Given an exceptional Lie group $G$ and maximal torus $T$ thereof, the inclusion $T \hookrightarrow G$ induces a map $BT \to BG$ of classifying spaces and a cohomological pullback $$H^*(BG) \cong H^*(BT)^W \hookrightarrow H^*(BT)$$ as the subring of Weyl group invariants. Except for that of $G_2$, these Weyl groups range from "large" to absurdly large, but it seems vaguely possible that these inclusions of polynomial rings might nevertheless be comprehensible. Is that the case?

Or if, more likely, explicit generators for $H^*(BG)$ wouldn't fit in this answer space, where could I find a reference for these inclusions?

NB: I am completely happy taking coefficients in $\mathbb Q$.


All of these polynomial algebras are known; this is probably very classical material but I don't know a reference. Their generators have cohomological degree $2d$ where $d$ runs over the numbers from this list. For example, $H^{\bullet}(BG_2, \mathbb{Q})$ is a polynomial algebra on generators of degrees $4$ and $12$. You might want to look up some material on Coxeter groups; IIRC there's some Coxeter group magic you can use to compute these degrees.

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    $\begingroup$ Thanks. I wasn't having any doubts that they were known, and know the list of degrees (each one more than those of generators for the exterior algebra $H^*(G;\mathbb Q)$), but still am not sure where to find them written out. $\endgroup$ – jdc Jul 1 '14 at 21:06
  • $\begingroup$ @jdc: ah, I see. I wasn't clear on what exactly you knew and what you were asking for. See, for example, mathoverflow.net/questions/37602/…. $\endgroup$ – Qiaochu Yuan Jul 1 '14 at 21:44
  • $\begingroup$ I realize now that I'm not quite sure if I've asked exactly the right question. Polynomial generators of $H^*(BG)$ as a subalgebra of $H^*(BT)$ aren't particularly unique, but I really want to see these generators as images of generators of $H^*(G)$ under transgression. It seems as if for that purpose I need to identify particular polynomial generators. Is that the case, or is the indeterminacy in generators in fact covered by the bottom rows of later pages of the Serre spectral sequence of $G \to EG \to BG$ being quotients of earlier ones? $\endgroup$ – jdc Jul 2 '14 at 4:50
  • $\begingroup$ @jdc: I don't understand the question. $\endgroup$ – Qiaochu Yuan Jul 2 '14 at 6:17
  • $\begingroup$ Sorry about that. I'll try to be clearer. In the Serre spectral sequence of the fibration $G \to EG \to BG$, the cohomology of the (weakly contractible) total space is trivial, so the generators of the bottom row of the $E_2$ page, essentially $H^*(BG)$, each are eventually mapped to under an edge homomorphism by some element of the leftmost column, $H^*(G)$. What I'm really interested is the images under $H^*(G) \to H^*(BG) \hookrightarrow H^*(BT)$ of these prescribed generators of $H^*(G)$. They will collectively generate the Weyl group invariants, but there is usually some choice ... $\endgroup$ – jdc Jul 3 '14 at 20:43

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