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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.

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    $\begingroup$ 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. $\endgroup$
    – Ben McKay
    Commented Dec 20, 2019 at 21:01
  • $\begingroup$ Well if it is in this book (i cant access at the moment), you dont mind to post the result here? $\endgroup$
    – nxir
    Commented Dec 20, 2019 at 21:49
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    $\begingroup$ 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 $\endgroup$
    – Victor Petrov
    Commented Dec 21, 2019 at 13:51
  • $\begingroup$ 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. $\endgroup$
    – nxir
    Commented Dec 21, 2019 at 14:30

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