Let $V$ and $W$ be two finite dimensional vector spaces over an algebraically closed field $K$ of characteristic zero. Consider the coordinate ring $K[\mathrm{End}(V\otimes W)]$ of the affine space of endomorphisms of $V\otimes W$. The group $G=\mathrm{PGL}(V)\times \mathrm{PGL}(W)$ acts on the affine space by conjugation, and therefore on the coordinate ring as well.

Question: What is known about the invariant subalgebra $K[\mathrm{End}(V\otimes W)]^G$?

In case the dimension of either $V$ (or $W$) is 1 we just get $K[\mathrm{End}(W)]^{\mathrm{PGL}(W)}$, and this is the algebra spanned by the coefficients of the characteristic polynomial. In particular, we get a polynomial algebra. Do we get a polynomial algebra also if $\dim(W)$ and $\dim(V)$ are different from 1? If not, is there a description of this algebra by finite set of generators and relations? (which must depend on the dimensions of $V$ and $W$, of course)

  • 3
    $\begingroup$ Is it useful to observe that $End(V\otimes W)\cong End(V)\otimes End(W)$? $\endgroup$ – Tom Goodwillie Mar 15 at 16:50
  • $\begingroup$ It is, but it is not enough. One can even go further with this observation and use Schur-Weyl duality to describe the invariants, but to understand something like a finite generating set, and all possible relations between them is way more difficult. $\endgroup$ – Ehud Meir Mar 15 at 17:12

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