JournalscmhVol. 87, No. 4pp. 861–889

Equivariant classes of matrix matroid varieties

  • László M. Fehér

    Eötvös Loránd University, Budapest, Hungary
  • András Némethi

    Hungarian Academy of Sciences, Budapest, Hungary
  • Richárd Rimányi

    University of North Carolina at Chapel Hill, USA
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Abstract

To each subset II of {1,,k}\{1,\dots,k\} associate an integer r(I)r(I). Denote by XX the collection of those n×kn\times k matrices for which the rank of a union of columns corresponding to a subset II is r(I)r(I), for all II. We study the equivariant cohomology class represented by the Zariski closure Y=XY=\overline{X}. This class is an invariant of the underlying matroid structure. Its calculation incorporates challenges similar to the calculation of the ideal of YY, namely, the determination of the geometric theorems for the matroid. This class also gives information on the degenerations and hierarchy of matroids. New developments in the theory of Thom polynomials of contact singularities (namely, a recently found stability property) help us to calculate these classes and present their basic properties. We also show that the coefficients of this class are solutions to problems in enumerative geometry, which are natural generalization of the linear Gromov–Witten invariants of projective spaces.

Cite this article

László M. Fehér, András Némethi, Richárd Rimányi, Equivariant classes of matrix matroid varieties. Comment. Math. Helv. 87 (2012), no. 4, pp. 861–889

DOI 10.4171/CMH/271