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Matrix orbit closures

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Abstract

Let G be the group \(\mathrm {GL}_r(\mathbf {C}) \times (\mathbf {C}^\times )^n.\) We conjecture that the finely-graded Hilbert series of a G orbit closure in the space of r-by-n matrices is wholly determined by the associated matroid. In support of this, we prove that the coefficients of this Hilbert series corresponding to certain hook-shaped Schur functions in the \(\mathrm {GL}_r(\mathbf {C})\) variables are determined by the matroid, and that the orbit closure has a set-theoretic system of ideal generators whose combinatorics are also so determined. We also discuss relations between these Hilbert series for related matrices, including their stabilizing behaviour as r increases.

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Acknowledgements

We thank Dave Anderson, Calin Chindris, Ezra Miller, Leonardo Mihalcea, Richard Rimányi, Seth Sullivant and Alex Woo for useful discussions, as well as the anonymous referees who pointed out errors in earlier claimed proofs of Conjecture 5.1. The use of Macaulay2 (Grayson and Stillman 1991) was invaluable in the early stages of the work presented here.

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Correspondence to Andrew Berget.

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Berget, A., Fink, A. Matrix orbit closures. Beitr Algebra Geom 59, 397–430 (2018). https://doi.org/10.1007/s13366-018-0402-x

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Keywords

  • Orbit closure
  • Matroid
  • Determinantal variety
  • Equivariant K-class

Mathematics Subject Classification

  • 14M15
  • 51M35
  • 14M12
  • 52B40
  • 55N91