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Over the last decade Nikita Semenov, Skip Garibaldi and others have made some progress in the theory of cohomological invariants, (Rost)-motives and motivic decompositions of algebraic groups. For example a motivic decomposition of $F_4/P$, with $P$ being a certain parabolic subgroup has been established by Semenov, Nikolenko and Zainoulline.

Looking at the Dynkin diagramm of $D_n$ one will notice that the case $n=4$ gives a diagramm with a special symmetry (cue: triality).

Question: Is the symmetry of the Dynkin diagramm of $D_4$ somehow reflected in the motive of $D_4$ or is there a motivic decomposition known at all?

It might be possible that the motivic structure can be derived from general results, but i dont too many papers dealing with these kinds of problems.

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  • $\begingroup$ Which motives do you mean? If you mean a projective homogeneous space under D4, that would be a projective quadric. Semenov "Hasse diagrams and motives of projective homogeneous varieties" would be relevant. If you mean a group of type D4, like Spin(2n), that's another story - and for the motive of a split reductive group you could look at Biglari's results. If you're after the non-split case, there is not so much known, in general. $\endgroup$ Mar 24, 2015 at 22:53
  • $\begingroup$ I think in the paper you mention there is done: Take three points from the Dynkin diagramm. They form a parabolic subgroup $P$. Then $D_4/P \cong SO(8)/P$ is a projective quadric. Isnt it possible to take just one or two points from the Dynkin diagramm, look at the respective factor group and establish a motivic decomposition? I was hoping to get infos on that and maybe the motive of $D_4$ itself. By pushing forward cycles from the CH()'s, motives of the small factor group should be contained in the bigger ones. Im not sure if the other subgroups of the Dynkin diagramm are parabolic as $P$. $\endgroup$
    – nxir
    Mar 25, 2015 at 2:13
  • $\begingroup$ Despite the title and abstract, in [arxiv.org/abs/1402.5520] you'll also find a motivic decomposition for the toroidal completion of a split semisimple group. Combining this with a localization sequence might give you the motive of D4, if you can work out how the boundary components intersect (which should be doable with a fair amount of work). $\endgroup$ Mar 25, 2015 at 12:58
  • $\begingroup$ Thanks. I will do my best, to figure things out. Considering my level it will take while i think. $\endgroup$
    – nxir
    Mar 25, 2015 at 14:08

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You can compute the motive of $SO(8)$ with rational coefficients in the category of derived mixed Tate motives with Biglari's theorem (after looking up the exponents of the Weyl group in Bourbaki's tables).

The exponents are $1,3,5,3$, which gives the numbers $2,4,6,4$ as dimensions of certain cohomology groups of weight graded parts of $M(SO(8))$.

The primitive motive of $SO(8)$ therefore is $$PM(SO(8)) = \mathbb{Q}(2)[3] \oplus \mathbb{Q}(4)[7] \oplus \mathbb{Q}(4)[7] \oplus \mathbb{Q}(6)[11]$$ and then we have $$M(SO(8))_{\mathbb{Q}} = Sym^\bullet PM(SO(8)).$$

I don't immediately see the triality operation in this description, but it should be there. Maybe you can get a better hold on triality by looking at $M(SO(8)/B)$ in terms of the Bruhat decomposition...

added in response to Matthias' comment: In Gross' article, he gives an ad-hoc definition for "the motive of $G$", which can be compared with the primitive motive of $G$ by the expression $$M^{Gross}(G)(1) = PM(G)$$ Also note that Gross uses another index convention.

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  • $\begingroup$ This is for the split case only. For the non-split case over number field and with rational coefficients, there should be a description in Gross's paper "On the motive of a reductive group", Invent. Math. 130 (1997). For non-split cases and statements about finite coefficients, with explicit trialitarian examples, there is also the paper of Quéginer-Mathieu, Semenov and Zainouilline: arxiv.org/abs/1104.1096 $\endgroup$ Mar 26, 2015 at 11:50
  • $\begingroup$ Thanks again (for mentioning Biglaris Theorem). This will take me a lot of research to look at and understand everything you suggested. Ill vote this as answered. $\endgroup$
    – nxir
    Mar 26, 2015 at 15:29
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    $\begingroup$ Just to clarify why I mentioned Gross' paper: whenever the group $G$ is an inner form the motive will be as you said. But the interesting thing here are probably the outer forms, where the Galois group acts nontrivially on the Dynkin diagram (this is the point where the very special very symmetric form of the Dynkin diagram plays a role). In this case, there will also be an action of the Galois group on $PM(SO(8))$, and Gross' paper explains how to relate the Galois-action on the character lattice and the Galois-action on $PM(SO(8))$. $\endgroup$ Mar 26, 2015 at 19:22

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