Schubert polynomials for the classical groups. (English) Zbl 0832.05098

Billey, Sara; Haiman, Mark

J. Am. Math. Soc. 8, No. 2, 443-482 (1995).

We present a general theory of Schubert polynomials, which are explicit representatives for Schubert classes in the cohomology ring of a flag variety with certain combinatorial properties. The starting point for this theory is a construction of Schubert classes in the cohomology ring of the flag variety of any semi-simple complex Lie group by Bernstein- Gelfand-Gelfand and Demazure. For the groups \(\text{SL}(n, \mathbb{C})\), Lascoux and Schützenberger made the crucial observation that one particular choice of representative of the top cohomology class yields Schubert polynomials simultaneously for all \(n\). In the present work we replicate the theory of \(\text{SL}(n, \mathbb{C})\) Schubert polynomials for the other infinite families of classical Lie groups and their flag varieties—the orthogonal groups \(\text{SO}(2n, \mathbb{C})\) and \(\text{SO}(2n+ 1,\mathbb{C})\) and the symplectic groups \(\text{Sp}(2n, \mathbb{C})\). We define Schubert polynomials to be elements in an inverse limit, which can be calculated as the unique solution of an infinite system of divided difference equations. The solution is derived using two equivalent formulas; one is an analog of the Billey-Jockusch-Stanley formula, while the other expresses our polynomials in terms of \(\text{SL}(n)\) Schubert polynomials and Schur \(Q\)- or \(P\)-functions. Our second formula involves the ‘shifted Edelman-Greene correspondences’ and analogs of the Stanley symmetric functions. The Schubert polynomials form a \(\mathbb{Z}\)-basis for the ring in which they are defined. The non-negative integer coefficients that appear when they are multiplied give intersection multiplicities for Schubert varieties directly, without the need to reduce the product modulo an ideal.

Reviewer: S.Billey (Cambridge)

MSC:

05E15	Combinatorial aspects of groups and algebras (MSC2010)
14M15	Grassmannians, Schubert varieties, flag manifolds

Keywords:

Schur functions; Schubert polynomials; Schubert classes; cohomology ring; flag variety; Lie group; orthogonal groups; symplectic groups; divided difference equations; Billey-Jockusch-Stanley formula; Stanley symmetric functions; Schubert varieties

References:

[1]	I. N. Bernšteĭn, I. M. Gel\(^{\prime}\)fand, and S. I. Gel\(^{\prime}\)fand, Schubert cells, and the cohomology of the spaces \?/\?, Uspehi Mat. Nauk 28 (1973), no. 3(171), 3 – 26 (Russian).
[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). · Zbl 0186.33001
[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. · Zbl 0790.05093 · doi:10.1023/A:1022419800503
[4]	Michel Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53 – 88 (French). Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I. · Zbl 0312.14009
[5]	Paul Edelman and Curtis Greene, Balanced tableaux, Adv. in Math. 63 (1987), no. 1, 42 – 99. · Zbl 0616.05005 · doi: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. · Zbl 0809.05091 · doi: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. · Zbl 0760.05093 · doi: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. · Zbl 0336.14001
[11]	S. L. Kleiman and Dan Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061 – 1082. · Zbl 0272.14016 · doi: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). · Zbl 0495.14031 · doi:10.1090/conm/088/1000001
[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. · Zbl 0784.05061
[14]	I. G. Macdonald, Symmetric functions and Hall polynomials, The Clarendon Press, Oxford University Press, New York, 1979. Oxford Mathematical Monographs. · Zbl 0487.20007
[15]	Piotr Pragacz, Algebro-geometric applications of Schur \?- and \?-polynomials, Topics in invariant theory (Paris, 1989/1990) Lecture Notes in Math., vol. 1478, Springer, Berlin, 1991, pp. 130 – 191. · Zbl 0783.14031 · doi:10.1007/BFb0083503
[16]	Bruce E. Sagan, Shifted tableaux, Schur \?-functions, and a conjecture of R. Stanley, J. Combin. Theory Ser. A 45 (1987), no. 1, 62 – 103. · Zbl 0661.05010 · doi: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. · Zbl 0587.20002 · doi: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.
[19]	Michelle L. Wachs, Flagged Schur functions, Schubert polynomials, and symmetrizing operators, J. Combin. Theory Ser. A 40 (1985), no. 2, 276 – 289. · Zbl 0579.05001 · doi:10.1016/0097-3165(85)90091-3
[20]	D. R. Worley, A theory of shifted Young tableaux, Ph.D. Thesis, M.I.T., 1984.