The symmetric functions catalog

#Thom polynomials

The Thom polynomial of a singularity class \(\eta\) is a universal polynomial in Chern classes whose value gives the cohomology class of the \(\eta\)-locus of a sufficiently generic map. The subject goes back to R. Thom’s work on singularities of differentiable maps [Tho56].

For maps \(f : M \to N\) of complex manifolds, the relevant Chern classes are usually the Chern classes of the virtual bundle \(f^\ast TN - TM.\) If we pass to Chern roots, this means that Thom polynomials are naturally symmetric polynomials in a difference of alphabets. This is why supersymmetric Schur functions, also called Schur functions in a difference of alphabets, are a natural basis for them.

P. Pragacz used Schur and supersymmetric Schur functions to compute and organize Thom polynomials for several singularity classes [Pra05, Pra07, Pra08]. One advantage of the Schur basis is that it can reveal structural restrictions and recurrences in the partitions that occur, which are much harder to see from monomials in Chern classes.

P. Pragacz and A. Weber proved positivity results for Schur function expansions of Thom polynomials [PW07]. In particular, for stable singularities satisfying their positivity hypotheses, the Schur expansion has non-negative coefficients. This places Thom polynomials in the same general circle of geometric Schur-positivity phenomena as degeneracy-locus formulas and Schubert calculus.

Bibliography

1. [Pra05]Piotr Pragacz. Thom polynomials and Schur functions I. arXiv:math/0509234, 2005.
   .bib

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   @article{Pragacz2005x,
     author = {Piotr Pragacz},
     title = {Thom polynomials and {S}chur functions {I}},
     year = {2005},
     eprint = {math/0509234},
     url = {https://arxiv.org/abs/math/0509234},
     journal = {arXiv e-prints}
   }
   

2. [Pra07]Piotr Pragacz. Thom polynomials and Schur functions: The singularities I 2, 2 (-). Annales de l’Institut Fourier, 57(5):1487–1508, 2007.
   .bib

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   @article{Pragacz2007,
     author = {Pragacz, Piotr},
     title = {Thom polynomials and {S}chur functions: the singularities {I} 2, 2
       (-)},
     year = {2007},
     journal = {Annales de l'Institut Fourier},
     volume = {57},
     number = {5},
     pages = {1487--1508},
     publisher = {MathDoc/Centre Mersenne},
     doi = {10.5802/aif.2302},
     url = {http://dx.doi.org/10.5802/aif.2302},
     issn = {1777-5310}
   }
   

3. [Pra08]Piotr Pragacz. Thom polynomials and Schur functions: Towards the singularities ${A}_i(-)$. arXiv:0810.2441, 2008.
   .bib

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   @article{Pragacz2008x,
     author = {Piotr Pragacz},
     title = {Thom polynomials and {S}chur functions: towards the singularities
       ${A}_i(-)$},
     year = {2008},
     eprint = {0810.2441},
     url = {https://arxiv.org/abs/0810.2441},
     journal = {arXiv e-prints}
   }
   

4. [PW07]Piotr Pragacz and Andrzej Weber. Positivity of Schur function expansions of Thom polynomials. Fundamenta Mathematicae, 195(1):85–95, 2007.
   .bib

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   @article{PragaczWeber2007,
     author = {Pragacz, Piotr and Weber, Andrzej},
     title = {Positivity of {S}chur function expansions of {T}hom polynomials},
     year = {2007},
     journal = {Fundamenta Mathematicae},
     volume = {195},
     number = {1},
     pages = {85--95},
     publisher = {Institute of Mathematics, Polish Academy of Sciences},
     doi = {10.4064/fm195-1-3},
     url = {https://doi.org/10.4064/fm195-1-3}
   }
   

5. [Tho56]René Thom. Les singularités des applications différentiables. Annales de l’Institut Fourier, 6:43–87, 1956.
   .bib

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   @article{Thom1956,
     author = {Thom, Ren{\'e}},
     title = {Les singularit{\'e}s des applications diff{\'e}rentiables},
     year = {1956},
     journal = {Annales de l'Institut Fourier},
     volume = {6},
     pages = {43--87},
     url = {https://www.numdam.org/item/AIF_1956__6__43_0/}
   }