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We show that the $\\tau$-functions of type $nT$ satisfy bilinear equations of length $3,4,\\dots,n+1$. The $2T$-system is, after a change of variables, the usual $3$ term $T$-system of type $A$.\n  Restriction from $GL_{\\infty}^{(n)}$ to a subgroup isomorphic to the loop group $LGL_{n}$, defines $nQ$-"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1611.10340","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-11-30T20:20:51Z","cross_cats_sorted":[],"title_canon_sha256":"6e0705e5fb4282a3701157150b07755050c26a0c66b6791f57190e72d6f2212a","abstract_canon_sha256":"9b1bbea9d8a3287dccd4946d388fbec85c384dcf82d1d68042aaffc4a40badb3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:15:42.954442Z","signature_b64":"+cJHBabPCJxPm9/sOog7CGW3/EjiKOwnkrcqZTSZtLU0LCFE0oOhDyZcAJ404LUoJ1yPpxPd8OwwddQShsVoCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76f87610c179a50a4797c9b4e9136637de69fd9de99280bcec36b07b7b076a44","last_reissued_at":"2026-05-18T00:15:42.953886Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:15:42.953886Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Difference Hierarchies for $nT$ $\\tau$-Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Darlayne Addabbo, Maarten Bergvelt","submitted_at":"2016-11-30T20:20:51Z","abstract_excerpt":"We introduce hierarchies of difference equations (referred to as $nT$-systems) associated to the action of a (centrally extended, completed) infinite matrix group $GL_{\\infty}^{(n)}$ on $n$-component fermionic Fock space. 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