Combinatorial 𝐵_{𝑛}-analogues of Schubert polynomials

Fomin, Sergey; Kirillov, Anatol

doi:10.1090/S0002-9947-96-01558-9

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ISSN 1088-6850(online) ISSN 0002-9947(print)

Combinatorial $B_{n}$ -analogues
of Schubert polynomials

Authors: Sergey Fomin and Anatol N. Kirillov
Journal: Trans. Amer. Math. Soc. 348 (1996), 3591-3620
MSC (1991): Primary 05E15; Secondary 05E05, 14M15
MathSciNet review: 1340174
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Abstract: Combinatorial $B_{n}$ -analogues of Schubert polynomials and corresponding symmetric functions are constructed and studied. The development is based on an exponential solution of the type $B$ Yang-Baxter equation that involves the nilCoxeter algebra of the hyperoctahedral group.

[BGG] 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 (55 #2941)
[B] 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 (39 #1590)
[Bo] Armand Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115–207 (French). MR 0051508 (14,490e)
[BH] S. C. Billey and M. Haiman, Schubert polynomials for the classical groups, manuscript, September 1993.
[BJS] 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 (94m:05197), http://dx.doi.org/10.1023/A:1022419800503
[Ch] I. Cherednik, Notes on affine Hecke algebras. I, Max-Planck-Institut Preprint MPI/91-14, 1991.
[FK1] S. Fomin and A. N. Kirillov, The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Proceedings of the 5th International Conference on Formal Power Series and Algebraic Combinatorics, Firenze, 1993, pp. 215--229, to appear in Discrete Mathematics.
[FK2] ------, Grothendieck polynomials and the Yang-Baxter equation, Proceedings of the 6th International Conference on Formal Power Series and Algebraic Combinatorics, DIMACS, 1994, pp. 183--190.
[FK3] ------, Universal exponential solution of the Yang-Baxter equation, preprint PARLPTHE 93/37, Université Paris VI, June 1993, to appear in Letters in Math. Physics.
[FS] Sergey Fomin and Richard P. Stanley, Schubert polynomials and the nil-Coxeter algebra, Adv. Math. 103 (1994), no. 2, 196–207. MR 1265793 (95f:05115), http://dx.doi.org/10.1006/aima.1994.1009
[Fu] W. Fulton, Schubert varieties in flag bundles for the classical groups, preprint.
[L1] Alain Lascoux, Anneau de Grothendieck de la variété de drapeaux, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 1–34 (French). MR 1106909 (92j:14064), http://dx.doi.org/10.1007/978-0-8176-4576-2_1
[L2] ------, Polynômes de Schubert. Une approche historique, Séries formelles et combinatoire algébrique (P. Leroux and C. Reutenauer, eds.), Montréal, LACIM, UQAM, 1992, pp. 283--296.
[LS] A.Lascoux, M.P.Schützenberger, Polynômes de Schubert, C. R. Ac. Sci. 294 (1982), 447.
[TKL1] Tao Kai Lam, $B_{n}$ Stanley symmetric functions, Séries formelles et combinatoire algébrique, Proceedings of the 6th International Conference on Formal Power Series and Algebraic Combinatorics, DIMACS, 1994, pp. 315--324.
[TKL2] ------, and analogues of stable Schubert polynomials and related insertion algorithms, Ph. D. thesis, M.I.T., 1994.
[M] I. G. Macdonald, Notes on Schubert polynomials, Laboratoire de combinatoire et d'informatique mathématique (LACIM), Université du Québec à Montréal, Montréal, 1991.
[P] P. Pragacz, Algebro-geometric applications of Schur - and -polynomials, ``Topics in Invariant Theory'' (M.-P. Malliavin, ed.), pp. 130--191, Lecture Notes in Math., vol. 1478, Springer-Verlag, Berlin, 1991. MR 93h:05190
[SS] 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 (86i:05011)
[S] 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 (86i:05011)
[V] A. M. Vershik, Local stationary algebras, Algebra and analysis (Kemerovo, 1988) Amer. Math. Soc. Transl. Ser. 2, vol. 148, Amer. Math. Soc., Providence, RI, 1991, pp. 1–13. MR 1109059 (92b:16060)

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