Squaring operations in the Hermitian symmetric spaces (Q2367488)

The author calculates explicitly squaring operations \(Sq^ i\) in the \(\mathbb{Z}_ 2\)-cohomology of the Hermitian compact symmetric spaces. \(Sq^ i\)-action on generators of \(\mathbb{Z}_ 2\)-cohomology algebra is given. Each of the irreducible compact Hermitian symmetric space is considered as a quotient of an appropriate compact simple Lie group. In the case of classical Lie groups the resulting formulae are obtained by means of the Wu formula. In the exceptional cases the author takes the spectral sequence of the fibration \(K/T\to G/T@>p>> G/K\) \((T\) is a maximal torus of \(G\), \(K\) is a centralizer of the \(I\)-dimensional torus in \(G)\). The spectral sequence evidently collapses and \(p^*:H^*(G/K)\to H^*(G/T)\) is injective for any coefficient ring. Therefore the action of \(Sq^ i\) in \(G/K\) is derived from that in \(G/T\). The action of \(Sq^ i\) in \(G/T\) is derived from the explicit expressions of \(H^*(G/T)\) [the author uses some previous results of himself and others, e.g. with \textit{A. Kono} in ``Algebraic topology'', Proc. Symp., Barcelona/Spain 1986, Lect. Notes Math. 1298, 192-206 (1987; Zbl 0656.57025)].