# Invariants of linear endomorphisms of tensor products

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)

• Is it useful to observe that $End(V\otimes W)\cong End(V)\otimes End(W)$? – Tom Goodwillie Mar 15 at 16:50
• 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. – Ehud Meir Mar 15 at 17:12