Schubert polynomials for the classical groups

Billey, Sara; Haiman, Mark

doi:10.1090/S0894-0347-1995-1290232-1

Schubert polynomials for the classical groups

Authors: Sara Billey and Mark Haiman
Journal: J. Amer. Math. Soc. 8 (1995), 443-482
MSC: Primary 05E15; Secondary 14M15
DOI: https://doi.org/10.1090/S0894-0347-1995-1290232-1
MathSciNet review: 1290232
Full-text PDF Free Access

[1] I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Schubert cells, and the cohomology of the spaces 𝐺/𝑃, Uspehi Mat. Nauk 28 (1973), no. 3(171), 3–26 (Russian). MR 0429933
[2] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). MR 0240238
[3] Sara C. Billey, William Jockusch, and Richard P. Stanley, Some combinatorial properties of Schubert polynomials, J. Algebraic Combin. 2 (1993), no. 4, 345–374. MR 1241505, https://doi.org/10.1023/A:1022419800503
[4] Michel Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88 (French). MR 354697
[5] Paul Edelman and Curtis Greene, Balanced tableaux, Adv. in Math. 63 (1987), no. 1, 42–99. MR 871081, https://doi.org/10.1016/0001-8708(87)90063-6
[6] S. Fomin and A. N. Kirillov, Combinatorial ${B_n}$ analogues of Schubert polynomials, Manuscript, M.I.T., 1993.
[7] Sergey Fomin and Richard P. Stanley, Schubert polynomials and the nil-Coxeter algebra, Adv. Math. 103 (1994), no. 2, 196–207. MR 1265793, https://doi.org/10.1006/aima.1994.1009
[8] I. Gessel and G. Viennot, Determinants and plane partitions, preprint, 1983.
[9] Mark D. Haiman, Dual equivalence with applications, including a conjecture of Proctor, Discrete Math. 99 (1992), no. 1-3, 79–113. MR 1158783, https://doi.org/10.1016/0012-365X(92)90368-P
[10] Steven L. Kleiman, Problem 15: rigorous foundation of Schubert’s enumerative calculus, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R. I., 1976, pp. 445–482. Proc. Sympos. Pure Math., Vol. XXVIII. MR 0429938
[11] S. L. Kleiman and Dan Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061–1082. MR 323796, https://doi.org/10.2307/2317421
[12] Alain Lascoux and Marcel-Paul Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 13, 447–450 (French, with English summary). MR 660739
[13] I. G. Macdonald, Schubert polynomials, Surveys in combinatorics, 1991 (Guildford, 1991) London Math. Soc. Lecture Note Ser., vol. 166, Cambridge Univ. Press, Cambridge, 1991, pp. 73–99. MR 1161461
[14] I. G. Macdonald, Symmetric functions and Hall polynomials, The Clarendon Press, Oxford University Press, New York, 1979. Oxford Mathematical Monographs. MR 553598
[15] Piotr Pragacz, Algebro-geometric applications of Schur 𝑆- and 𝑄-polynomials, Topics in invariant theory (Paris, 1989/1990) Lecture Notes in Math., vol. 1478, Springer, Berlin, 1991, pp. 130–191. MR 1180989, https://doi.org/10.1007/BFb0083503
[16] Bruce E. Sagan, Shifted tableaux, Schur 𝑄-functions, and a conjecture of R. Stanley, J. Combin. Theory Ser. A 45 (1987), no. 1, 62–103. MR 883894, https://doi.org/10.1016/0097-3165(87)90047-1
[17] Richard P. Stanley, On the number of reduced decompositions of elements of Coxeter groups, European J. Combin. 5 (1984), no. 4, 359–372. MR 782057, https://doi.org/10.1016/S0195-6698(84)80039-6
[18] Richard P. Stanley, Some combinatorial aspects of the Schubert calculus, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) Springer, Berlin, 1977, pp. 217–251. Lecture Notes in Math., Vol. 579. MR 0465880
[19] Michelle L. Wachs, Flagged Schur functions, Schubert polynomials, and symmetrizing operators, J. Combin. Theory Ser. A 40 (1985), no. 2, 276–289. MR 814415, https://doi.org/10.1016/0097-3165(85)90091-3
[20] D. R. Worley, A theory of shifted Young tableaux, Ph.D. Thesis, M.I.T., 1984.