KO-theory of complex partial flag manifolds (Q2922425)

A flag in \(\mathbb C^n\) is a sequence of vector subspaces NEWLINE\[NEWLINE 0 = V_0 \subset V_1 \subset V_2 \subset \dots \subset V_l = \mathbb C^{n}. NEWLINE\]NEWLINE If \(\dim V_i - \dim V_{i-1} = n_i\) for \(i=1,\dots, l\), the above flag is called of type \((n_1, n_2,\dots, n_l)\), where \(n= n_1 + n_2 +\dots+n_l\). If \(n_i = 1\) for each \(i\), the flag is called \(\mathbf{complete}\) otherwise it is called \(\mathbf{partial}\). The space of all flags in \(\mathbb C^n\) of type \((n_1, n_2,\dots, n_l)\) is identified with the homogeneous space NEWLINE\[NEWLINE F(n_1, n_2,\dots, n_l) = \frac{U(n)}{U(n_1)\times \dots \times U(n_l)}, NEWLINE\]NEWLINE which is called the \(\mathbf{partial \,flag \,manifold \,of\,type}\) \((n_1, n_2,\dots, n_l)\). \(F(1,\dots,1)\) is the flag manifold and \(F(n_1 , n_2)\) is the Grassmannian.NEWLINENEWLINEIn this paper the authors explicitly calculate the real \(KO\)-groups of partial flag manifolds by the Atiyah Hirzebruch spectral sequence. All calculations are clear.