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

Abstract

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 $K[\mathrm{End}(V{)}^{\oplus m}{]}^{{\mathrm{GL}}_{\mathbf{d}}}$ of $m$ endomorphisms of $V$ under the adjoint action of ${\mathrm{GL}}_{\mathbf{d}}:=\mathrm{GL}(V_1)\times \cdots \times \mathrm{GL}(V_n)$. 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 $\sigma \in {\mathcal{S}}_m^n$ is a choice of $n$ permutations of $[m]$. We find that $K[\mathrm{End}(V{)}^{\oplus m}{]}^{{\mathrm{GL}}_{\mathbf{d}}}$ is generated by the ${\mathrm{Tr}}_{\sigma }^M$ of degree at most $\frac{3}{8}m\dim (V{)}^6$.

This paper has been withdrawn due to a an error in the proof of the main result that could not be fixed

References in corpus (1)

• Topological phases and multiqubit entanglement

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