0
$\begingroup$

Take a real vector space $R$ transforming in the adjoint representation of the ${\rm E}_7(7)$ Lie group as $R \rightarrow G R G^{-1}$. One can define invariants using traces of products of $R$ as ${\rm Tr}[R^k]$.

I heard that a basis of invariants is given by $k = 2,6,8,10,12,14,18$. Is this correct? Which theorem states this?

Thanks for your help

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ The question is out of focus, starting with "the $E_7(7)$ Lie group". It's not clear what this means, but the invariant theory of the associated Weyl group does lead to a polynomial ring of invariants (Chevalley) geneerated by algebraically independent homogeneous polynomials of degrees $d$ which you've listed with the index $k$. This in turn provides a picture of invariants in the simple complex Lie algebra of type $E_7$ via Harish-Chandra's isomorphism. All of this is found in standard textbooks, so I think the question is not currently at research-level. $\endgroup$
    – Jim Humphreys
    Commented Oct 8, 2015 at 14:25
  • $\begingroup$ $E_{7(7)}$ refers to the split real form, not the complex $E_7$. $\endgroup$
    – David Chow
    Commented Oct 8, 2015 at 15:43
  • $\begingroup$ Among "standard textbooks" let me recommend Jim Humphreys' "Reflection groups and Coxeter groups". $\endgroup$
    – Allen Knutson
    Commented Oct 8, 2015 at 18:28

0

Reset to default

You must log in to answer this question.