Multiplicity on a Richardson variety in a cominuscule 𝐺/𝑃

Balan, Michaël

doi:10.1090/S0002-9947-2013-05630-9

Multiplicity on a Richardson variety in a cominuscule $G/P$
HTML articles powered by AMS MathViewer

by Michaël Balan PDF

Trans. Amer. Math. Soc. 365 (2013), 3971-3986 Request permission

Abstract:

We show that in a cominuscule partial flag variety $G/P$, the multiplicity of an arbitrary point on a Richardson variety $X_{w}^{v}=X_w \cap X^v \subset G/P$ is the product of its multiplicities on the Schubert varieties $X_w$ and $X^v$.

References

Sara Billey and Izzet Coskun, Singularities of generalized Richardson varieties, Comm. Algebra 40 (2012), no. 4, 1466–1495. MR 2912998, DOI 10.1080/00927872.2011.551903
Sara C. Billey and Gregory S. Warrington, Maximal singular loci of Schubert varieties in $\textrm {SL}(n)/B$, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3915–3945. MR 1990570, DOI 10.1090/S0002-9947-03-03019-8
Michel Brion, Positivity in the Grothendieck group of complex flag varieties, J. Algebra 258 (2002), no. 1, 137–159. Special issue in celebration of Claudio Procesi’s 60th birthday. MR 1958901, DOI 10.1016/S0021-8693(02)00505-7
M. Brion and V. Lakshmibai, A geometric approach to standard monomial theory, Represent. Theory 7 (2003), 651–680. MR 2017071, DOI 10.1090/S1088-4165-03-00211-5
Aurélie Cortez, Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire, Adv. Math. 178 (2003), no. 2, 396–445 (French, with English summary). MR 1994224, DOI 10.1016/S0001-8708(02)00081-6
Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math. 79 (1985), no. 3, 499–511. MR 782232, DOI 10.1007/BF01388520
Sudhir R. Ghorpade and K. N. Raghavan, Hilbert functions of points on Schubert varieties in the symplectic Grassmannian, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5401–5423. MR 2238920, DOI 10.1090/S0002-9947-06-04037-2
K. R. Goodearl and M. Yakimov, Poisson structures on affine spaces and flag varieties. II, Trans. Amer. Math. Soc. 361 (2009), no. 11, 5753–5780. MR 2529913, DOI 10.1090/S0002-9947-09-04654-6
W. V. D. Hodge and D. Pedoe, Methods of algebraic geometry. Vol. II, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994. Book III: General theory of algebraic varieties in projective space; Book IV: Quadrics and Grassmann varieties; Reprint of the 1952 original. MR 1288306, DOI 10.1017/CBO9780511623899
Takeshi Ikeda, Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian, Adv. Math. 215 (2007), no. 1, 1–23. MR 2354984, DOI 10.1016/j.aim.2007.04.008
Takeshi Ikeda and Hiroshi Naruse, Excited Young diagrams and equivariant Schubert calculus, Trans. Amer. Math. Soc. 361 (2009), no. 10, 5193–5221. MR 2515809, DOI 10.1090/S0002-9947-09-04879-X
Christian Kassel, Alain Lascoux, and Christophe Reutenauer, The singular locus of a Schubert variety, J. Algebra 269 (2003), no. 1, 74–108 (English, with English and French summaries). MR 2015302, DOI 10.1016/S0021-8693(03)00014-0
A. Knutson, T. Lam, D. Speyer, Positroid varieties: Juggling and Geometry, Preprint http://arxiv.org/abs/1111.3660
A. Knutson, T. Lam, D. Speyer, Projections of Richardson Varieties, Preprint http://arxiv.org/abs/1008.3939
Allen Knutson and Terence Tao, Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J. 119 (2003), no. 2, 221–260. MR 1997946, DOI 10.1215/S0012-7094-03-11922-5
Vijay Kodiyalam and K. N. Raghavan, Hilbert functions of points on Schubert varieties in Grassmannians, J. Algebra 270 (2003), no. 1, 28–54. MR 2015929, DOI 10.1016/S0021-8693(03)00518-0
C. Krattenthaler, On multiplicities of points on Schubert varieties in Grassmannians, Sém. Lothar. Combin. 45 (2000/01), Art. B45c, 11. MR 1817336
C. Krattenthaler, On multiplicities of points on Schubert varieties in Graßmannians. II, J. Algebraic Combin. 22 (2005), no. 3, 273–288. MR 2181366, DOI 10.1007/s10801-005-4527-2
Victor Kreiman, Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence, J. Algebraic Combin. 27 (2008), no. 3, 351–382. MR 2393260, DOI 10.1007/s10801-007-0093-0
V. Kreiman, Schubert Classes in the Equivariant K-Theory and Equivariant Cohomology of the Grassmannian, Preprint http://arxiv.org/abs/math/0512204
V. Kreiman, Schubert Classes in the Equivariant K-Theory and Equivariant Cohomology of the Lagrangian Grassmannian, Preprint http://arxiv.org/abs/math/0602245
Victor Kreiman and V. Lakshmibai, Richardson varieties in the Grassmannian, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 573–597. MR 2058620
Victor Kreiman and V. Lakshmibai, Multiplicities of singular points in Schubert varieties of Grassmannians, Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000) Springer, Berlin, 2004, pp. 553–563. MR 2037109
V. Lakshmibai and P. Littelmann, Richardson varieties and equivariant $K$-theory, J. Algebra 260 (2003), no. 1, 230–260. Special issue celebrating the 80th birthday of Robert Steinberg. MR 1973584, DOI 10.1016/S0021-8693(02)00634-8
V. Lakshmibai, K. N. Raghavan, and P. Sankaran, Equivariant Giambelli and determinantal restriction formulas for the Grassmannian, Pure Appl. Math. Q. 2 (2006), no. 3, Special Issue: In honor of Robert D. MacPherson., 699–717. MR 2252114, DOI 10.4310/PAMQ.2006.v2.n3.a5
V. Lakshmibai and J. Weyman, Multiplicities of points on a Schubert variety in a minuscule $G/P$, Adv. Math. 84 (1990), no. 2, 179–208. MR 1080976, DOI 10.1016/0001-8708(90)90044-N
A. Lascoux, Foncteurs de Schur et grassmannienne, thèse, Université de Paris VII, 1977
Li Li and Alexander Yong, Some degenerations of Kazhdan-Lusztig ideals and multiplicities of Schubert varieties, Adv. Math. 229 (2012), no. 1, 633–667. MR 2854186, DOI 10.1016/j.aim.2011.09.010
L. Manivel, Le lieu singulier des variétés de Schubert, Internat. Math. Res. Notices 16 (2001), 849–871 (French). MR 1853139, DOI 10.1155/S1073792801000423
David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732
K. N. Raghavan and Shyamashree Upadhyay, Initial ideals of tangent cones to Schubert varieties in orthogonal Grassmannians, J. Combin. Theory Ser. A 116 (2009), no. 3, 663–683. MR 2500164, DOI 10.1016/j.jcta.2008.10.005
K. N. Raghavan and Shyamashree Upadhyay, Hilbert functions of points on Schubert varieties in orthogonal Grassmannians, J. Algebraic Combin. 31 (2010), no. 3, 355–409. MR 2610290, DOI 10.1007/s10801-009-0188-x
R. W. Richardson, Intersections of double cosets in algebraic groups, Indag. Math. (N.S.) 3 (1992), no. 1, 69–77. MR 1157520, DOI 10.1016/0019-3577(92)90028-J
Joachim Rosenthal and Andrei Zelevinsky, Multiplicities of points on Schubert varieties in Grassmannians, J. Algebraic Combin. 13 (2001), no. 2, 213–218. MR 1826954, DOI 10.1023/A:1011253800374
T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713, DOI 10.1007/978-0-8176-4840-4
Richard P. Stanley, Some combinatorial aspects of the Schubert calculus, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) Lecture Notes in Math., Vol. 579, Springer, Berlin, 1977, pp. 217–251. MR 0465880
Torgny Svanes, Coherent cohomology on Schubert subschemes of flag schemes and applications, Advances in Math. 14 (1974), 369–453. MR 419469, DOI 10.1016/0001-8708(74)90039-5
S. Upadhyay, Initial ideals of tangent cones to Richardson varieties in the Orthogonal Grassmannian via a Orthogonal-Bounded-RSK-Correspondence, Preprint http://arxiv.org/abs/0909.1424
A. Woo, Multiplicities of the Most Singular Point on Schubert varieties on $Gl(n)/B$ for $n=5,6$, Preprint http://arxiv.org/abs/math/0407158
Alexander Woo and Alexander Yong, Governing singularities of Schubert varieties, J. Algebra 320 (2008), no. 2, 495–520. MR 2422304, DOI 10.1016/j.jalgebra.2007.12.016
Alexander Woo and Alexander Yong, A Gröbner basis for Kazhdan-Lusztig ideals, Amer. J. Math. 134 (2012), no. 4, 1089–1137. MR 2956258, DOI 10.1353/ajm.2012.0031