JOURNAL OF
ALGEBRAIC
COMBINATORICS

Home > Vol 2, No 4 (1993)

Sara C. Billey , William Jockusch and Richard P. Stanley

Schubert polynomials were introduced by Bernstein et al. and Demazure, and were extensively developed by Lascoux, Schützenberger, Macdonald, and others. We give an explicit combinatorial interpretation of the Schubert polynomial \mathfrak S _w \mathfrak{S}_ω in terms of the reduced decompositions of the permutation w. Using this result, a variation of Schensted's correspondence due to Edelman and Greene allows one to associate in a natural way a certain set M _w \mathcal{M}_ω of tableaux with w, each tableau contributing a single term to \mathfrak S _w \mathfrak{S}_ω . This correspondence leads to many problems and conjectures, whose interrelation is investigated. In Section 2 we consider permutations with no decreasing subsequence of length three (or 321-avoiding permutations). We show for such permutations that \mathfrak S _w \mathfrak{S}_ω is a flag skew Schur function. In Section 3 we use this result to obtain some interesting properties of the rational function 8 _{l/ m} (1, q, q ² , \frac{1}{4} ) 8_{λ/μ} (1,q,q^2 , \ldots ) , where 8 _{l/ m} 8_{λ/μ} denotes a skew Schur function.

1. E. Bender and D. Knuth, "Enumeration of plane partitions," J. Combin. Theory 13 (1972), 40-54.
2. N. Bergeron, "A combinatorial construction of the Schubert polynomials," J. Combin. Theory Series A, 60 (1992), 168-182.
3. I.N. Bernstein, I.M. Gelfand, and S.I. Gelfand, "Schubert cells and cohomology of the spaces G/P," Russian Math, Surveys 28, 3, (1973), 1-26. Translated from Uspekhi Mat. Nauk 28 (1973), 3-26.
4. M. Demazure, "Desingularisation des varietes Schubert generalisees," Ann. Sc. E.N.S.(4) 7 (1974), 53-88.
5. P. Edelman and C. Greene, "Balanced tableaux," Adv. Math. 63 (1987), 42-99.
6. D. Foata, "On the Netto inversion number of a sequence," Proc. Amer. Math. Soc. 19 (1968), 236-240.
7. S. Fomin and R. Stanley, "Schubert polynomials and the nilCoxeter algebra," Adv. Math., to appear.
8. F.R. Gantmacher, The Theory of Matrices, Vol. 1, Chelsea, New York, 1959.
9. P.M. Goodman, P. de la Harpe, and V.F.R. Jones, Coxeter Graphs and Towers of Algebras, Springer - Verlag, New York, 1989.
10. D.E. Knuth, The Art of Computer Programming, vol. 3, Addison-Wesley, Reading, MA, 1973.
11. W. Kraskiewicz and P. Pragacz, "Schubert polynomials and Schubert functors," preprint.
12. A. Lascoux and M. P. Schiitzenberger, "Structure de Hopf de 1'anneau de cohomologie et de 1'anneau de Grothendieck d'une variete de drapeaux," C. R. Acad. Sc. Paris 295 (1982), 629-633.
13. I.G. Macdonald, Notes on Schubert Polynomials, Laboratoire de combinatoire et d'informatique mathematique (LACIM), Universite du Quebec a Montreal, Montreal, 1991.
14. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1979.
15. H. Matsumoto, "Generateurs et relations des groupes de Weyl generalisees," C. R. Acad. Sci. Paris 258 (1964), 3419-3422.
16. V. Reiner and M. Shimozono, "Key polynomials and a flagged Littlewood-Richardson rule", preprint.
17. R. Simion and F.W. Schmidt, "Restricted permutations," European J. Combin. 6 (1985), 383-406.
18. R. Stanley, Enumerative Combinatorics, Vol. 1, Wadsworth & Brooks/Cole, Belmont, CA, 1986.
19. R. Stanley, "On the number of reduced decompositions of elements of Coxeter groups," European J. Combin. S (1984), 359-372.
20. R. Stanley, "Ordered structures and partitions," Mem. Amer. Math. Soc. 119, 1972.
21. R. Stanley, "Theory and application of plane partitions," Parts 1 and 2, Studies in Appl. Math. 50 (1971), 167-188, 259-279.
22. J. Tits, "Le probleme des mots dans les groupes de Coxeter," Symposia Math., Vol. 1, Academic Press, London, 1969, 175-185. 23 M.L. Wachs, "Flagged Schur functions, Schubert polynomials, and symmetrizing operators," /. Combin. Theory Series A 40 (1985), 276-289.