Geodesic
Schubert calculus and Gelfand–Zetlin polytopes
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 67 (2012) no. 4, pp. 685-719 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A new approach is described to the Schubert calculus on complete flag varieties, using the volume polynomial associated with Gelfand–Zetlin polytopes. This approach makes it possible to compute the intersection products of Schubert cycles by intersecting faces of a polytope. Bibliography: 23 titles.
Keywords: Flag variety, Gelfand–Zetlin polytope
Mots-clés : Schubert calculus, volume polynomial. 
@article{RM_2012_67_4_a1,
     author = {V. A. Kirichenko and E. Yu. Smirnov and V. A. Timorin},
     title = {Schubert calculus and {Gelfand{\textendash}Zetlin} polytopes},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {685--719},
     year = {2012},
     volume = {67},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2012_67_4_a1/}
}
TY  - JOUR
AU  - V. A. Kirichenko
AU  - E. Yu. Smirnov
AU  - V. A. Timorin
TI  - Schubert calculus and Gelfand–Zetlin polytopes
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2012
SP  - 685
EP  - 719
VL  - 67
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/RM_2012_67_4_a1/
LA  - en
ID  - RM_2012_67_4_a1
ER  - 
%0 Journal Article
%A V. A. Kirichenko
%A E. Yu. Smirnov
%A V. A. Timorin
%T Schubert calculus and Gelfand–Zetlin polytopes
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2012
%P 685-719
%V 67
%N 4
%U http://geodesic.mathdoc.fr/item/RM_2012_67_4_a1/
%G en
%F RM_2012_67_4_a1
V. A. Kirichenko; E. Yu. Smirnov; V. A. Timorin. Schubert calculus and Gelfand–Zetlin polytopes. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 67 (2012) no. 4, pp. 685-719. http://geodesic.mathdoc.fr/item/RM_2012_67_4_a1/

[1] H. H. Andersen, “Schubert varieties and Demazure's character formula”, Invent. Math., 79:3 (1985), 611–618 | DOI | MR | Zbl

[2] N. Bergeron, S. Billey, “RC-graphs and Schubert polynomials”, Experiment. Math., 2:4 (1993), 257–269 | DOI | MR | Zbl

[3] I. N. Bernstein, I. M. Gel'fand, S. I. Gel'fand, “Schubert cells and cohomology of the spaces $G/P$”, Russian Math. Surveys, 28:3 (1973), 1–26 | DOI | MR | Zbl

[4] M. Brion, “The structure of the polytope algebra”, Tohoku Math. J. (2), 49:1 (1997), 1–32 | MR | Zbl

[5] I. Coskun, “A Littlewood–Richardson rule for two-step flag varieties”, Invent. Math., 176:2 (2009), 325–395 | DOI | MR | Zbl

[6] M. Demazure, “Désingularisation des variétés de Schubert généralisées”, Ann. Sci. École Norm. Sup. (4), 7 (1974), 53–88 | MR | Zbl

[7] S. Fomin, A. N. Kirillov, “The Yang–Baxter equation, symmetric functions, and Schubert polynomials”, Discrete Math., 153:1-3 (1996), 123–143 | DOI | MR | Zbl

[8] W. Fulton, Young tableaux, With application to representation theory and geometry, London Math. Soc. Stud. Texts, 35, Cambridge Univ. Press, Cambridge, 1997, x+260 pp. | MR | Zbl

[9] K. Kaveh, “Note on cohomology rings of spherical varieties and volume polynomial”, J. Lie Theory, 21:2 (2011), 263–283, arXiv: math/0312503 | MR | Zbl

[10] K. Kaveh, A. G. Khovanskii, “Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory”, Ann. of Math. (2), 176:2 (2012), 925–978, arXiv: 0904.3350 | DOI

[11] A. G. Khovanskii, “Newton polyhedron, Hilbert polynomial, and sums of finite sets”, Funct. Anal. Appl., 26:4 (1992), 276–281 | DOI | MR | Zbl

[12] V. Kiritchenko, “Gelfand–Zetlin polytopes and geometry of flag varieties”, Int. Math. Res. Not. IMRN, 2010, no. 13, 2512–2531 | DOI | MR | Zbl

[13] S. L. Kleiman, “The transversality of a general translate”, Compositio Math., 28 (1974), 287–297 | MR | Zbl

[14] A. Knutson, E. Miller, “Gröbner geometry of Schubert polynomials”, Ann. of Math. (2), 161:3 (2005), 1245–1318 | DOI | MR | Zbl

[15] M. Kogan, Schubert geometry of flag varieties and Gelfand–Cetlin theory, Ph.D. thesis, Massachusetts Institute of Technology, 2000 | MR

[16] M. Kogan, E. Miller, “Toric degeneration of Schubert varieties and Gelfand–Tsetlin polytopes”, Adv. Math., 193:1 (2005), 1–17 | DOI | MR | Zbl

[17] A. Lascoux, M.-P. Schützenberger, “Polynômes de Schubert”, C. R. Acad. Sci. Paris Sér. I Math., 294:13 (1982), 447–450 | DOI | MR | Zbl

[18] L. Manivel, Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence, Cours Spec., 3, Société Mathématique de France, Paris, 1998, vi+179 pp. | MR | Zbl

[19] E. Miller, “Mitosis recursion for coefficients of Schubert polynomials”, J. Comb. Theory Ser. A, 103:2 (2003), 223–235 | DOI | MR | Zbl

[20] H. Pittie, A. Ram, “A Pieri–Chevalley formula in the $K$-theory of a $G/B$-bundle”, Electron. Res. Announc. Amer. Math. Soc., 5 (1999), 102–107 | DOI | MR | Zbl

[21] A. Postnikov, R. P. Stanley, “Chains in the Bruhat order”, J. Algebraic Combin., 29:2 (2009), 133–174 | DOI | MR | Zbl

[22] A. V. Pukhlikov, A. G. Khovanskii, “The Riemann–Roch theorem for integrals and sums of quasipolynomials on virtual polytopes”, St. Petersburg Math. J., 4:4 (1993), 789–812 | MR | Zbl

[23] V. A. Timorin, “An analogue of the Hodge–Riemann relations for simple convex polytopes”, Russian Math. Surveys, 54:2 (1999), 381–426 | DOI | MR | Zbl