The Invariant Ring Of m Matrices Under The Adjoint Action By a Product Of General Linear Groups

Let $V=V_1\otimes \cdots \otimes V_n$ be a vector space over an algebraically closed field $K$ of characteristic zero with $\dim (V_i)=d_i$. We study the ring of polynomial invariants $K[\operatorname{End}(V)^{\oplus m}]^{\operatorname{GL}{$\mathbf{d}$}}$of$m$endomorphismsof$V$ under the adjoint action of $\operatorname{GL}{$\mathbf{d}$}:=\operatorname{GL}(V_1) \times \cdots \times \operatorname{GL}(V_n)$. We find that the ring is generated by certain generalized trace monomials ${\mathrm{Tr}}_{\sigma }^M$ where $M$ is a multiset with entries in $[m]=\{1,\dots ,m\}$ and $\sigma \in $\mathcal{S}$m^n$isachoiceof$n$permutationsof$[m]$\mathrm{.}Wefindthat$K[\operatorname{End}(V)^{\oplus m}]^{\operatorname{GL}{$\mathbf{d}$}}$ is generated by the ${\mathrm{Tr}}_{\sigma }^M$ of degree at most $\frac{3}{8}m\dim (V{)}^6$.