{"paper":{"title":"The Invariant Ring Of m Matrices Under The Adjoint Action By a Product Of General Linear Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Jacob Turner, Jason Morton","submitted_at":"2013-10-01T16:16:39Z","abstract_excerpt":"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 \\math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.0370","kind":"arxiv","version":8},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}