PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2009 | 7 | 3 | 429-441
Tytuł artykułu

The incidence class and the hierarchy of orbits

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
R. Rimányi defined the incidence class of two singularities η and ζ as [η]|ζ, the restriction of the Thom polynomial of η to ζ. He conjectured that (under mild conditions) [η]|ζ ≠ 0 ⇔ ζ ⊂ $$ \bar \eta $$. Generalizing this notion we define the incidence class of two orbits η and ζ of a representation. We give a sufficient condition (positivity) for ζ to have the property that [η]|ζ ≠ 0 ⇔ ζ ⊂ $$ \bar \eta $$ for any other orbit η. We show that for many interesting cases, e.g. the quiver representations of Dynkin type positivity holds for all orbits. In other words in these cases the incidence classes completely determine the hierarchy of the orbits. We also study the case of singularities where positivity doesn’t hold for all orbits.
Wydawca
Czasopismo
Rocznik
Tom
7
Numer
3
Strony
429-441
Opis fizyczny
Daty
wydano
2009-09-01
online
2009-08-12
Twórcy
Bibliografia
  • [1] Arnol’d V.I., Guseĭn-Zade S.M., Varchenko A.N., Singularities of differentiable maps II, Monographs in Mathematics, Birkhauser Boston Inc., Boston, MA, 1988
  • [2] Buch A.S., Rimanyi R., Specializations of Grothendieck polynomials, C. R. Math. Acad. Sci. Paris, 2004, 339(1), 1–4
  • [3] Edidin D., Graham W., Equivariant intersection theory, Invent. Math., 1998, 131(3), 595–634 http://dx.doi.org/10.1007/s002220050214
  • [4] Fehér L.M., Némethi A., Rimányi R., The degree of the discriminant of irreducible representations, J. Algebraic Geometry, 2008, 17, 751–780
  • [5] Fehér L., Rimányi R., Classes of degeneracy loci for quivers: the Thom polynomial point of view, Duke Math. J., 2002, 114(2), 193–213 http://dx.doi.org/10.1215/S0012-7094-02-11421-5
  • [6] Fehér L.M., Rimányi R., Thom polynomials with integer coefficients, Illinois J. Math., 2002, 46(4), 1145–1158
  • [7] Fehér L.M., Rimányi R., Schur and Schubert polynomials as Thom polynomials-cohomology of moduli spaces, Cent. Eur. J. Math., 2003, 1(4), 418–434 http://dx.doi.org/10.2478/BF02475176
  • [8] Fehér L.M., Rimányi R., Calculation of Thom polynomials and other cohomological obstructions for group actions, In: Real and complex singularities, Contemp. Math., Amer. Math. Soc., Providence, RI, 2004, 354, 69–93
  • [9] Fulton W., Young tableaux, London Mathematical Society Student Texts, Vol. 35, Cambridge University Press, Cambridge, 1997
  • [10] Goldin R.F., The cohomology ring of weight varieties and polygon spaces, Adv. in Math., 2001, 160, 175–204 http://dx.doi.org/10.1006/aima.2001.1984
  • [11] Kazarian M.É., Characteristic classes of singularity theory, In: The Arnold-Gelfand mathematical seminars, pages, Birkhäuser Boston, Boston, MA, 1997, 325–340
  • [12] Knutson A., Miller E., Gröbner geometry of Schubert polynomials, Annals of Math., 2005, 2(3), 1245–1318 http://dx.doi.org/10.4007/annals.2005.161.1245
  • [13] Knutson A., Miller E., Shimozono M., Four positive formulae for type a quiver polynomials, Invent. Math., 2006, 166, 229–325 http://dx.doi.org/10.1007/s00222-006-0505-0
  • [14] Knutson A., Shimozono M., Kempf collapsing and quiver loci, preprint available at http://arxiv.org/abs/math/0608327
  • [15] Kumar S., The nil hecke ring and singularity of Schubert varieties, Invent. Math., 1996, 123(3), 471–506 http://dx.doi.org/10.1007/s002220050038
  • [16] Lascoux A., Schützenberger M.-P., Décompositions dans l’algébre des differences divisées, Discrete Math., 1992, 99, 165–179 http://dx.doi.org/10.1016/0012-365X(92)90372-M
  • [17] Mather J., Stability of C ∞ mappings. VI. the nice dimensions, In: Liverpool Singularities-Symposium I, number 192 in SLNM, 1971, 207–253
  • [18] Miller E., Sturmfels B., Combinatorial commutative algebra, Springer, Berlin, 2004
  • [19] Patakfalvi Z., Orbit structures and incidence, Master’s thesis, Eotvos University, Budapest, 2006
  • [20] Porteous I., Simple singularities of maps, In: Liverpool Singularities-Symposium I, number 192 in SLNM, 1971, 286–307
  • [21] Rimányi R., Generalized Pontrjagin-Thom construction for singular maps, PhD thesis, Eotvos University, Budapest, 1999
  • [22] Rimányi R., Thom polynomials, symmetries and incidences of singularities, Invent. Math., 2001, 143(3), 499–521 http://dx.doi.org/10.1007/s002220000113
  • [23] Wall C.T.C., Nets of conics, Math. Proc. Cambridge Philos. Soc., 1977, 81(3), 351–364 http://dx.doi.org/10.1017/S0305004100053421
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-009-0026-3
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.