Explicit Formulae for Some Kazhdan-Lusztig Polynomials
Francesco Brenti
and Rodica Simion
dagger
DOI: 10.1023/A:1008741113381
Abstract
We consider the Kazhdan-Lusztig polynomials P u,v (q) indexed by permutations u, v having particular forms with regard to their monotonicity patterns. The main results are the following. First we obtain a simplified recurrence relation satisfied by P u,v (q) when the maximum value of v 
S n occurs in position n - 2 or n - 1. As a corollary we obtain the explicit expression for P e,3 4 ... n 1 2(q) (where e denotes the identity permutation), as a q-analogue of the Fibonacci number. This establishes a conjecture due to M. Haiman. Second, we obtain an explicit expression for P e, 3 4 ... ( n - 2) n ( n - 1) 1 2(q). Our proofs rely on the recurrence relation satisfied by the Kazhdan-Lusztig polynomials when the indexing permutations are of the form under consideration, and on the fact that these classes of permutations lend themselves to the use of induction. We present several conjectures regarding the expression for P u,v (q) under hypotheses similar to those of the main results.
S n occurs in position n - 2 or n - 1. As a corollary we obtain the explicit expression for P e,3 4 ... n 1 2(q) (where e denotes the identity permutation), as a q-analogue of the Fibonacci number. This establishes a conjecture due to M. Haiman. Second, we obtain an explicit expression for P e, 3 4 ... ( n - 2) n ( n - 1) 1 2(q). Our proofs rely on the recurrence relation satisfied by the Kazhdan-Lusztig polynomials when the indexing permutations are of the form under consideration, and on the fact that these classes of permutations lend themselves to the use of induction. We present several conjectures regarding the expression for P u,v (q) under hypotheses similar to those of the main results.Pages: 187–196
Keywords: Kazhdan-Lusztig polynomial; q-Fibonacci number; Bruhat order
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References
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2. F. Brenti, “Combinatorial properties of the Kazhdan-Lusztig R-polynomials for Sn,” Advances in Math. 126 (1997), 21-51.
3. F. Brenti, “Kazhdan-Lusztig and R-polynomials from a combinatorial point of view,” Discrete Math. 193 (1998), 93-116.
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11. I.G. Macdonald, Notes on Schubert Polynomials, Publ. LACIM, UQAM, Montreal, 1991.
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13. R.P. Stanley, Enumerative Combinatorics , Vol. 1, Wadsworth and Brooks/Cole, Monterey, CA, 1986; second printing, Cambridge University Press, Cambridge/New York, 1997.
14. A. Zelevinski, “Small resolutions of singularities of Schubert varieties,” Funct. Anal. Appl. 17 (1983), 142- 144.
2. F. Brenti, “Combinatorial properties of the Kazhdan-Lusztig R-polynomials for Sn,” Advances in Math. 126 (1997), 21-51.
3. F. Brenti, “Kazhdan-Lusztig and R-polynomials from a combinatorial point of view,” Discrete Math. 193 (1998), 93-116.
4. F. Brenti, “Lattice paths and Kazhdan-Lusztig polynomials,” J. Amer. Math. Soc. 11 (1998), 229-259.
5. V.V. Deodhar, “A combinatorial setting for questions in Kazhdan-Lusztig theory,” Geom. Dedicata 36 (1990), 95-119.
6. C. Ehresmann, “Sur la topologie de certains espaces homog`enes,” Ann. Math. 35 (1934), 396-443.
7. J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Univ. Press, Cambridge,
1990. Cambridge Studies in Advanced Mathematics, No. 29.
8. D. Kazhdan and G. Lusztig, “Representations of Coxeter groups and Hecke algebras,” Invent. Math. 53 (1979), 165-184.
9. A. Lascoux, “Polyn\hat omes de Kazhdan-Lusztig pour les variétés de Schubert vexillaires,” C. R. Acad. Sci. Paris, Ser. I, Math. 321 (1995), 667-670.
10. A. Lascoux and M.-P. Sch\ddot utzenberger, “Polyn\hat omes de Kazhdan & Lusztig pour les grassmanniennes, Young tableaux and Schur functions in algebra and geometry,” Astérisque 87/88 (1981), 249-266.
11. I.G. Macdonald, Notes on Schubert Polynomials, Publ. LACIM, UQAM, Montreal, 1991.
12. B. Shapiro, M. Shapiro, and A. Vainshtein, “Kazhdan-Lusztig polynomials for certain varieties of incomplete flags,” Discrete Math. 180 (1998), 345-355.
13. R.P. Stanley, Enumerative Combinatorics , Vol. 1, Wadsworth and Brooks/Cole, Monterey, CA, 1986; second printing, Cambridge University Press, Cambridge/New York, 1997.
14. A. Zelevinski, “Small resolutions of singularities of Schubert varieties,” Funct. Anal. Appl. 17 (1983), 142- 144.