Chow ring of $E_7$ varieties

Consider a split algebraic group $$G$$ of type $$E_7$$ over a field of characteristic zero. It is known that some subgroups $$P_i$$ of $$G$$, which are called parabolic, have the property that the object $$G/P_i =:X_i$$ is a projective, homogeneous variety over $$k$$. These parabolic subgroups correspond to subsets of nodes in the Dynkin diagram of $$G$$. We use Bourbaki enumeration. We write Ch$$(X)$$ for the Chow ring of $$X$$ modulo rational equivalence and with $$\mathbb{F}_2$$ coefficients.

Question: What is the structure of Ch$$(X_3)$$ ?

I can find references for $$X_1$$ and $$X_7$$, but for the Chow ring of $$X_3$$, which is probably way more complicated, i cant find any results. It would already be great to know how many generators there are and what their dimension is.

• Over the complex numbers, this is discussed in Baston and Eastwood, The Penrose Transform. The computation involves the Hasse diagram of $G$, if I remember correctly. Dec 20, 2019 at 21:01
• Well if it is in this book (i cant access at the moment), you dont mind to post the result here?
– nxir
Dec 20, 2019 at 21:49
• If you need concrete computations with Schubert cycles and not a presentation in terms of generators and relations, you can try our Maple package (dirty written, but hope the examples may help): mathematik.uni-muenchen.de/~semenov/software.php Dec 21, 2019 at 13:51
• I need to know all about the generators for computing the coaction presented in your latest paper to prove that that there are no Rost motives, when $E_7$ is versal.
– nxir
Dec 21, 2019 at 14:30