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Studying real enumerative questions I noticed, that the even-even Schubert varieties of the real Grassmannian $Gr:=Gr_{2k}(2n,\mathbb R)$ behave analogously to the complex case.

I call a partition even-even if it is of the form $DDa:=(2a_1,2a_1,...,2a_k,2a_k)$. Already Ehresmann showed that the corresponding Schubert varieties generate additively the rational cohomology of $Gr$.

The cohomology ring of $Gr$ is generated by the Pontryagin classes $p_1,...,p_{n-k}$, and I noticed that $[\sigma_{DDa}]=\Delta_a(p)$ in other words, the same Schur-polynomial expresses the even-even real schubert class for $DDa$, as in the complex complex case for $\sigma_a$ just not in Chern classes but in Pontryagin classes. (This also implies that the rational cohomology ring of $Gr$ is isomorphic to the rational cohomology ring of the complex Grassmannian $Gr_k(n,\mathbb C)$ where $p_i$ corresponds to $c_i$: we have the same "Schubert calculus"). I can prove it but I am not happy with the proof. And most likely this is well-known.

Can anyone give me a reference? Thank you.

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I found a proof I am satisfied with, but I would be still grateful for a reference if the result is known. –  László Fehér Feb 1 '13 at 19:53

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