# degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$

How can we find the degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$ , $dimV=8$ by the model in the Lie algebra $E_7$.

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You don't really need Vinberg theory to find the degrees of the invariants here. This representation of $SL_8=SL(V)$ can be constructed by considering a maximal rank involution of an algebraic group of type $E_7$, which has fixed points isomorphic to $SL(V)/\{\pm I\}$ and such that the $(-1)$ eigenspace in the Lie algebra is exactly $\Lambda^4(V)$.

Well-known results about representations associated to symmetric spaces, as found for example in Kostant-Rallis' 1971 paper in the American Journal of Mathematics, then tell you that the invariants are given by restricting the invariants for the adjoint representation of the full $E_7$ group, and that these restrictions are non-zero. So the degrees are exactly the degrees of the invariants for the adjoint representation of $E_7$.

If you want to know how to see $SL_8/\{\pm I\}$ inside the group of type $E_7$, just take the subgroup whose Lie algebra is spanned by a Cartan subalgebra and all root spaces which have an even coefficient of $\alpha_2$ (in Bourbaki numbering); the representation $\Lambda^4(V)$ is just the span of all the root-spaces which have an odd coefficient of $\alpha_2$.

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NB: It is possible that this is also done in the earlier paper by L. V. Antonyan, Classification of $4$-vectors of $8$-dimensional space, Trudy Sem. Vektor. Tenzor. Anal. 20 (1981), pp. 144-161, but I don't have access to this paper and, in any case, I don't read Russian. It could be that this paper just deals with the normal forms and doesn't discuss the ring of invariants.
The 7 numbers 2,6,8,10,12,14,18 are the degrees of polynomial invariants of the Weyl reflection group of $E_7$. These generate (freely) the full ring of polynomial invariants of the group in the reflection rep. These can be calculated many ways; look at invariant theory of reflection or Coxeter groups. See also en.wikipedia.org/wiki/Coxeter_element –  Y Macdisi Jan 23 '13 at 6:03