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 $m$ endomorphisms of $V$ 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 $\operatorname{Tr}^M_{\sigma}$ 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 $K[\operatorname{End}(V)^{\oplus m}]^{\operatorname{GL}_{\mathbf{d}}}$ is generated by the $\operatorname{Tr}^M_\sigma$ of degree at most $\frac{3}{8}m\dim(V)^6$.

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