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EQUIVARIANT CHOW CLASSES OF MATRIX ORBIT CLOSURES

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Abstract

Let G be the product GL r (C) × (C ×)n. We show that the G-equivariant Chow class of a G orbit closure in the space of r-by-n matrices is determined by a matroid. To do this, we split the natural surjective map from the G equvariant Chow ring of the space of matrices to the torus equivariant Chow ring of the Grassmannian. The splitting takes the class of a Schubert variety to the corresponding factorial Schur polynomial, and also has the property that the class of a subvariety of the Grassmannian is mapped to the class of the closure of those matrices whose row span is in the variety.

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References

  1. D. Anderson, Introduction to equivariant cohomology in algebraic geometry, in: Contributions to Algebraic Geometry, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2012, pp. 71–92.

  2. M. Brion, Equivariant Chow groups for torus actions. Transform. Groups 2 (1997), no. 3, 225–267.

  3. J. Edmonds, Submodular functions, matroids, and certain polyhedra, in: Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969), Gordon and Breach, New York, 1970, pp. 69–87.

  4. D. Edidin, W. Graham, Equivariant intersection theory, Invent. Math. 131 (1998), no. 3, 595–634.

  5. D. Edidin, W. Graham, Localization in equivariant intersection theory and the Bott residue formula, Amer. J. Math. 120 (1998), no. 3, 619–636.

  6. L. M. Fehér, A. Némethi, R. Rimányi, Equivariant classes of matrix matroid varieties, Comment. Math. Helv. 87 (2012), no. 4, 861–889.

  7. L. M. Fehér, R. Rimányi, On the structure of Thom polynomials of singularities, Bull. Lond. Math. Soc. 39 (2007), no. 4, 541–549.

  8. A. Fink, D. E. Speyer, K-classes for matroids and equivariant localization, Duke Math. J. 161 (2012), no. 14, 2699–2723.

  9. W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1992), no. 3, 381–420.

  10. W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, Vol. 131, Princeton University Press, Princeton, NJ, 1993.

  11. I. M. Gel0fand, R. M. Goresky, R. D. MacPherson, V. V. Serganova, Combinatorial geometries, convex polyhedra, and Schubert cells, Adv. in Math. 63 (1987), no. 3, 301–316.

  12. M. Goresky, R. Kottwitz, R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), no. 1, 25–83.

  13. D. Halpern-Leistner, The derived category of a GIT quotient, J. Amer. Math. Soc. 28 (2015), no. 3, 871–912.

  14. F. C. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematical Notes, Vol. 31, Princeton University Press, Princeton, NJ, 1984.

  15. A. A. Клячкo, Opбиты мaкcимaльнoгo тopa нa пpocтpaнcтвe флaгoв, Функц. aнaлиз и eгo пpил. 19 (1985), vyp. 1, 77–78. Engl. transl.: A. A. Klyachko, Orbits of a maximal torus on a flag space, Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 65–66.

  16. A. Knutson, E. Miller, Gröbner geometry of Schubert polynomials, Ann. of Math. (2) 161 (2005), no. 3, 1245–1318.

  17. A. Knutson, E. Miller, M. Shimozono, Four positive formulae for type A quiver polynomials, Invent. Math. 166 (2006), no. 2, 229–325.

  18. A. Knutson, E. Miller, A. Yong, Gröbner geometry of vertex decompositions and of flagged tableaux, J. Reine Angew. Math. 630 (2009), 1–31.

  19. A. Knutson, Puzzles, positroid varieties, and equivariant K-theory of grassmannians, arXiv:1008.4302 (2010).

  20. I. G. Macdonald. Symmetric Functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky, Oxford Science Publications.

  21. E. Miller, B. Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, Vol. 227, Springer-Verlag, New York, 2005.

  22. D. E. Speyer, A matroid invariant via the K-theory of the Grassmannian, Adv. Math. 221 (2009), no. 3, 882–913.

  23. N. L. White, The basis monomial ring of a matroid, Adv. Math. 24 (1977), no. 3, 292–297.

  24. B. J. Wyser, A. Yong, Polynomials for GLp × GLq orbit closures in the flag variety, Selecta Math. (N.S.) 20 (2014), no. 4, 1083–1110.

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Authors and Affiliations

  1. Department of Mathematics, Western Washington University, Bellingham, Washington, USA

    ANDREW BERGET

  2. School of Mathematical Sciences, Queen Mary University of London, London, United Kingdom

    ALEX FINK

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  1. ANDREW BERGET
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  2. ALEX FINK
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Correspondence to ANDREW BERGET.

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BERGET, A., FINK, A. EQUIVARIANT CHOW CLASSES OF MATRIX ORBIT CLOSURES. Transformation Groups 22, 631–643 (2017). https://doi.org/10.1007/s00031-016-9406-5

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  • DOI: https://doi.org/10.1007/s00031-016-9406-5

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