am. Math. Soc. 133, 2525-2531 (2005)
14. Stembridge, J.R.: On the fully commutative elements of Coxeter groups. J. Algebr. Comb. 5, 353-385 (1996)
15. Xi, N.: The leading coefficient of certain Kazhdan-Lusztig polynomials of the permutation group Sn.