Symmetric polynomials and divided differences in formulas of intersection theory

 Access to full text

 Full (PDF)

1. [A-C] E. Akyildiz, J. B. Carrell, An algebraic formula for the Gysin homomorphism from G/B to G/P, Illinois J. Math. 31 (1987), 312-320. Zbl0629.57030
2. [A-L-P] E. Akyildiz, A. Lascoux, P. Pragacz, Cohomology of Schubert subvarieties of $G{L}_n/P$, J. Differential Geom. 35 (1992), 511-519. 
3. [B-G-G] I. N. Bernstein, I. M. Gel'fand, S. I. Gel'fand, Schubert cells and cohomology of the spaces G/P, Russian Math. Surv. 28 (1973), 1-26. 
4. [Bou] N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 et 6, Herrmann, Paris, 1968. 
5. [B] W. Bruns, Die Divisorenklassengruppe der Restklassenringe von Polynomringen nach Determinantenidealen, Rev. Roumaine Math. Pures Appl. 20 (1975) 1109-1111. Zbl0347.13008
6. [Ch] C. Chevalley, Sur les Décompositions Cellulaires des Espaces G/B, Proc. Sympos. Pure Math. 56(1) (1994), 1-23. 
7. [DC-P] C. De Concini, P. Pragacz, On the class of Brill-Noether loci for Prym varieties, Math. Ann. 302 (1995), 687-697. Zbl0829.14021
8. [D-P-S] J. P. Demailly, T. Peternell, M. Schneider, Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994), 295-345. Zbl0827.14027
9. [D1] M. Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287-301. Zbl0269.22010
10. [D2] M. Demazure, Désingularisation des variétés de Schubert géneralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53-88. Zbl0312.14009
11. [F1] W. Fulton, Intersection Theory, Springer-Verlag, 1984. 
12. [F2] W. Fulton, Flags, Schubert polynomials, degeneracy loci and determinantal formulas, Duke Math. J. 65 (1992), 381-420. Zbl0788.14044
13. [F-L] W. Fulton, R. Lazarsfeld, Positive polynomials for ample vector bundles, Ann. of Math. (2) 118 (1983), 35-60. Zbl0537.14009
14. [G-V] I. Gessel, G. Viennot, Binomial determinants, paths, and hook length formulae, Adv. Math. 58 (1985), 300-321. Zbl0579.05004
15. [G1] G. Z. Giambelli, Risoluzione del problema degli spazi secanti, Mem. Accad. Sci. Torino (2) 52 (1903), 171-211. Zbl34.0615.02
16. [G2] G. Z. Giambelli, Il problema della correlazione negli iperspazi, Mem. Reale Istituto Lombardo 19 (1903), 155-194. 
17. [G3] G. Z. Giambelli, Ordine della varietà rappresentata coll'annullare tutti i minori di dato ordine estratti da una data matrice di forme, Acc. Nazion. dei Lincei, Roma, Classe di Science Fis., Mat. e Nat., Rendiconti 12 (1903), 294-297. Zbl34.0616.01
18. [G4] G. Z. Giambelli, Ordine di una varieta piu ampia di quella rappresentata coll'annulare tutti i minori di dato ordine, Memorie Reale Istituto Lombardo 20 (1904), 101-135. 
19. [G5] G. Z. Giambelli, Risoluzione del problema generale numerativo per gli spazi plurisecanti di una curva algebrica, Mem. Accad. Sci. Torino (2) 59 (1909), 433-508. Zbl40.0612.04
20. [H-T1] J. Harris, L. Tu, On symmetric and skew-symmetric determinantal varieties, Topology 23 (1984), 71-84. Zbl0534.55010
21. [H-T2] J. Harris, L. Tu, Chern numbers of kernel and cokernel bundles, Invent. Math. 75 (1984), 467-475. Zbl0542.14015
22. [Ha] R. Hartshorne, Ample vector bundles, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 63-94. Zbl0173.49003
23. [He-T] J. Herzog, Ngô Viêt Trung, Gröbner bases and multiplicity of determinantal and Pfaffian ideals, Adv. Math. 96 (1992), 1-37. 
24. [H-B] H. Hiller, B. Boe, Pieri formula for $S{O}_{2}n+1/U_n$ and $S{p}_n/U_n$, Adv. Math. 62 (1986), 49-67. 
25. [H] F. Hirzebruch, Topological Methods in Algebraic Geometry, Grundlehren der Math. Wissenschaften, Springer-Verlag, 1966; also Collected Papers, vol. I, Springer-Verlag, 1987, 151-334. 
26. [H-H] P. N. Hoffman, J. F. Humphreys, Projective representations of symmetric groups, Oxford University Press, 1992. ßk Zbl0777.20005
27. [J-L] C. G. Jacobi, A. Lascoux, De quibusdam rationibus universalibus ad determinantia functionalia expedienda, an unpublished manuscript. ßk 
28. [J-L-P] T. Józefiak, A. Lascoux, P. Pragacz, Classes of determinantal varieties associated with symmetric and antisymmetric matrices (in Russian), Izv. Akad. Nauk SSSR 45 (1981), 662-673. Zbl0471.14028
29. [K-L] G. Kempf, D. Laksov, The determinantal formula of Schubert calculus, Acta Math. 132 (1974), 153-162. Zbl0295.14023
30. [La] D. Laksov, Remarks on Giovanni Zeno Giambelli's work and life, Rend. Circ. Mat. Palermo (2) Suppl. 36 (1994), 207-218. Zbl0926.01007
31. [L-L-P-T] D. Laksov, A. Lascoux, P. Pragacz, A. Thorup, a book in preparation. 
32. [La-La-T] D. Laksov, A. Lascoux, A. Thorup, On Giambelli's theorem for complete correlations, Acta Math. 162 (1989), 143-199. Zbl0695.14023
33. [L1] A. Lascoux, Puissances extérieures, déterminants et cycles de Schubert, Bull. Soc. Math. France 102 (1974), 161-179. Zbl0295.14024
34. [L2] A. Lascoux, Classes de Chern d'un produit tensoriel, C. R. Acad. Sci. Paris Sér. I Math. 286 (1978), 385-387. Zbl0379.55011
35. [L3] A. Lascoux, La résultante de deux polynômes, in: Séminaire d'Algèbre Dubreil-Malliavin 1985 (M.-P. Malliavin, ed.), Lecture Notes in Math. 1220,Springer, 1986, 56-72. 
36. [L4] A. Lascoux, Interpolation de Lagrange, in: Second International Symposium (Segovia 1986) 'On Orthogonal Polynomials and their Applications', Monograf. Acad. Ci. Exact. Fís.-Quím. Nat. Zaragoza 1 (1988), 95-101. 
37. [L5] A. Lascoux, Classes de Chern des variétés de drapeaux, C. R. Acad. Sci. Paris 25 (1982), 393-398. Zbl0495.14032
38. [L6] A. Lascoux, Polynômes de Schubert; une approche historique, Discrete Math. 139 (1995), 303-317. 
39. [La-Le-T1] A. Lascoux, B. Leclerc, J.-Y. Thibon, Une nouvelle expression de functions P de Schur, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 221-224. 
40. [La-Le-T2] A. Lascoux, B. Leclerc, J.-Y. Thibon, Fonctions de Hall-Littlewood et polynômes de Kostka-Foulkes aux racines de l'unité, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 1-6. Zbl0769.05095
41. [L-P] A. Lascoux, P. Pragacz, Divided differences and ideals generated by symmetric polynomials, Discrete Math. 126 (1994), 209-215. Zbl0789.05094
42. [L-S1] A. Lascoux, M.-P. Schützenberger, Formulairé raisonné des functions symétriques, Prepublication L.I.T.P., Université Paris 7, 1985. 
43. [L-S2] A. Lascoux, M.-P. Schützenberger, Symmetry and flag manifolds, in: Invariant Theory (F. Gherardelli, ed.), Lecture Notes in Math. 996, Springer, 1983, 118-144. 
44. [L-S3] A. Lascoux, M.-P. Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), 447-450. 
45. [L-S4] A. Lascoux, M.-P. Schützenberger, Symmetrizing operators on polynomial rings, Functional Anal. Appl. 21 (1987), 77-78. 
46. [L-S5] A. Lascoux, M.-P. Schützenberger, Décompositions dans l'algèbre des differences divisées, Discrete Math. 99 (1992), 165-179. 
47. [L-S6] A. Lascoux, M.-P. Schützenberger, Schubert and Grothendieck polynomials, Notes of the talk given by the first author at Moscow University (November 1987), Preprint L.I.T.P., 1988. 
48. [M1] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 1979. 
49. [M2] I. G. Macdonald, Notes on Schubert polynomials, Publ. LACIM 6, UQUAM, Montréal, 1991. 
50. [Ma] L. Manivel, Un théoreme d'annulation 'à la Kawamata-Viehweg', Manuscripta Math. 83 (1994), 387-404. Zbl0820.14012
51. [Na] V. Navarro Aznar, On the Chern classes and the Euler characteristic for nonsingular complete intersections, Proc. Amer. Math. Soc. 78 (1980), 143-148. Zbl0473.14020
52. [N] I. Newton, Philosophiæ Naturalis Principia Mathematica, London, 1687. 
53. [Ni] H. A. Nielsen, Tensor Functors of Complexes, Aarhus Univ. Preprint 15, 1977/78. 
54. [P-P1] A. Parusiński, P. Pragacz, Characteristic numbers of degeneracy loci, in: Enumerative Algebraic Geometry (Copenhagen, 1989), (S. Kleiman, A. Thorup, eds.) Contemp. Math. 123 (1991), 189-197. Zbl0756.32019
55. [P-P2] A. Parusiński, P. Pragacz, Chern-Schwartz-MacPherson classes and the Euler characteristic of degeneracy loci and special divisors, J. Amer. Math. Soc. 8 (1995), 793-817. Zbl0857.14003
56. [Po] I. R. Porteous, Simple singularities of maps, in: Proceedings of Liverpool Singularities Symposium I, Lecture Notes in Math. 192, Springer, 1971, 286-307. 
57. [P1] P. Pragacz, Determinantal varieties and symmetric polynomials, Functional Anal. Appl. 21 (1987), 249-250. Zbl0633.14029
58. [P2] P. Pragacz, A note on elimination theory, Indag. Math. (N.S.) 49 (1987), 215-221. Zbl0632.12002
59. [P3] P. Pragacz, Enumerative geometry of degeneracy loci, Ann. Sci. École Norm. Sup. (4) 21 (1988), 413-454. Zbl0687.14043
60. [P4] P. Pragacz, Algebro-geometric applications of Schur S- and Q-polynomials, in: Topics in Invariant Theory -- Séminaire d'Algèbre Dubreil-Malliavin 1989-1990 (M.-P. Malliavin, ed.), Lecture Notes in Math. 1478, Springer, 1991, 130-191. 
61. [P5] P. Pragacz, Cycles of isotropic subspaces and formulas for symmetric degeneracy loci, Topics in Algebra, Banach Center Publ. 26(2), 1990, 189-199. Zbl0743.14009
62. [P-R1] P. Pragacz, J. Ratajski, Polynomials homologically supported on degeneracy loci, Preprint of the University of Bergen No. 61, 1991; to appear in: Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4). Zbl0876.14014
63. [P-R2] P. Pragacz, J. Ratajski, Pieri type formula for isotropic Grassmannians; the operator approach, Manuscripta Math. 79 (1993), 127-151. Zbl0789.14041
64. [P-R3] P. Pragacz, J. Ratajski, A Pieri-type formula for Sp(2m)/P and SO(2m+1)/P, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 1035-1040. Zbl0812.14034
65. [P-R4] P. Pragacz, J. Ratajski, A Pieri-type theorem for Lagrangian and odd Orthogonal Grassmannians, Preprint of the Max-Planck Institut für Mathematik 94-15, 1994; to appear in J. Reine Angew. Math. Zbl0847.14029
66. [P-R5] P. Pragacz, J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; the $\tilde{Q}$-polynomials approach, Preprint of the Max-Planck Institut für Mathematik, 1994, alg-geom/9602019; to appear in Compositio Math. 
67. [P-T] P. Pragacz, A. Thorup, On a Jacobi-Trudi formula for supersymmetric polynomials, Adv. Math. 95 (1992), 8-17. Zbl0774.05101
68. [R] J. Ratajski, Thesis, Math. Inst. Polish Acad. Sci., Warsaw, 1995. 
69. [Se] S. Sertöz, A triple intersection theorem for the varieties $SO\left(n\right)/P_d$, Fund. Math. 142 (1993), 201-220. Zbl0837.14040
70. [S] H. Schubert, Allgemeine Anzahlfunctionen für Kegelschnitte, Flächen und Raüme zweiten Grades in $n$ Dimensionen, Math. Ann. 45 (1894), 153-206. Zbl25.1038.03
71. [T] R. Thom, Les ensembles singuliers d'une application différentiable et leurs propriétés homologiques, in: Seminaire de Topologie de Strasbourg, December 1957. 
72. [Th] A. Thorup, Parameter spaces for quadrics, in this volume. 
73. [Tu] L. Tu, Degeneracy loci, Proceedings of the International Conference on Algebraic Geometry (Berlin 1985), Teubner Verlag, Leipzig, 1986, 296-305. 

1. Piotr Pragacz, Thom polynomials and Schur functions: the singularities $I_{2,2}(-)$