Matrix orbit closures

Original Paper


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.


Orbit closure Matroid Determinantal variety Equivariant K-class 

Mathematics Subject Classification

14M15 51M35 14M12 52B40 55N91 



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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Copyright information

© The Managing Editors 2018

Authors and Affiliations

  1. 1.Department of MathematicsWestern Washington UniversityBellinghamUSA
  2. 2.School of Mathematical SciencesQueen Mary University of LondonLondonUK

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