The Abstract

	Annals of Combinatorics 10 (2006) Permutation Diagrams, Fixed Points and Kazhdan-Lusztig R-Polynomials Federico Incitti Department of Mathematics, Royal Institute of Technology, Lindstedts väg 25, 10044 Stockholm, Sweden incitti@math.kth.se Received May 13, 2005 AMS Subject Classification: 05E15, 20F55 Abstract. In this paper, we give an algorithm for computing the Kazhdan-Lusztig, R-polynomials in the symmetric group. The algorithm is described in terms of permutation diagrams. In particular we focus on how the computation of the polynomial is affected by certain fixed points. As a consequence of our methods, we obtain explicit formulas for the R-polynomials associated with some general classes of intervals, generalizing results in [3] and [9]. Keywords: permutation diagram, fixed point, Bruhat order, Kazhdan-Lusztig polynomial, R-polynomial References 1. A. Björner and F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, Vol. 231, Springer-Verlag, New York, 2005. 2. A. Björner and M. Wachs, Bruhat order of Coxeter groups and shellability, Adv. Math. 43 (1982) 87–100. 3. F. Brenti, Combinatorial properties of the Kazhdan-Lusztig R-polynomials for S_n, Adv. Math. 126 (1997) 21–51. 4. M.J. Dyer, Hecke algebras and shellings of Bruhat intervals, Compositio Math. 89 (1993) 91–115. 5. M.J. Dyer, On minimal lengths of expressions of Coxeter group elements as products of reflections, Proc. Amer. Math. Soc. 129 (9) (2001) 2591–2595. 6. J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, 1990. 7. C. Kassel, A. Lascoux, and C. Reutenauer, The singular locus of a Schubert variety, J. Algebra 269 (2003) 74–108. 8. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979) 165–184. 9. M. Pagliacci, Explicit fromulae for some Kazhdan-Lusztig R-polynomials, J. Combin. Theory Ser. A 95 (2001) 74–87. 10. R.P. Stanley, Enumerative Combinatorics, Vol. 1, Wadsworth and Brooks/Cole, Pacific Grove, CA, 1986. Get the DVI \| PS \| PDF file of this abstract. back

Annals of Combinatorics 10 (2006)

Permutation Diagrams, Fixed Points and Kazhdan-Lusztig R-Polynomials

Federico Incitti

Department of Mathematics, Royal Institute of Technology, Lindstedts väg 25, 10044 Stockholm, Sweden
incitti@math.kth.se

Received May 13, 2005

AMS Subject Classification: 05E15, 20F55

Abstract. In this paper, we give an algorithm for computing the Kazhdan-Lusztig, R-polynomials in the symmetric group. The algorithm is described in terms of permutation diagrams. In particular we focus on how the computation of the polynomial is affected by certain fixed points. As a consequence of our methods, we obtain explicit formulas for the R-polynomials associated with some general classes of intervals, generalizing results in [3] and [9].

Keywords: permutation diagram, fixed point, Bruhat order, Kazhdan-Lusztig polynomial, R-polynomial

References

1. A. Björner and F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, Vol. 231, Springer-Verlag, New York, 2005.

2. A. Björner and M. Wachs, Bruhat order of Coxeter groups and shellability, Adv. Math. 43 (1982) 87–100.

3. F. Brenti, Combinatorial properties of the Kazhdan-Lusztig R-polynomials for S_n, Adv. Math. 126 (1997) 21–51.

4. M.J. Dyer, Hecke algebras and shellings of Bruhat intervals, Compositio Math. 89 (1993) 91–115.

5. M.J. Dyer, On minimal lengths of expressions of Coxeter group elements as products of reflections, Proc. Amer. Math. Soc. 129 (9) (2001) 2591–2595.

6. J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, 1990.

7. C. Kassel, A. Lascoux, and C. Reutenauer, The singular locus of a Schubert variety, J. Algebra 269 (2003) 74–108.

8. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979) 165–184.

9. M. Pagliacci, Explicit fromulae for some Kazhdan-Lusztig R-polynomials, J. Combin. Theory Ser. A 95 (2001) 74–87.

10. R.P. Stanley, Enumerative Combinatorics, Vol. 1, Wadsworth and Brooks/Cole, Pacific Grove, CA, 1986.

Get the DVI | PS | PDF file of this abstract.

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