2017/10/01 by Faten Abu-Shoga, Abu-Shoga, Faten S.
Computer Science · Mathematics · #14M15 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Coding theory and cryptography #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1710.00306
openalex publication_date 2017/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A real form G0 of a complex semisimple Lie group G has only finitely many orbits in any given compact G-homogeneous projective algebraic manifold Z=G/Q. A maximal compact subgroup K0 of G0 has special orbits C which are complex sub-manifolds in the open orbits of G0. These are referred to as cycles. The cycles intersect Shubert varieties S transversely in finitely many points. In particular, determining these points of intersection yields a description of the topological class of the given cycle. This was carried out for all real forms of SL(n,ℂ) in the work of A. Brecan. Our work here is devoted to the real forms of the other classical groups, Sp(2n,ℂ) and SO(n,ℂ). For the manifold Z=G/B of complete flags the points of intersection in S∩ C are described, in particular the number of such is computed. For certain real forms, e.g., Sp(2n,ℝ) and SO^*(2n), remarkably simple formulas are proved. In other cases the results are algorithmic in nature.