no. 5
Thom polynomials and Schur functions: the singularities I 2,2 (-)
Pragacz, Piotr1
1 Institute of Mathematics of Polish Academy of Sciences Sniadeckich 8 00-956 Warszawa (Poland)
Annales de l'Institut Fourier, Volume 57 (2007) no. 5, pp. 1487-1508.

We give the Thom polynomials for the singularities I 2,2 associated with maps (ℂ • ,0)→(ℂ •+k ,0) with parameter k≥0. Our computations combine the characterization of Thom polynomials via the “method of restriction equations” of Rimanyi et al. with the techniques of Schur functions.

Nous présentons les polynômes de Thom pour les singularités I 2,2 associées aux applications (ℂ • ,0)→(ℂ •+k ,0) de paramètre k≥0. Nos calculs combinent la caractérisation des polynômes de Thom via la « méthode des équations restreintes » de Rimanyi et al. avec les techniques des fonctions de Schur.

DOI: 10.5802/aif.2302
Classification: 05E05, 14N10, 57R45
Keywords: Thom polynomials, singularities, global singularity theory, classes of degeneracy loci, Schur functions, resultants
Mot clés : polynômes de Thom, singularités, fonctions de Schur

Pragacz, Piotr 1

1 Institute of Mathematics of Polish Academy of Sciences Sniadeckich 8 00-956 Warszawa (Poland) 
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Pragacz, Piotr. Thom polynomials and Schur functions: the singularities $I_{2,2}(-)$. Annales de l'Institut Fourier, Volume 57 (2007) no. 5, pp. 1487-1508. doi : 10.5802/aif.2302. https://aif.centre-mersenne.org/articles/10.5802/aif.2302/

[1] Arnold, V.; Vasilev, V.; Goryunov, V.; Lyashko, O. Singularities. Local and global theory, Enc. Math. Sci., Volume 6 (Dynamical Systems VI), Springer, 1993 | Zbl

[2] Berczi, G.; Feher, L.; Rimanyi, R. Expressions for resultants coming from the global theory of singularities, Topics in algebraic and noncommutative geometry, (L.McEwan et al. eds.), Contemporary Math., Volume 324, Amer. Math. Soc., 2003, pp. 63-69 | MR | Zbl

[3] Berele, A.; Regev, A. Hook Young diagrams with applications to combinatorics and to representation theory of Lie superalgebras, Adv. in Math., Volume 64 (1987), pp. 118-175 | DOI | MR | Zbl

[4] Damon, J. Thom polynomials for contact singularities, Harvard (1972) (Ph. D. Thesis)

[5] Du Plessis, A.; Wall, C. T. C. The geometry of topological stability, Oxford Math. Monograph, 1995 | MR | Zbl

[6] Feher, L.; Komuves, B. On second order Thom-Boardman singularities, Fund. Math., Volume 191 (2006), pp. 249-264 | DOI | MR | Zbl

[7] Fehér, L.; M., László; Rimányi, R. Calculation of Thom polynomials and other cohomological obstructions for group actions, Real and complex singularities (Contemp. Math.), Volume 354, Amer. Math. Soc., 2004, pp. 69-93 | MR | Zbl

[8] Feher, L.; Rimányi, R. On the structure of Thom polynomials of singularities (Preprint, September 2005)

[9] Fulton, William; Pragacz, Piotr Schubert varieties and degeneracy loci, Lecture Notes in Mathematics, 1689, Springer-Verlag, Berlin, 1998 (Appendix J by the authors in collaboration with I. Ciocan-Fontanine) | MR | Zbl

[10] Jänich, K. Symmetry properties of singularities of C ∞ -functions, Math. Ann., Volume 238 (1979), pp. 147-156 | DOI | MR | Zbl

[11] Kazarian, M. E. (Letter to the author, dated July 4, 2006)

[12] Kazarian, M. E. Characteristic classes of singularity theory, The Arnold-Gelfand mathematical seminars: Geometry and singularity theory (1997), pp. 325-340 | MR | Zbl

[13] Kazarian, M. E. Classifying spaces of singularities and Thom polynomials, New developments in singularity theory, NATO Sci. Ser. II Math. Phys. Chem., Volume 21, Kluwer Acad. Publ., Dordrecht, 2001, pp. 117-134 | MR | Zbl

[14] Kleiman, S. The enumerative theory of singularities, Real and complex singularities, Oslo 1976 (P. Holm ed.) (1978), pp. 297-396 | MR | Zbl

[15] Laksov, D.; Lascoux, A.; Thorup, A. On Giambelli’s theorem for complete correlations, Acta Math., Volume 162 (1989), pp. 143-199 | DOI | Zbl

[16] Lascoux, A. Symmetric functions and combinatorial operators on polynomials, CBMS/AMS Lectures Notes, Volume 99, Providence, 2003 | MR | Zbl

[17] Lascoux, A.; Schützenberger, M.-P. Formulaire raisonné de fonctions symétriques, Université Paris 7, 1985

[18] Lascoux, Alain Addition of ±1: application to arithmetic, Sém. Lothar. Combin., Volume 52 (2004/05), pp. Art. B52a, 9pp. (electronic) | MR | Zbl

[19] Macdonald, I. G. Symmetric functions and Hall-Littlewood polynomials, Oxford Math. Monographs, Amer. Math. Soc., 1995 (2nd ed.)

[20] Ozturk, O. On Thom polynomials for A 4 (-) via Schur functions Preprint, IMPAN Warszawa 2006 (670), to appear in Serdica Math.J.

[21] Porteous, I. Simple singularities of maps, Proc. Liverpool Singularities I, Springer Lecture Notes in Math., Volume 192, Springer, 1971, pp. 286-307 | MR | Zbl

[22] Pragacz, P. Thom polynomials and Schur functions I (math.AG/0509234)

[23] Pragacz, P. Thom polynomials and Schur functions: the singularities A 3 (-) (in preparation)

[24] Pragacz, P. Thom polynomials and Schur functions: towards the singularities A i (-) Preprint MPIM Bonn 2006 (139)

[25] Pragacz, P. Note on elimination theory, Indagationes Math., Volume 49 (1987), pp. 215-221 | DOI | MR | Zbl

[26] Pragacz, P. Enumerative geometry of degeneracy loci, Ann. Sci. École Norm. Sup., Volume 21 (1988), pp. 413-454 | Numdam | MR | Zbl

[27] Pragacz, P. Algebro-geometric applications of Schur S- and Q-polynomials, Topics in invariant theory, Séminaire d’Algèbre Dubreil-Malliavin 1989-1990 (M.-P. Malliavin ed.), Springer Lecture Notes in Math., Volume 1478, Springer, 1991, pp. 130-191 | Zbl

[28] Pragacz, P. Symmetric polynomials and divided differences in formulas of intersection theory, Parameter spaces (P. Pragacz ed.), Volume 36, Banach Center Publications, 1996, pp. 125-177 | MR | Zbl

[29] Pragacz, P.; Thorup, A. On a Jacobi-Trudi identity for supersymmetric polynomials, Adv. in Math., Volume 95 (1992), pp. 8-17 | DOI | MR | Zbl

[30] Pragacz, P.; Weber, A. Positivity of Schur function expansions of Thom polynomials Preprint, math.AG/0605308 and MPIM Bonn 2006 (60), to appear in Fund. Math.

[31] Rimanyi, R. Thom polynomials, symmetries and incidences of singularities, Inv. Math., Volume 143 (2001), pp. 499-521 | DOI | MR | Zbl

[32] Rimanyi, R.; Szücs, A. Generalized Pontrjagin-Thom construction for maps with singularities, Topology, Volume 37 (1998), pp. 1177-1191 | DOI | MR | Zbl

[33] Schubert, H. Allgemeine Anzahlfunctionen für Kegelschnitte, Flächen und Raüme zweiten Grades in n Dimensionen, Math. Ann., Volume 45 (1894), pp. 153-206 | DOI | MR

[34] Stembridge, J. A characterization of supersymmetric polynomials, J. Algebra, Volume 95 (1985), pp. 439-444 | DOI | MR | Zbl

[35] Thom, R. Les singularités des applications différentiables, Ann. Inst. Fourier, Volume 6 (1955–56), pp. 43-87 | DOI | Numdam | MR | Zbl

[36] Wall, C. T. C. A second note on symmetry of singularities, Bull. London Math. Soc., Volume 12 (1980), pp. 347-354 | DOI | MR | Zbl

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