On Schur function expansions of Thom polynomials

@article{Ozturk2012OnSF,
  title={On Schur function expansions of Thom polynomials},
  author={Ozer Ozturk and Piotr Pragacz},
  journal={arXiv: Algebraic Geometry},
  year={2012},
  pages={443-479}
}
We discuss computations of the Thom polynomials of singularity classes of maps in the basis of Schur functions. We survey the known results about the bound on the length and a rectangle containment for partitions appearing in such Schur function expansions. We describe several recursions for the coefficients. For some singularities, we give old and new computations of their Thom polynomials. 

Positivity of Thom polynomials and Schubert calculus

We describe the positivity of Thom polynomials of singularities of maps, Lagrangian Thom polynomials and Legendrian Thom polynomials. We show that these positivities come from Schubert calculus.

Characteristic Classes

  • A. Ranicki
  • Mathematics
    Lectures on the Geometry of Manifolds
  • 2020
The goal of this lecture notes is to introduce to Characteristic Classes. This is an important tool of the contemporary mathematics, indispensable to work in geometry and topology, and also useful in

References

SHOWING 1-10 OF 44 REFERENCES

Thom polynomials of invariant cones, Schur functions, and positivity

We generalize the notion of Thom polynomials from singularities of maps between two complex manifolds to invariant cones in representations, and collections of vector bundles. We prove that the

Thom polynomials and Schur functions: the singularities $III_{2,3}(-)$

We give a closed formula for the Thom polynomials of the singularities III2,3(−) in terms of Schur functions. Our computations combine the characterization of the Thom polynomials via the “method of

On second order Thom–Boardman singularities

We derive closed formulas for the Thom polynomials of two families of second order Thom-Boardman singularitiesi,j . The formulas are given as linear combi- nations of Schur polynomials, and all

Thom polynomials and Schur functions I

We give the Thom polynomials for the singularities I_2,2 and A_3 associated with maps (C^n,0) -> (C^{n+k},0) with parameter k>=0. We give the Schur function expansions of these Thom polynomials.

Positivity of Thom polynomials II: the Lagrange singularities

We show that Thom polynomials of Lagrangian singularities have nonnegative coefficients in the basis consisting of Q-functions. The main tool in the proof is nonnegativity of cone classes for

Thom polynomials and Schur funcions: the singularities $A_3(-)$

Combining the "method of restriction equations" of Rim\'anyi et al. with the techniques of symmetric functions, we establish the Schur function expansions of the Thom polynomials for the Morin

Positivity of Legendrian Thom polynomials

We study Legendrian singularities arising in complex contact geometry. We define a one-parameter family of bases in the ring of Legendrian characteristic classes such that any Legendrian Thom

Thom series of contact singularities

Thom polynomials measure how global topology forces singularities. The power of Thom polynomials predestine them to be a useful tool not only in differential topology, but also in algebraic geometry

Algebro — Geometric applications of schur s- and q-polynomials

Introduction 1. Three definitions of Schur Sand Q-polynomials. 2. Algebraic properties and characterizations of Sand Q-polynomials. 3. Symmetrizing operators and formulas for Gysin push forwards. 4.

Thom polynomials, symmetries and incidences of singularities

Abstract.As an application of the generalized Pontryagin-Thom construction [RSz] here we introduce a new method to compute cohomological obstructions of removing singularities — i.e. Thom polynomials