Table of Contents
ISRN Geometry
Volume 2012, Article ID 254235, 34 pages
http://dx.doi.org/10.5402/2012/254235
Research Article

Poset Pinball, the Dimension Pair Algorithm, and Type 𝐴 Regular Nilpotent Hessenberg Varieties

Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada L8S 4K1

Received 28 January 2012; Accepted 15 March 2012

Academic Editors: L. C. Jeffrey, A. Morozov, and E. H. Saidi

Copyright © 2012 Darius Bayegan and Megumi Harada. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. Harada and J. Tymoczko, “Poset pinball, GKM-compatible subspaces, and Hessenberg varieties,” http://arxiv.org/abs/1007.2750.
  2. J. S. Tymoczko, “An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson,” in Snowbird Lectures in Algebraic Geometry, vol. 388 of Contemporary Mathematics, pp. 169–188, American Mathematical Society, Providence, RI, USA, 2005. View at Publisher · View at Google Scholar
  3. F. Y. C. Fung, “On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory,” Advances in Mathematics, vol. 178, no. 2, pp. 244–276, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. N. Shimomura, “The fixed point subvarieties of unipotent transformations on the flag varieties,” Journal of the Mathematical Society of Japan, vol. 37, no. 3, pp. 537–556, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. N. Spaltenstein, “The fixed point set of a unipotent transformation on the flag manifold,” Proceedings. Akadamie van Wetenschappen Series A, vol. 38, no. 5, pp. 452–456, 1976. View at Google Scholar · View at Zentralblatt MATH
  6. F. de Mari, C. Procesi, and M. A. Shayman, “Hessenberg varieties,” Transactions of the American Mathematical Society, vol. 332, no. 2, pp. 529–534, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. B. Kostant, “Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight ρ,” Selecta Mathematica. New Series, vol. 2, no. 1, pp. 43–91, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. K. Rietsch, “Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties,” Journal of the American Mathematical Society, vol. 16, no. 2, pp. 363–392, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. J. Fulman, “Descent identities, Hessenberg varieties, and the Weil conjectures,” Journal of Combinatorial Theory A, vol. 87, no. 2, pp. 390–397, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. M. Brion and J. B. Carrell, “The equivariant cohomology ring of regular varieties,” The Michigan Mathematical Journal, vol. 52, no. 1, pp. 189–203, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. J. B. Carrell and K. Kaveh, “On the equivariant cohomology of subvarieties of a B-regular variety,” Transformation Groups, vol. 13, no. 3-4, pp. 495–505, 2008. View at Publisher · View at Google Scholar
  12. D. Bayegan and M. Harada, “A Giambelli formula for the S1-equivariant cohomology of type A Peterson varieties,” Accepted for publication in Involve, http://arxiv.org/abs/1012.4053.
  13. M. Harada and J. Tymoczko, “A positive Monk formula in the S1-equivariant cohomology of type A Peterson varieties,” Proceedings of the London Mathematical Society Series III, vol. 103, no. 1, pp. 40–72, 2011. View at Publisher · View at Google Scholar
  14. A. Mbirika, “A Hessenberg generalization of the Garsia-Procesi basis for the cohomology ring of Springer varieties,” Electronic Journal of Combinatorics, vol. 17, no. 1, article R153, 2010, Research paper no. 153. View at Google Scholar · View at Zentralblatt MATH
  15. B. Dewitt and M. Harada, “Poset pinball, highest forms, and (n-2,2) Springer varieties,” Electronic Journal of Combinatorics, vol. 19, no. 1, #P56, 2012, http://arxiv.org/abs/1012.5265. View at Google Scholar
  16. J. S. Tymoczko, “Linear conditions imposed on flag varieties,” American Journal of Mathematics, vol. 128, no. 6, pp. 1587–1604, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. A. Björner and F. Brenti, Combinatorics of Coxeter Groups, vol. 231 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2005.
  18. S. Billey, “Kostant polynomials and the cohomology ring for G/B,” Duke Mathematical Journal, vol. 96, no. 1, pp. 205–224, 1999. View at Publisher · View at Google Scholar