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Polynomials for \(\mathrm{GL}_p\times \mathrm{GL}_q\) orbit closures in the flag variety. (English) Zbl 1303.05212

Summary: The subgroup \(K=\mathrm{GL}_p \times \mathrm{GL}_q\) of \(\mathrm{GL}_{p+q}\) acts on the (complex) flag variety \(\mathrm{GL}_{p+q}/B\) with finitely many orbits. We introduce a family of polynomials specializing representatives for cohomology classes of the orbit closures in the Borel model. We define and study \(K\)-orbit determinantal ideals to support the geometric naturality of these representatives. Using a modification of these ideals, we describe an analogy between two local singularity measures: the \(H\)-polynomials and the Kazhdan-Lusztig-Vogan polynomials.

MSC:

05E18 Group actions on combinatorial structures
05E05 Symmetric functions and generalizations
14M15 Grassmannians, Schubert varieties, flag manifolds
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[1] Anderson, D., Griffeth, S., Miller, E.: Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces. J. Eur. Math. Soc. 13, 57-84 (2011) · Zbl 1213.19003 · doi:10.4171/JEMS/244
[2] Bergeron, N., Billey, S.: RC-graphs and Schubert poylnomials. Exp. Math. 2, 257-269 (1993) · Zbl 0803.05054 · doi:10.1080/10586458.1993.10504567
[3] Bernstein, I.N., Gelfand, I.M., Gelfand, S.I.: Schubert cells, and the cohomology of the spaces \[G/P\] G/P. Uspehi Mat. Nauk 28, 3(171)-26 (1973)
[4] Borel, A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogeènes de groupes de Lie compacts. Ann. Math. 57, 115-207 (1953) · Zbl 0052.40001 · doi:10.2307/1969728
[5] Brion, M.: On orbit closures of spherical subgroups in flag varieties. Comment. Math. Helv. 76(2), 263-299 (2001) · Zbl 1043.14012 · doi:10.1007/PL00000379
[6] Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, Berlin (1995) · Zbl 0819.13001
[7] Fomin, S., Gelfand, S., Postnikov, A.: Quantum Schubert polynomials. J. Am. Math. Soc. 10(3), 565-596 (1997) · Zbl 0912.14018 · doi:10.1090/S0894-0347-97-00237-3
[8] Fomin, S., Kirillov, A.: The Yang-Baxter equation, symmetric functions, and Schubert polynomials. Discrete Math. 153 123-143 (1996), Proceedings of Fifth Conference Formal Power Series and Algebraic Combinatorics (Florence, 1993) · Zbl 0852.05078
[9] Fomin, S., Kirillov, A.: Grothendieck polynomials and the Yang-Baxter equation. In: Proceedings of 6th International Conference on Formal Power Series and Algebraic Combinatorics, DIMACS, pp. 183-190 (1994) · Zbl 1284.14068
[10] Fulton, W.: With Applications to Representation Theory and Geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge (1997) · Zbl 0878.14034
[11] Fulton, W., Lascoux, A.: A Pieri formula in the Grothendieck ring of the flag bundle. Duke Math. J. 76(3), 711-729 (1994) · Zbl 0840.14007 · doi:10.1215/S0012-7094-94-07627-8
[12] Harada, M., Landweber, G.: Surjectivity for Hamiltonian \[G\] G-spaces in \[KK\]-theory. Trans. AMS 359, 6001-6025 (2007) · Zbl 1128.53057 · doi:10.1090/S0002-9947-07-04164-5
[13] Insko, E., Yong, A.: Patch ideals and Peterson varieties. Transf. Groups 17(4), 1011-1036 (2012) · Zbl 1267.14066 · doi:10.1007/s00031-012-9201-x
[14] Knutson, A.: Frobenius splitting and Möbius inversion, preprint, arXiv:1209.4146 (2009) · Zbl 1128.53057
[15] Knutson, A., Miller, E.: Gröbner geometry of Schubert polynomials. Ann. Math. 161, 1245-1318 (2005) · Zbl 1089.14007 · doi:10.4007/annals.2005.161.1245
[16] Knutson, A., Miller, E., Yong, A.: Gröbner geometry of vertex decompositions and of flagged tableaux. J. Reine Angew. Math. 630, 1-31 (2009) · Zbl 1169.14033 · doi:10.1515/CRELLE.2009.033
[17] Knutson, A., Woo, A., Yong, A.: Singularities of Richardson varieties, Math. Res. Lett. to appear arXiv:1209.4146 (2013) · Zbl 1298.14053
[18] Kostant, B., Kumar, \[S.: T\] T-equivariant \[KK\]-theory of generalized flag varieties. J. Differ. Geom. 32(2), 549-603 (1990) · Zbl 0731.55005
[19] Lascoux, A., Schützenberger, M.-P.: Polynômes de Schubert. C. R. Acad. Sci. Paris Sér. I Math. 295, 629-633 (1982) · Zbl 0542.14030
[20] Lenart, C., Sottile, F.: Skew Schubert polynomials. Proc. Am. Math. Soc. 131, 3319-3328 (2003) · Zbl 1033.05097 · doi:10.1090/S0002-9939-03-06919-3
[21] Li, L., Yong, A.: Kazhdan-Lusztig polynomials and drift configurations. Algeb. Number Theory J. 5(5), 595-626 (2011) · Zbl 1273.14100 · doi:10.2140/ant.2011.5.595
[22] Lusztig, G., Vogan, D.: Singularities of closures of \[KK\]-orbits on flag manifolds. Invent. Math. 71(2), 365-379 (1983) · Zbl 0544.14035 · doi:10.1007/BF01389103
[23] Manivel, L.: Symmetric Functions, Schubert Polynomials and Degeneracy Loci. American Mathematical Society, Providence, RI (2001) · Zbl 0998.14023
[24] Matsuki, T.: The orbits of affine symmetric spaces under the action of minimal parabolic subgroups. J. Math. Soc. Jpn. 31(2), 331-357 (1979) · Zbl 0396.53025 · doi:10.2969/jmsj/03120331
[25] Matsuki, T., Oshima, T.: Embeddings of discrete series into principal series. In: The Orbit Method in Representation Theory (Copenhagen, 1988), Volume 82 of Progr. Math., pp. 147-175. Birkhäuser, Boston, MA (1990) · Zbl 0746.22011
[26] McGovern, W.M.: Closures of \[KK\]-orbits in the flag variety for \[U(p, q)U\](p,q). J. Algeb. 322(8), 2709-2712 (2009) · Zbl 1185.14040 · doi:10.1016/j.jalgebra.2009.07.005
[27] McGovern, W. M.: Upper semicontinuity of KLV polynomials for certain blocks of Harish-Chandra modules. Preprint. arXiv:1311.0911 (2013) · Zbl 1341.22009
[28] McGovern, W.M., Trapa, P.: Pattern avoidance and smoothness of closures for orbits of a symmetric subgroup in the flag variety. J. Algeb. 322(8), 2713-2730 (2009) · Zbl 1187.22018 · doi:10.1016/j.jalgebra.2009.07.006
[29] Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227. Springer, New York (2004)
[30] Richardson, R.W.: Intersections of double cosets in algebraic groups. Indag. Math. (N.S.) 3(1), 69-77 (1992) · Zbl 0833.22001 · doi:10.1016/0019-3577(92)90028-J
[31] Richardson, R.W., Springer, T.A.: The Bruhat order on symmetric varieties. Geom. Dedic. 35, 389-436 (1990) · Zbl 0704.20039 · doi:10.1007/BF00147354
[32] Springer, T.A.: Some results on algebraic groups with involutions. Algebraic groups and related topics, Vol. 6 of Adv. Stud. Pure Math, pp. 525-543 · Zbl 1146.14026
[33] Vogan, D.: Irreducible characters of semisimple Lie groups. III. Proof of Kazhdan-Lusztig conjecture in the integral case. Invent. Math. 71(2), 381-417 (1983) · Zbl 0505.22016 · doi:10.1007/BF01389104
[34] Wyser, \[B.: KK\]-orbit closures on \[G/B\] G/B as universal degeneracy loci for flagged vector bundles splitting as direct sums. Preprint. arXiv:1301.1713 (2013) · Zbl 1284.14068
[35] Wyser, \[B.: KK\]-orbit closures on \[G/B\] G/B as universal degeneracy loci for flagged vector bundles with symmetric or skew-symmetric bilinear form. Transf. Groups 18(2), 557-594 (2013) · Zbl 1284.14068 · doi:10.1007/s00031-013-9221-1
[36] Wyser, B.: Schubert calculus of Richardson varieties stable under spherical Levi subgroups. J. Algeb. Comb. 38(4), 829-850 (2013) · Zbl 1284.14069 · doi:10.1007/s10801-013-0427-z
[37] Wyser, B., Yong, A.: Polynomials for symmetric orbit closures in the flag variety. Preprint. arXiv:1310.7271 (2013) · Zbl 1400.14130
[38] Yamamoto, A.: Orbits in the flag variety and images of the moment map for classical groups. I. Represent. Theory 1, 329-404 (1997). (electronic) · Zbl 0887.22017 · doi:10.1090/S1088-4165-97-00007-1
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