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What are the generators of $\mathbb C[V^m]^W$, where $W$ is the Weyl group of type $E_6, E_7, E_8$, V^m denote 'm' (m > 1) copies of the Cartan subalgebra and the action is the diagonal action?

Is there any reference where I can find the generators explicitly?

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    $\begingroup$ Are there any restrictions on generators that you want? Why exceptional types? This is a very hard problem even for $A_n$ (symmetric group), at least, if you want a minimal system of generators. $\endgroup$
    – Victor Protsak
    Commented May 13, 2010 at 20:53
  • $\begingroup$ Looks like another shooter in the dark... $\endgroup$
    – Victor Protsak
    Commented May 14, 2010 at 4:08
  • $\begingroup$ When m=1, if the Weyl group is from a classical Lie algebra then this is a known result, and it is not hard to imagine that somewhere the case of m copies has been worked out. I suspect this is why the asker is more interested in exceptional type. $\endgroup$
    – Q.Q.J.
    Commented May 15, 2010 at 4:44
  • $\begingroup$ No, any system of generator will work for me, not necessarily a minimal system. For type $A_n, B_n, C_n, D_n$ and $G_2$ I know a set of generators but I do not have any clue for other exceptional types. Actually I am much more interested in the degrees of the generators. $\endgroup$
    – user6079
    Commented May 15, 2010 at 11:48
  • $\begingroup$ @Q.Q.J. Could you please give me a reference for you claim concerning the case $m=1$? $\endgroup$
    – Anthony Conway
    Commented Feb 3, 2014 at 19:14

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I don't think the answer is known. The paper [Hunziker, Classical invariant theory for finite reflection groups. Transform. Groups 2 (1997), no. 2, 147–163] is relevant. The author conjectures an answer and shows his answer is correct for $F_4$.

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