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Matrix orbit closures

  • Andrew Berget
  • Alex Fink
Original Paper

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.

Keywords

Orbit closure Matroid Determinantal variety Equivariant K-class 

Mathematics Subject Classification

14M15 51M35 14M12 52B40 55N91 

Notes

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.

References

  1. Ardila, F., Klivans, C.: The Bergman complex of a matroid and phylogenetic trees. J. Comb. Theory Ser. B 96(1), 38–49 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Ardila, F., Postnikov, A.: Combinatorics and geometry of power ideals. Trans. Am. Math. Soc. 362(8), 4357–4384 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Berget, A.: Symmetries of tensors. Ph.D. Thesis, University of Minnesota (2009)Google Scholar
  4. Berget, A.: Products of linear forms and Tutte polynomials. Eur. J. Comb. 31(7), 1924–1935 (2010a)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Berget, A.: Tableaux in the Whitney module of a matroid. Sém. Lothar. Comb. 63, Art. no. B63f (2010b)Google Scholar
  6. Berget, A.: Equality of symmetrized tensors and the flag variety. Linear Algebra Appl. 438(2), 658–656 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Berget, A., Fink, A.: Equivariant Chow classes of matrix orbit closures. Transform. Groups 22(3), 631–643 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Bernstein, I.N., Gelfand, I.M., Gelfand, S.I.: Schubert cells and cohomology of the spaces \(G/P\). Russ. Math. Surv. 28, 1–26 (1973)MathSciNetCrossRefGoogle Scholar
  9. Demazure, M.: Désingularization des variétés de Schubert généralisées. Ann. Sc. ENS sér. 4(7), 53–88 (1974)zbMATHGoogle Scholar
  10. Dias da Silva, J.A.: On the \(\mu \)-colorings of a matroid. Linear Multilinear Algebra 27(1), 25–32 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Eagon, J., Hochster, M.: Cohen–Macaulay rings, invariant theory, and the generic perfection of determinantal loci. Am. J. Math. 93(4), 1020–1058 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schonheim, J. (eds.) Combinatorial Structures and Their Applications, pp. 69–87. Gordon and Breach, New York (1970)Google Scholar
  13. Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, Berlin (1995)zbMATHGoogle Scholar
  14. Fehér, L., Némethi, A., Rimányi, R.: Equivariant classes of matrix matroid varieties. Comment. Math. Helv. 87, 861–889 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Fulton, W.: Flags, Schubert polynomials, degeneracy loci, and determinantal formulas. Duke Math. J. 65(3), 381–420 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Fulton, W., Harris, J.: Representation Theory. A First Course. Springer, Berlin (1991)zbMATHGoogle Scholar
  17. Gamas, C.: Conditions for a symmetrized decomposable tensor to be zero. Linear Algebra Appl. 108, 83–119 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Gel’fand, I.M., Goresky, R.M., MacPherson, R.D., Serganova, V.V.: Combinatorial geometries, convex polyhedra, and Schubert cells. Adv. Math. 63(3), 301–316 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/ (1991). Accessed 24 May 2018
  20. Kapranov, M.: Chow quotients of Grassmannians. I. In: Gelfand, I.M. (ed.) Seminar. Advances in Soviet Mathematics, vol. 16, pp. 29–110. American Mathematical Society, Providence (1993)Google Scholar
  21. Klyachko, A.: Orbits of a maximal torus on a flag space. Funktsional. Anal. i Prilozhen. 19(1), 77–78 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Knutson, A.: Puzzles, positroid varieties, and equivariant k-theory of Grassmannians. Preprint. arXiv:1008.4302 (2010a)
  23. Knutson, A.: Introduction to geometric representation theory. Course notes. http://www.math.cornell.edu/~allenk/courses/10fall/notes.pdf (2010b). Accessed 24 May 2018
  24. Knutson, A., Miller, E.: Gröbner geometry of Schubert polynomials. Ann. Math. 2(161), 1245–1318 (2005)CrossRefzbMATHGoogle Scholar
  25. Lee, S.H., Vakil, R.: Mnëv–Sturmfels universality for schemes. In: A Celebration of Algebraic Geometry. Clay Mathematics Proceedings, vol. 18, pp. 457–468. American Mathematical Society, Providence, RI (2013)Google Scholar
  26. Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Springer, Berlin (2005)zbMATHGoogle Scholar
  27. Sokal, A.D.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In: Surveys in Combinatorics 2005. London Mathematical Society Lecture Note Series, vol. 327, pp. 173–226. Cambridge University Press, Cambridge (2005).  https://doi.org/10.1017/CBO9780511734885.009
  28. Speyer, D.: A matroid invariant via the \(K\)-theory of the Grassmannian. Adv. Math. 221(3), 882–913 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Sturmfels, B.: On the matroid stratification of Grassmann varieties, specialization of coordinates, and a problem of N. White. Adv. Math. 75, 202–211 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  30. White, N. (ed.): Theory of Matroids. Encyclopedia of Mathematics and Its Applications, vol. 26. Cambridge University Press, Cambridge (1986)Google Scholar

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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