The theory of Schur and Schubert polynomials is revisited in this paper from the point of view of generalized Thom polynomials. When we apply a general method to compute Thom polynomials for this case we obtain a new definition for (double versions of) Schur and Schubert polynomials: they will be solutions of interpolation problems.
@article{bwmeta1.element.doi-10_2478_BF02475176,
author = {L\'aszl\'o Feh\'er and Rich\'ard Rim\'anyi},
title = {Schur and Schubert polynomials as Thom polynomials-cohomology of moduli spaces},
journal = {Open Mathematics},
volume = {1},
year = {2003},
pages = {418-434},
zbl = {1038.57008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02475176}
}
László Fehér; Richárd Rimányi. Schur and Schubert polynomials as Thom polynomials-cohomology of moduli spaces. Open Mathematics, Tome 1 (2003) pp. 418-434. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02475176/
[1] [AB82] M.F. Atiyah and R. Bott: “The Yang-Mills equations over Riemann surfaces”, Philos. Trans. Roy. Soc. London Ser. A, Vol. 308, (1982), pp. 523–615. | Zbl 0509.14014
[2] [AVGL91] V.I. Arnold, V.A. Vassiliev, V.V. Goryunov, O.V. Lyashko: “Singularities. Local and global theory”, Enc. Math. Sci. Dynamical Systems VI, Springer-Verlag, Berlin, 1991.
[3] [BF99] A. Buch and W. Fulton: “Chern class formulas for quiver varieties”, Inv. Math., Vol. 135, (1999), pp. 665–687. http://dx.doi.org/10.1007/s002220050297[Crossref] | Zbl 0942.14027
[4] [CLL02] W. Chen, B. Li, J.D. Louck: “The flagged double schur function”, J. Algebraic Combin., Vol. 15, (2002), pp. 7–26. http://dx.doi.org/10.1023/A:1013217015135[Crossref] | Zbl 0990.05133
[5] [FP89] W. Fulton and P. Pragacz: Schubert Varieties and Degeneracy Loci, Springer-Verlag, Berlin, 1998.
[6] [FR02] L. Fehér and R. Rimányi: “Classes of degeneraci loci for quivers-the Thom polynomial point of view”, Duke J. Math., Vol. 114, (2002), pp. 193–213. http://dx.doi.org/10.1215/S0012-7094-02-11421-5[Crossref] | Zbl 1054.14010
[7] [FR03] L. M. Fehér and R. Rimányi: “Calculation of Thom polynomials and other cohomological obstructions for group actions”, to appear in Sao Carlos Singularities 2002, Cont. Math. AMS, 2003.
[8] [FRN03] L. Fehér, A. Némethi, R. Rimányi: Coincident root loci of binary forms, http://www.math.ohio-state.edu/∼rimanyi/cikkek, 2003. | Zbl 1115.14046
[9] [Ful92] W. Fulton: “Flags, Schubert polynomials, degeneracy loci, and determinantal formulas”, Duke Math. J., Vol. 65, (1992), pp. 381–420. http://dx.doi.org/10.1215/S0012-7094-92-06516-1[Crossref] | Zbl 0788.14044
[10] [Ful98] W. Fulton: Intersection Theory. Springer, Berlin, 1984, 1998. [WoS]
[11] [Kaz95] M. Kazarian: “Characteristic classes of Lagrange and Legendre singularities”, Russian Math. Surv., Vol. 50, (1995), pp. 701–726. http://dx.doi.org/10.1070/RM1995v050n04ABEH002579[Crossref] | Zbl 0868.58030
[12] [Kaz97] M.É. Kazarian: “Characteristic classes of singularity theory”, In: V.I. Arnold et al., (Eds): The Arnold-Gelfand mathematical seminars: geometry and singularity theory, Birkhauser Boston, Boston MA, 1997, pp. 325–340.
[13] [Kaz00] M. Kazarian: “Thom polynomials for Lagrange, Legendre and critical point singularities”, Isaac Newton Institute for Math. Sci., preprint, 2000.
[14] [Kir84] F. Kirwan: Cohomology of quotients in symplectic and algebraic geometry. No. 31, in Mathematical Notes, Princeton University Press, Princeton NY, 1984.
[15] [Kir92] F. Kirwan: “The cohomology rings of moduli spaces of bundles over Riemann surfaces”, J. Amer. Math. Soc., Vol. 5, (1992), pp. 853–906. http://dx.doi.org/10.2307/2152712[Crossref]
[16] [KL74] G. Kempf and D. Laksov: “The determinantal formula of Schubert calculus”, Acta. Math., Vol. 132, (1974), pp. 153–162. http://dx.doi.org/10.1007/BF02392111[Crossref] | Zbl 0295.14023
[17] [Las74] Alain Lascoux: “Puissances extérieures, déterminants et cycles de Schubert”, Bull. Soc. Math. France, Vol. 102, (1974), pp. 161–179. | Zbl 0295.14024
[18] [LS82] Lascoux and Schützenberger: “Polynômes de Schubert”, C. R. Acad. Sci. Paris, Vol. 294, (1982), pp. 447–450. | Zbl 0495.14031
[19] [Mac91] I.G. MacDonald: Notes on Schubert polynomials, LACIM 6, 1991.
[20] [Pra88] Piotr Pragacz: “Enumerative geometry of degeneracy loci”, Ann. Sci. École Norm. Sup. (4), Vol. 21, (1988), pp. 413–454. | Zbl 0687.14043
[21] [Rim01] R. Rimányi: “Thom polynomials, symmetries and incidences of singularities”, Inv. Math., Vol. 143, (2001), pp. 499–521. http://dx.doi.org/10.1007/s002220000113[Crossref] | Zbl 0985.32012
[22] [Sti36] E. Stiefel: “Richtungsfelder und Fernparallelismus in Mannigfaltigkeiten”, Comm. Math. Helv., Vol. 8, (1936), pp. 3–51. | Zbl 62.0662.02
[23] [Szű79] A. Szűcs: “Analogue of the Thom space for mapping with singularity of type ∑1”, Math. Sb. (N. S.), Vol. 108, (1979), pp. 438–456.
[24] [Vas88] V.A. Vassiliev: Lagrange and Legendre Characteristic Classes, Gordon and Breach, New York, 1988.