{"paper":{"title":"Presentations and Representations of the Multi-Virtual Twin Group and Associated Subgroups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The multi-virtual twin group M_kVT_n has exactly eight types of homogeneous 2-local representations into GL_n(C) for n at least 3.","cross_cats":["math.GR","math.RT"],"primary_cat":"math.GT","authors_text":"Madeti Prabhakar, Mohamad N. Nasser, Taher I. Mayassi, Vaibhav Keshari","submitted_at":"2026-05-13T07:01:00Z","abstract_excerpt":"Motivated by the notion of the multi-virtual braid group introduced by L. Kauffman and by the study of extensions of the well-known twin group T_n, n >= 2, we introduce a new group called the multi-virtual twin group M_kVT_n, where k >= 1 and n >= 2, together with two associated subgroups: the multi-virtual pure twin group M_kVPT_n and the multi-virtual semi-pure twin group M_kVHT_n.We classify all homogeneous 2-local representations of M_kVT_n into GL_n(C) for all k >= 1 and n >= 3, and show that they fall into exactly eight distinct types. We also investigate their main properties, including"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We classify all homogeneous 2-local representations of M_kVT_n into GL_n(C) for all k >= 1 and n >= 3, and show that they fall into exactly eight distinct types.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The multi-virtual twin group is presented by a specific set of generators and relations that correctly capture the intended multi-virtual and twin structure; the notion of homogeneous 2-local representation is the right restriction for obtaining a complete finite classification.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The multi-virtual twin group M_kVT_n admits exactly eight distinct homogeneous 2-local representations into GL_n(C) for n >= 3; these are generally unfaithful but irreducible under explicit conditions, with induced non-local representations constructed for the pure subgroup.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The multi-virtual twin group M_kVT_n has exactly eight types of homogeneous 2-local representations into GL_n(C) for n at least 3.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d0631cff66cb02ecdd516432bca6380921910cd83da37389876d8a8702dba777"},"source":{"id":"2605.13090","kind":"arxiv","version":1},"verdict":{"id":"d42968fe-8f45-4c47-9cfa-8fe802d8d510","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T01:45:06.757347Z","strongest_claim":"We classify all homogeneous 2-local representations of M_kVT_n into GL_n(C) for all k >= 1 and n >= 3, and show that they fall into exactly eight distinct types.","one_line_summary":"The multi-virtual twin group M_kVT_n admits exactly eight distinct homogeneous 2-local representations into GL_n(C) for n >= 3; these are generally unfaithful but irreducible under explicit conditions, with induced non-local representations constructed for the pure subgroup.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The multi-virtual twin group is presented by a specific set of generators and relations that correctly capture the intended multi-virtual and twin structure; the notion of homogeneous 2-local representation is the right restriction for obtaining a complete finite classification.","pith_extraction_headline":"The multi-virtual twin group M_kVT_n has exactly eight types of homogeneous 2-local representations into GL_n(C) for n at least 3."},"references":{"count":18,"sample":[{"doi":"","year":2019,"title":"V. Bardakov, M. Singh, and A. Vesnin,Structural aspects of twin and pure twin groups, Geometriae Dedicata, 203, 135–154, (2019)","work_id":"9093e86f-38f5-4ce2-9721-7f8c59c94d94","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"C. Caprau and M. Nasser,The virtual singular twin monoid and group: presentations and representations, arXiv:2601.01707, (2026)","work_id":"9de54cbc-a1a3-4e0c-8b7b-e12d6f2a786c","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"M. Chreif, M. Dally,On the irreducibility of local representations of the Braid groupB n, Arab. J. Math., (2024)","work_id":"cf12f3b4-a320-4fd5-bd8a-d139ef877f4b","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Kauffman,Multi-virtual knot theory, Journal of Knot Theory and Its Ramifications, 34, 2540002, (2025)","work_id":"86d2ef4f-244b-4761-a894-ad7252ca989d","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"V.Keshari, M.Nasser, andM.Prabhakar,On representations of the multi- virtual braid groupM kV Bn and the multi-welded braid groupMkW Bn, arXiv:2508.04168, (2025). 26","work_id":"8ce6b241-5189-44d0-a361-50984866b225","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":18,"snapshot_sha256":"635d6245f07a1c41406da2d9c76e26a4b28dfddb42811c845482740e00f5cc9f","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"5f8cb51ff3ea9d20a4fab0b42c63af2cd811b7e5c1851898985895128a70a3e8"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}