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EQUIVARIANT CHOW CLASSES OF MATRIX ORBIT CLOSURES

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. [And12]

    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. [Bri97]

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

  3. [Edm70]

    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. [EG98a]

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

  5. [EG98b]

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

  6. [FNR12]

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

  7. [FR07]

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

  8. [FS12]

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

  9. [Ful92]

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

  10. [Ful93]

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

  11. [GGMS87]

    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. [GKM98]

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

  13. [HL15]

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

  14. [Kir84]

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

  15. [Kly85]

    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. [KM05]

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

  17. [KMS06]

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

  18. [KMY09]

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

  19. [Knu10]

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

  20. [Mac95]

    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. [MS05]

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

  22. [Spe09]

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

  23. [Whi77]

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

  24. [WY14]

    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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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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