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. We study the ring of polynomial invariants $K[\operatorname{End}(V)^{\oplus m}]^{\operatorname{G}}$ of $m$ endomorphisms of $V$ under the adjoint action of $\operatorname{GLd}:=\operatorname{GL}(V_1) \times \cdots \times \operatorname{GL}(V_n)$. We find that the ring is generated by certain generalized trace monomials $\operatorname{Tr}^M_{\sigma}$ where $M$ is a multiset with entries in $[m]=\{1,\dots, m\}$ and $\sigma \in \mathcal{S}_m^n$ is a choice of $n$ permutations of $[m]$. We find that $\mathbb{C}[\operatorname{End}(V)]^{\operatorname{GLd}}$ is generated by the $\operatorname{Tr}^M_\sigma$ of degree at most $\operatorname{dim}(V)^2$.