×

Matrix orbit closures. (English) Zbl 1471.14103

Consider the vector space \(V\) of \(r\times n\) matrices, acted upon by the group \(G=\mathrm{GL}_r(C) \times (C^{\times})^n\). Most points of \(V\) correspond to ordered \(n\)-tuples of points in the projective space \(P^{r-1}\). Such a tuple determines a matroid, a so-called representable matroid, via the dimensions of the spans of various subsets of the tuple. That matroid is invariant under the \(G\) action.
The paper under review is one of the many works that explore the relation between the matroid and certain aspects of the geometry of the orbit. In particular, consider the class of the orbit closure in the \(G\)-equivariant \(K\)-theory of the vector space \(V\). The authors conjecture (Conjecture 5.1) that the matroid determines this class.
Besides this conjecture the paper has two main results. In the first one the authors prove that certain coefficients of the \(K\)-class of the orbit closure are indeed determined by the matroid. In this result, the combinatorics of hook-shaped partitions/Schur functions play a role.
The other main result is the description of the ideal of the orbit closure using only the matroid – up to radical. If the ideal was reduced this would prove the conjecture, and it indeed does prove it in special cases when the ideal is reduced. For example, rank 2 or corank 2 uniform matroids satisfy this property.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
51M35 Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations)
14M12 Determinantal varieties
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
55N91 Equivariant homology and cohomology in algebraic topology

Software:

Macaulay2
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ardila, F; Klivans, C, The Bergman complex of a matroid and phylogenetic trees, J. Comb. Theory Ser. B, 96, 38-49, (2006) · Zbl 1082.05021 · doi:10.1016/j.jctb.2005.06.004
[2] Ardila, F; Postnikov, A, Combinatorics and geometry of power ideals, Trans. Am. Math. Soc., 362, 4357-4384, (2010) · Zbl 1226.05019 · doi:10.1090/S0002-9947-10-05018-X
[3] Berget, A.: Symmetries of tensors. Ph.D. Thesis, University of Minnesota (2009) · Zbl 1183.15022
[4] Berget, A, Products of linear forms and Tutte polynomials, Eur. J. Comb., 31, 1924-1935, (2010) · Zbl 1219.05032 · doi:10.1016/j.ejc.2010.01.006
[5] Berget, A.: Tableaux in the Whitney module of a matroid. Sém. Lothar. Comb. 63, Art. no. B63f (2010b) · Zbl 1205.05037
[6] Berget, A, Equality of symmetrized tensors and the flag variety, Linear Algebra Appl., 438, 658-656, (2013) · Zbl 1258.15009 · doi:10.1016/j.laa.2011.03.036
[7] Berget, A; Fink, A, Equivariant Chow classes of matrix orbit closures, Transform. Groups, 22, 631-643, (2017) · Zbl 1401.14030 · doi:10.1007/s00031-016-9406-5
[8] Bernstein, IN; Gelfand, IM; Gelfand, SI, Schubert cells and cohomology of the spaces \(G/P\), Russ. Math. Surv., 28, 1-26, (1973) · Zbl 0289.57024 · doi:10.1070/RM1973v028n03ABEH001557
[9] Demazure, M, Désingularization des variétés de Schubert généralisées, Ann. Sc. ENS sér., 4, 53-88, (1974) · Zbl 0312.14009
[10] Dias da Silva, JA, On the \(μ \)-colorings of a matroid, Linear Multilinear Algebra, 27, 25-32, (1990) · Zbl 0739.05020 · doi:10.1080/03081089008817990
[11] Eagon, J; Hochster, M, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Am. J. Math., 93, 1020-1058, (1971) · Zbl 0244.13012 · doi:10.2307/2373744
[12] Edmonds, J; Guy, R (ed.); Hanani, H (ed.); Sauer, N (ed.); Schonheim, J (ed.), Submodular functions, matroids, and certain polyhedra, 69-87, (1970), New York · Zbl 0268.05019
[13] Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, Berlin (1995) · Zbl 0819.13001
[14] Fehér, L; Némethi, A; Rimányi, R, Equivariant classes of matrix matroid varieties, Comment. Math. Helv., 87, 861-889, (2012) · Zbl 1267.14069 · doi:10.4171/CMH/271
[15] Fulton, W, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J., 65, 381-420, (1992) · Zbl 0788.14044 · doi:10.1215/S0012-7094-92-06516-1
[16] Fulton, W., Harris, J.: Representation Theory. A First Course. Springer, Berlin (1991) · Zbl 0744.22001
[17] Gamas, C, Conditions for a symmetrized decomposable tensor to be zero, Linear Algebra Appl., 108, 83-119, (1988) · Zbl 0652.15023 · doi:10.1016/0024-3795(88)90180-2
[18] Gel’fand, IM; Goresky, RM; MacPherson, RD; Serganova, VV, Combinatorial geometries, convex polyhedra, and Schubert cells, Adv. Math., 63, 301-316, (1987) · Zbl 0622.57014 · doi:10.1016/0001-8708(87)90059-4
[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; Gelfand, IM (ed.), Chow quotients of Grassmannians. I, No. 16, 29-110, (1993), Providence · Zbl 0811.14043
[21] Klyachko, A, Orbits of a maximal torus on a flag space, Funktsional. Anal. i Prilozhen., 19, 77-78, (1985) · Zbl 0581.14038 · doi:10.1007/BF01086033
[22] Knutson, A.: Puzzles, positroid varieties, and equivariant k-theory of Grassmannians. Preprint. arXiv:1008.4302 (2010a) · Zbl 0739.05020
[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 · Zbl 0312.14009
[24] Knutson, A; Miller, E, Gröbner geometry of Schubert polynomials, Ann. Math., 2, 1245-1318, (2005) · Zbl 1089.14007 · doi:10.4007/annals.2005.161.1245
[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) · Zbl 1317.14080
[26] Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Springer, Berlin (2005) · Zbl 1090.13001
[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 · Zbl 1110.05020
[28] Speyer, D, A matroid invariant via the \(K\)-theory of the Grassmannian, Adv. Math., 221, 882-913, (2009) · Zbl 1222.14131 · doi:10.1016/j.aim.2009.01.010
[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) · Zbl 0726.05028 · doi:10.1016/0001-8708(89)90037-6
[30] White, N. (ed.): Theory of Matroids. Encyclopedia of Mathematics and Its Applications, vol. 26. Cambridge University Press, Cambridge (1986) · Zbl 0579.00001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.