Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Excited Young diagrams and equivariant Schubert calculus


Authors: Takeshi Ikeda and Hiroshi Naruse
Journal: Trans. Amer. Math. Soc. 361 (2009), 5193-5221
MSC (2000): Primary 05E15; Secondary 14N15, 14M15, 05E05
DOI: https://doi.org/10.1090/S0002-9947-09-04879-X
Published electronically: April 30, 2009
MathSciNet review: 2515809
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We describe the torus-equivariant cohomology ring of isotropic Grassmannians by using a localization map to the torus fixed points. We present two types of formulas for equivariant Schubert classes of these homogeneous spaces. The first formula involves combinatorial objects which we call ``excited Young diagrams'', and the second one is written in terms of factorial Schur $ Q$- or $ P$-functions. As an application, we give a Giambelli-type formula for the equivariant Schubert classes. We also give combinatorial and Pfaffian formulas for the multiplicity of a singular point in a Schubert variety.


References [Enhancements On Off] (What's this?)

  • [1] H. H. Andersen, J. C. Jantzen, and W. Soergel, Representations of quantum groups at a $ p$th root of unity and of semisimple groups in characteristic $ p$: independence of $ p$, Asterisque No. 220 (1994), 321 pp. MR 1272539 (95j:20036)
  • [2] A. Arabia, Cycles de Schubert et cohomologie équivariante de $ K/T$, Invent. Math. 85 (1986), 39-52. MR 842047 (87g:32036)
  • [3] N. Bergeron and S. Billey, RC-graphs and Schubert polynomials, Exparemental Math. 2 (1993), no. 4, 257-269. MR 1281474 (95g:05107)
  • [4] S. Billey, Kostant polynomials and the cohomology ring for $ G/B$, Duke Math. J. 96 (1999), no. 1, 205-224. MR 1663931 (2000a:14060)
  • [5] S. Billey and V. Lakshmibai, Singular loci of Schubert varieties, Progress in Mathematics, 182. Birkhäuser Boston, Inc., Boston, MA, 2000. xii+251 pp. ISBN: 0-8176-4092-4 MR 1782635 (2001j:14065)
  • [6] N. Bourbaki, Groupes et algèbres de Lie, Ch. IV, V, VI, Hermann, Paris, 1968. MR 0240238 (39:1590)
  • [7] S. Fomin and A. Kirillov, The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Discrete Math. 153 (1996), 123-143. MR 1394950 (98b:05101)
  • [8] S. 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 (electronic). MR 2238920 (2007d:14088)
  • [9] W. Graham, Positivity in equivariant Schubert calculus, Duke Math. J. 109 (2001), no. 3, 599-614. MR 1853356 (2002h:14083)
  • [10] T. Ikeda, Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian, Adv. Math. 215 (2007), 1-23. MR 2354984
  • [11] V. N. Ivanov, Interpolation analogues of Schur $ Q$-functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 307 (2004), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 10, 99-119, 281-282; translation in J. Math. Sci. (N. Y.) 131 (2005), no. 2, 5495-5507. MR 2050689 (2004m:05268)
  • [12] V. Kodiyalam and K. N. Raghavan, Hilbert functions of points on Schubert varieties in Grassmannians, Journal of Alg. 270 (2003), 28-54. MR 2015929 (2005d:14067)
  • [13] A. Knutson and E. Miller, Gröbner geometry of Schubert polynomials, Ann. Math. 161 (2005), no. 3, 1245-1318. MR 2180402 (2006i:05177)
  • [14] A. Knutson, E. Miller, and A. Yong, Gröbner geometry of vertex decompositions and flagged tableaux, preprint arXiv: math.AG/0502144
  • [15] A. Knutson and T. Tao, Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J. 119 (2003), no. 2, 221-260. MR 1997946 (2006a:14088)
  • [16] B. Kostant and S. Kumar, The nil Hecke ring and cohomology of $ G/P$ for a Kac-Moody group $ G$, Adv. in Math. 62 (1986) 187-237. MR 866159 (88b:17025b)
  • [17] V. Kreiman, Schubert classes in the equivariant K-theory and equivariant cohomology of the Grassmannian, preprint math.AG/0512204
  • [18] V. Kreiman, Schubert classes in the equivariant K-theory and equivariant cohomology of the Lagrangian Grassmannian, preprint math.AG/0602245
  • [19] V. Kreiman and V. Lakshmibai, Multiplicities of singular points in Schubert varieties of Grassmannians. Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), 553-563, Springer, Berlin, 2004. MR 2037109 (2005c:14060)
  • [20] A. Kresch and H. Tamvakis, Double Schubert polynomials and degeneracy loci for the classical groups, Annales de l'institut Fourier, 52 (2002), no. 6, 1681-1727. MR 1952528 (2004d:14078)
  • [21] S. Kumar, Kac-Moody groups, their flag varieties and representation theory, Progress in Math. 204, Birkhäuser, Boston, 2002. MR 1923198 (2003k:22022)
  • [22] V. Lakshmibai, K. N. Raghavan, and P. Sankaran, Equivariant Giambelli and determinantal restriction formulas for the Grassmannian, in: Special issue: In honor of Robert MacPherson, Part 1 of 3, Pure Appl. Math. Quart. 2 (3) (2006) 699-717. MR 2252114 (2007h:14084)
  • [23] 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 (92a:14058)
  • [24] A. Lascoux and M.-P. Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris, 294 (1982), 447-450. MR 660739 (83e:14039)
  • [25] A. Molev and B. E. A. Sagan, A Littlewood-Richardson rule for factorial Schur functions, Trans. Amer. Math. Soc. 351 (1999), no. 11, 4429-4443. MR 1621694 (2000a:05212)
  • [26] P. Pragacz, Algebro-geometric applications of Schur $ S$- and $ Q$-polynomials, Topics in invariant theory (Paris, 1989/1990), 130-191, Lecture Notes in Math., 1478, Springer, Berlin, 1991. MR 1180989 (93h:05170)
  • [27] W. Rossmann, Equivariant multiplicities on complex varieties, Orbites unipotentes et représentations, III, Astérisque No. 173-174, (1989), 11, 313-330. MR 1021516 (91g:32042)
  • [28] K. N. Raghavan and S. Upadhyay, Hilbert functions of points on Schubert varieties in orthogonal Grassmannians, preprint arXiv: 0704.0542 [math.CO]
  • [29] J. R. Stembridge, Nonintersecting paths, Pfaffians, and plane partitions, Adv. Math. 83 (1990), no. 1, 96-131. MR 1069389 (91h:05014)
  • [30] J. R. Stembridge, On the fully commutative elements of Coxeter groups, J. Algebraic Combin. 5 (1996), no. 4, 353-385. MR 1406459 (97g:20046)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 05E15, 14N15, 14M15, 05E05

Retrieve articles in all journals with MSC (2000): 05E15, 14N15, 14M15, 05E05


Additional Information

Takeshi Ikeda
Affiliation: Department of Applied Mathematics, Okayama University of Science, Okayama 700-0005, Japan
Email: ike@xmath.ous.ac.jp

Hiroshi Naruse
Affiliation: Graduate School of Education, Okayama University, Okayama 700-8530, Japan
Email: rdcv1654@cc.okayama-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-09-04879-X
Received by editor(s): September 4, 2007
Published electronically: April 30, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society