General isotropic flags are general (for Grassmannian Schubert calculus)
Author:
Frank Sottile
Journal:
J. Algebraic Geom. 19 (2010), 367-370
DOI:
https://doi.org/10.1090/S1056-3911-09-00518-9
Published electronically:
July 9, 2009
MathSciNet review:
2580679
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Abstract |
References |
Additional Information
Abstract: We show that general isotropic flags for odd-orthogonal and symplectic groups are general for Schubert calculus on the classical Grassmannian in that Schubert varieties defined by such flags meet transversally. This strengthens a result of Belkale and Kumar.
References
- P. Belkale and S. Kumar, Eigencone, saturation and Horn problems for symplectic and odd orthogonal groups, arXiv:0708.0398. J. Alg. Geom., to appear.
- D. Eisenbud and J. Harris, Divisors on general curves and cuspidal rational curves, Invent. Math. 74 (1983), no. 3, 371–418. MR 724011, DOI https://doi.org/10.1007/BF01394242
- Wm. Fulton and P. Pragacz, Schubert varieties and degeneracy loci, Lecture Notes in Mathematics, vol. 1689, Springer-Verlag, Berlin, 1998, Appendix J by the authors in collaboration with I. Ciocan-Fontanine.
- Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. MR 360616
- E. Mukhin, V. Tarasov, and A. Varchenko, The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Annals of Math., to appear.
- ---, Schubert calculus and representations of the general linear group, J. Amer. Math. Soc., to appear.
- Jim Ruffo, Yuval Sivan, Evgenia Soprunova, and Frank Sottile, Experimentation and conjectures in the real Schubert calculus for flag manifolds, Experiment. Math. 15 (2006), no. 2, 199–221. MR 2253007
- Frank Sottile, Real Schubert calculus: polynomial systems and a conjecture of Shapiro and Shapiro, Experiment. Math. 9 (2000), no. 2, 161–182. MR 1780204
- Frank Sottile, Some real and unreal enumerative geometry for flag manifolds, Michigan Math. J. 48 (2000), 573–592. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786506, DOI https://doi.org/10.1307/mmj/1030132734
References
- P. Belkale and S. Kumar, Eigencone, saturation and Horn problems for symplectic and odd orthogonal groups, arXiv:0708.0398. J. Alg. Geom., to appear.
- D. Eisenbud and J. Harris, Divisors on general curves and cuspidal rational curves, Invent. Math. 74 (1983), no. 3, 371–418. MR 724011 (85h:14019)
- Wm. Fulton and P. Pragacz, Schubert varieties and degeneracy loci, Lecture Notes in Mathematics, vol. 1689, Springer-Verlag, Berlin, 1998, Appendix J by the authors in collaboration with I. Ciocan-Fontanine.
- S. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. MR 0360616 (50:13063)
- E. Mukhin, V. Tarasov, and A. Varchenko, The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Annals of Math., to appear.
- ---, Schubert calculus and representations of the general linear group, J. Amer. Math. Soc., to appear.
- J. Ruffo, Y. Sivan, E. Soprunova, and F. Sottile, Experimentation and conjectures in the real Schubert calculus for flag manifolds, Experiment. Math. 15 (2006), no. 2, 199–221. MR 2253007 (2007g:14066)
- F. Sottile, Real Schubert calculus: polynomial systems and a conjecture of Shapiro and Shapiro, Experiment. Math. 9 (2000), no. 2, 161–182. MR 1780204 (2001e:14054)
- ---, Some real and unreal enumerative geometry for flag manifolds, Michigan Math. J. 48 (2000), 573–592. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786506 (2002d:14085)
Additional Information
Frank Sottile
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843
MR Author ID:
355336
ORCID:
0000-0003-0087-7120
Email:
sottile@math.tamu.edu
Received by editor(s):
January 16, 2008
Received by editor(s) in revised form:
July 21, 2008
Published electronically:
July 9, 2009
Additional Notes:
Work of Sottile supported by NSF grant DMS-0701050
