Leading coefficients of Kazhdan-Lusztig polynomials for Deodhar elements
Brant C. Jones
University of California Department of Mathematics One Shields Avenue Davis CA 95616 USA
DOI: 10.1007/s10801-008-0131-6
Abstract
We show that the leading coefficient of the Kazhdan-Lusztig polynomial P x, w ( q) known as μ ( x, w) is always either 0 or 1 when w is a Deodhar element of a finite Weyl group. The Deodhar elements have previously been characterized using pattern avoidance in Billey and Warrington (J. Algebraic Combin. 13(2):111-136, [ 2001]) and Billey and Jones (Ann. Comb. [ 2008], to appear). In type A, these elements are precisely the 321-hexagon avoiding permutations. Using Deodhar's algorithm (Deodhar in Geom. Dedicata 63(1):95-119, [ 1990]), we provide some combinatorial criteria to determine when μ ( x, w)=1 for such permutations w.
Pages: 229–260
Keywords: keywords Kazhdan-Lusztig polynomial; 321-hexagon; 0-1 conjecture; pattern avoidance
Full Text: PDF
References
1. Billey, S., Warrington, G.S.: Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations. J. Algebraic Combin. 13(2), 111-136 (2001)
2. Billey, S.C., Jones, B.C.: Embedded factor patterns for Deodhar elements in Kazhdan-Lusztig theory. Ann. Comb. 11(3/4), 285-333 (2007)
3. Deodhar, V.V.: A combinatorial setting for questions in Kazhdan-Lusztig theory. Geom. Dedicata 36(1), 95-119 (1990)
4. Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53(2), 165-184 (1979)
5. Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol.
231. Springer, New York (2005)
6. McLarnan, T.J., Warrington, G.S.: Counterexamples to the 0-1 conjecture, Represent. Theory 7, 181- 195 (2003) (electronic) J Algebr Comb (2009) 29: 229-260
7. Lusztig, G.: Cells in affine Weyl groups. In: Algebraic groups and related topics, Kyoto/Nagoya,
1983. Adv. Stud. Pure Math., vol. 6, pp. 255-287. North-Holland, Amsterdam (1985)
8. Lascoux, A.: Polynômes de Kazhdan-Lusztig pour les variétés de Schubert vexillaires. (French) [Kazhdan-Lusztig polynomials for vexillary Schubert varieties]. C.R. Acad. Sci. Paris Sér. I Math. 321(6), 667-670 (1995)
9. Lascoux, A., Schützenberger, M.-P.: Polynômes de Kazhdan et Lusztig pour les grassmanniennes. In: Young tableaux and Schur functors in algebra and geometry, Toruń,
1980. Astérisque, Soc. Math., vol. 87, pp. 249-266. France, Paris (1981)
10. Kazhdan, D., Lusztig, G.: Schubert varieties and Poincaré duality. Proc. Symp. Pure. Math., A.M.S. 36, 185-203 (1980)
11. Lakshmibai, V., Sandhya, B.: Criterion for smoothness of Schubert varieties in Sl(n)/B. Proc. Indian Acad. Sci. Math. Sci. 100(1), 45-52 (1990)
12. Ryan, K.M.: On Schubert varieties in the flag manifold of Sl(n, C). Math. Ann. 276(2), 205-224 (1987)
13. Wolper, J.S.: A combinatorial approach to the singularities of Schubert varieties. Adv. Math. 76(2), 184-193 (1989)
14. Graham, J.J.: Modular representations of Hecke algebras and related algebras, Ph.D. thesis, University of Sydney (1995)
15. Marietti, M.: Boolean elements in Kazhdan-Lusztig theory. J. Algebra 295(1), 1-26 (2006)
16. Xi, N.: The leading coefficient of certain Kazhdan-Lusztig polynomials of the permutation group Sn.
2. Billey, S.C., Jones, B.C.: Embedded factor patterns for Deodhar elements in Kazhdan-Lusztig theory. Ann. Comb. 11(3/4), 285-333 (2007)
3. Deodhar, V.V.: A combinatorial setting for questions in Kazhdan-Lusztig theory. Geom. Dedicata 36(1), 95-119 (1990)
4. Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53(2), 165-184 (1979)
5. Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol.
231. Springer, New York (2005)
6. McLarnan, T.J., Warrington, G.S.: Counterexamples to the 0-1 conjecture, Represent. Theory 7, 181- 195 (2003) (electronic) J Algebr Comb (2009) 29: 229-260
7. Lusztig, G.: Cells in affine Weyl groups. In: Algebraic groups and related topics, Kyoto/Nagoya,
1983. Adv. Stud. Pure Math., vol. 6, pp. 255-287. North-Holland, Amsterdam (1985)
8. Lascoux, A.: Polynômes de Kazhdan-Lusztig pour les variétés de Schubert vexillaires. (French) [Kazhdan-Lusztig polynomials for vexillary Schubert varieties]. C.R. Acad. Sci. Paris Sér. I Math. 321(6), 667-670 (1995)
9. Lascoux, A., Schützenberger, M.-P.: Polynômes de Kazhdan et Lusztig pour les grassmanniennes. In: Young tableaux and Schur functors in algebra and geometry, Toruń,
1980. Astérisque, Soc. Math., vol. 87, pp. 249-266. France, Paris (1981)
10. Kazhdan, D., Lusztig, G.: Schubert varieties and Poincaré duality. Proc. Symp. Pure. Math., A.M.S. 36, 185-203 (1980)
11. Lakshmibai, V., Sandhya, B.: Criterion for smoothness of Schubert varieties in Sl(n)/B. Proc. Indian Acad. Sci. Math. Sci. 100(1), 45-52 (1990)
12. Ryan, K.M.: On Schubert varieties in the flag manifold of Sl(n, C). Math. Ann. 276(2), 205-224 (1987)
13. Wolper, J.S.: A combinatorial approach to the singularities of Schubert varieties. Adv. Math. 76(2), 184-193 (1989)
14. Graham, J.J.: Modular representations of Hecke algebras and related algebras, Ph.D. thesis, University of Sydney (1995)
15. Marietti, M.: Boolean elements in Kazhdan-Lusztig theory. J. Algebra 295(1), 1-26 (2006)
16. Xi, N.: The leading coefficient of certain Kazhdan-Lusztig polynomials of the permutation group Sn.