Real schubert calculus

Question

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.

Answer 1

This is not really a literature reference, but I'll outline how I think this can be deduced from the standard results.

It is easier to write down the cohomology of complex Grassmannian $H^\bullet({\rm Gr}_k(n,\mathbb{C}),\mathbb{Z})$ in the following form: there are two complex bundles on the Grassmannian, one of rank $k$ and one of rank $n-k$ and their Whitney sum is trivial. These have Chern classes $c_1,\dots,c_k$ and $c_1^\perp,\dots,c_{n-k}^\perp$ and the cohomology of the Grassmannian has the presentation $$ \mathbb{Z}[c_1,\dots,c_k,c_1^\perp,\dots,c_{n-k}^\perp]/(c\cdot c^\perp-1) $$ where the relation $c\cdot c^\perp-1$ means that the product of the total Chern classes of the two bundles should be 1 (Whitney sum formula). The same presentation holds for the rational cohomology (even integral cohomology modulo 2-torsion) of the real Grassmannian ${\rm Gr}_{2k}(2n,\mathbb{R})$, see my answer to this MO-question and the references there. From this point of view, it is immediately clear that there is a ring homomorphism as claimed, sending $c_i$ to $p_i$ -- the presentations are simply the same.

It remains to answer the question what this has to do with the doubled partitions. We can check what the image of the Pontryagin class $p_i$ in $H^\bullet({\rm Gr}_{2k}(2n,\mathbb{R}),\mathbb{Z}/2)$ is. If we can understand this in terms of Schubert cells, we are done. The computations of cohomology of ${\rm B}{\rm SO}(n)$ (due to Brown) show that the Pontryagin class $p_i$ reduces to the Stiefel-Whitney class $w_{2i}^2$. By the Giambelli formula, the class of the doubled partition in the mod 2 cohomology of the real Grassmannian is $w_{2i}^2-w_{2i-1}w_{2i+1}$. The odd Stiefel-Whitney classes have lifts to the integral cohomology of the Grassmannian, but these are 2-torsion. In particular, up to 2-torsion, the Pontryagin class is a lift of the class corresponding to the Schubert cell for the doubled partition in the mod 2 cohomology ring of the real Grassmannian.