Abstract
In this paper we obtain a formula for the boundary map of the cellular ℤ-homology of the real maximal isotropic Grassmannians of type B, C and D which are obtained as minimal flag manifolds of split real forms of semi-simple Lie algebras of the respective type. We use the model of shifted Young diagrams presented in [9] to describe the Schubert varieties and the cellular decompositions of such flag manifolds. The results are applied to determine the orientability of the real maximal isotropic Grassmannians.
Communicated by: I. Coskun
Acknowledgements
I thank my PhD supervisor Luiz A. B. San Martin and the anonymous referee for useful suggestions during the preparation of this paper.
References
[1] A. Altomani, C. Medori, M. Nacinovich, On the topology of minimal orbits in complex flag manifolds. Tohoku Math. J. (2) 60(2008), 403–422. MR2453731 (2009j:32027) Zbl 1160.5703310.2748/tmj/1223057736Search in Google Scholar
[2] M. Brion, Lectures on the geometry of flag varieties. In: Topics in cohomological studies of algebraic varieties, 33–85, Birkhäuser 2005. MR2143072 (2006f:14058)10.1007/3-7643-7342-3_2Search in Google Scholar
[3] D. Burghelea, T. Hangan, H. Moscovici, A. Verona, Introducere in topologia diferentiala. Ed. Siintifica Bucuresti, 1973.Search in Google Scholar
[4] L. Casian, Y. Kodama, On the cohomology of real Grassmannian Manifolds. 2013. arXiv 1309.5520Search in Google Scholar
[5] L. Casian, R. J. Stanton, Schubert cells and representation theory. Invent. Math. 137(1999), 461–539. MR1709874 (2001e:14049) Zbl 0954.2201110.1007/s002220050334Search in Google Scholar
[6] J. J. Duistermaat, J. A. C. Kolk, V. S. Varadarajan, Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups. Compositio Math. 49(1983), 309–398. MR707179 (85e:58150) Zbl 0524.43008Search in Google Scholar
[7] A. Hatcher, Algebraic topology. Cambridge Univ. Press 2002. MR1867354 (2002k:55001) Zbl 1044.55001Search in Google Scholar
[8] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, volume 80 of Pure and Applied Mathematics. Academic Press 1978. MR514561 (80k:53081) Zbl 0451.53038Search in Google Scholar
[9] T. Ikeda, H. Naruse, Excited Young diagrams and equivariant Schubert calculus. Trans. Amer. Math. Soc. 361(2009), 5193–5221. MR2515809 (2010i:05351) Zbl 1229.0528710.1090/S0002-9947-09-04879-XSearch in Google Scholar
[10] A. W. Knapp, Lie groups beyond an introduction. Birkhäuser 2002. MR1920389 (2003c:22001) Zbl 1075.22501Search in Google Scholar
[11] R. R. Kocherlakota, Integral homology of real flag manifolds and loop spaces of symmetric spaces. Adv. Math. 110(1995), 1–46. MR1310389 (96a:57066) Zbl 0832.2202010.1006/aima.1995.1001Search in Google Scholar
[12] M. Patrão, L. A. B. San Martin, L. J. dos Santos, L. Seco, Orientability of vector bundles over real flag manifolds. Topology Appl. 159(2012), 2774–2786. MR2923447 Zbl 1267.5703010.1016/j.topol.2012.03.002Search in Google Scholar
[13] L. Rabelo, L. A. B. San Martin, Cellular Homology of Real Flag Manifolds. Submitted, 2014.Search in Google Scholar
[14] L. A. B. San Martin, Álgebras de Lie. Ed. Unicamp, 2010.Search in Google Scholar
[15] L. Seco, A note on the Bruhat decomposition of semisimple Lie groups. J. Lie Theory18(2008), 725–731. MR2493064 (2010c:22018) Zbl 1228.22016Search in Google Scholar
[16] G. Warner, Harmonic analysis on semi-simple Lie groups. I. Springer 1972. MR0498999 (58 #16979) Zbl 0265.2202010.1007/978-3-642-51640-5_3Search in Google Scholar
[17] M. Wiggerman, The fundamental group of a real flag manifold. Indag. Math. (N.S.) 9(1998), 141–153. MR1618211 (99e:57052) Zbl 0921.2201110.1016/S0019-3577(97)87572-6Search in Google Scholar
© 2016 by Walter de Gruyter Berlin/Boston
