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Matrix orbit closures

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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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  1. Ardila, F., Klivans, C.: The Bergman complex of a matroid and phylogenetic trees. J. Comb. Theory Ser. B 96(1), 38–49 (2006)

  2. Ardila, F., Postnikov, A.: Combinatorics and geometry of power ideals. Trans. Am. Math. Soc. 362(8), 4357–4384 (2010)

  3. Berget, A.: Symmetries of tensors. Ph.D. Thesis, University of Minnesota (2009)

  4. Berget, A.: Products of linear forms and Tutte polynomials. Eur. J. Comb. 31(7), 1924–1935 (2010a)

  5. Berget, A.: Tableaux in the Whitney module of a matroid. Sém. Lothar. Comb. 63, Art. no. B63f (2010b)

  6. Berget, A.: Equality of symmetrized tensors and the flag variety. Linear Algebra Appl. 438(2), 658–656 (2013)

  7. Berget, A., Fink, A.: Equivariant Chow classes of matrix orbit closures. Transform. Groups 22(3), 631–643 (2017)

  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)

  9. Demazure, M.: Désingularization des variétés de Schubert généralisées. Ann. Sc. ENS sér. 4(7), 53–88 (1974)

  10. Dias da Silva, J.A.: On the \(\mu \)-colorings of a matroid. Linear Multilinear Algebra 27(1), 25–32 (1990)

  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)

  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)

  13. Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, Berlin (1995)

  14. Fehér, L., Némethi, A., Rimányi, R.: Equivariant classes of matrix matroid varieties. Comment. Math. Helv. 87, 861–889 (2012)

  15. Fulton, W.: Flags, Schubert polynomials, degeneracy loci, and determinantal formulas. Duke Math. J. 65(3), 381–420 (1992)

  16. Fulton, W., Harris, J.: Representation Theory. A First Course. Springer, Berlin (1991)

  17. Gamas, C.: Conditions for a symmetrized decomposable tensor to be zero. Linear Algebra Appl. 108, 83–119 (1988)

  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)

  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)

  21. Klyachko, A.: Orbits of a maximal torus on a flag space. Funktsional. Anal. i Prilozhen. 19(1), 77–78 (1985)

  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)

  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)

  26. Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Springer, Berlin (2005)

  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)

  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)

  30. White, N. (ed.): Theory of Matroids. Encyclopedia of Mathematics and Its Applications, vol. 26. Cambridge University Press, Cambridge (1986)

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

Mathematics Subject Classification

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