{"paper":{"title":"Linear representations of manifolds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Linear representations of G-manifolds as matrix maps supply explicit minimal dimensions for Mostow-Palais equivariant embeddings.","cross_cats":["math.RT"],"primary_cat":"math.DG","authors_text":"Ke Ye, Lek-Heng Lim, Rongbiao Thomas Wang","submitted_at":"2026-05-13T18:26:45Z","abstract_excerpt":"A finite-dimensional linear representation of a group or an algebra may be regarded as a map into a space of matrices, endowing abstract elements with coordinates, and encoding algebraic operations as matrix products. With this in mind, we define a linear representation of a $\\mathsf{G}$-manifold $\\mathcal{M} $ as a map into a space of matrices, representing points as matrices and the $\\mathsf{G}$-action as matrix products. We show that this generalizes group representations to any $\\mathsf{G}$-manifold that may not have a group structure, with homogeneous spaces $\\mathsf{G}/\\mathsf{H}$ an imp"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We will give explicit values for dim V and show that our bounds are sharp. Furthermore, our method is constructive, giving explicit expressions for these minimal-dimensional Mostow-Palais embeddings.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the newly defined linear representations of the G-manifold exist and can be chosen so that the induced map into the space of matrices produces a Mostow-Palais embedding whose dimension is controlled by the representation dimension.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Linear representations of G-manifolds generalize group representations and deliver explicit sharp bounds for Mostow-Palais G-equivariant embeddings into finite-dimensional modules.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Linear representations of G-manifolds as matrix maps supply explicit minimal dimensions for Mostow-Palais equivariant embeddings.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f390e84ed89f096dcb4ba91aab511a5d1226520bdead53f69f1d2af8cf88427d"},"source":{"id":"2605.14013","kind":"arxiv","version":1},"verdict":{"id":"003ecb89-95ed-46f8-a665-acf9e931534d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:44:56.234237Z","strongest_claim":"We will give explicit values for dim V and show that our bounds are sharp. Furthermore, our method is constructive, giving explicit expressions for these minimal-dimensional Mostow-Palais embeddings.","one_line_summary":"Linear representations of G-manifolds generalize group representations and deliver explicit sharp bounds for Mostow-Palais G-equivariant embeddings into finite-dimensional modules.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the newly defined linear representations of the G-manifold exist and can be chosen so that the induced map into the space of matrices produces a Mostow-Palais embedding whose dimension is controlled by the representation dimension.","pith_extraction_headline":"Linear representations of G-manifolds as matrix maps supply explicit minimal dimensions for Mostow-Palais equivariant embeddings."},"references":{"count":42,"sample":[{"doi":"","year":1997,"title":"A. Altland and M. R. Zirnbauer. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures.Phys. Rev. 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Equivariant Gromov theory.Topology, 13:327–345, 1974","work_id":"428bb4cd-4f12-4742-811e-e620d475633e","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2013,"title":"Bump.Lie groups, volume 225 ofGraduate Texts in Mathematics","work_id":"21001335-6eb2-4706-82aa-6d350eb63733","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":42,"snapshot_sha256":"9120563df59aab6ae1f8ae5739ca40bca51d9b495ebfc904b472f0e964223201","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"49487deb290bf5e4eae7d6b426d69ced10c640acef1fa39eb02d62cc5f506468"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}