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We study here these cones in the case where $H=F_N$ is the Fourier matrix, $(w^{ij})$ with $w=e^{2\\pi i/N}$, our main result being a simple description of $\\widetilde{T}_HC_N$. As a consequence, the rationality conjecture $dim_\\mathbb R(\\widetilde{T}_HC_N)=dim_\\mathbb Q(\\widetilde{T}_HC_N\\cap M_N("},"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":"1302.4153","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-02-18T03:03:58Z","cross_cats_sorted":["math.AG","math.OA"],"title_canon_sha256":"f07fac76a249bd87da4305710ea00699465b8d42d9b168daa97ace7575a3b441","abstract_canon_sha256":"07459ef410432d2944fc8681063f96ebfbcb20434add612915766571538e6f7c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:51:35.869942Z","signature_b64":"iLmHQtIXodpl8hrLR8cl41leK5DiRgE5LTuTrOCiAjwcwtjYI0T5+oJrbqzCxMCV38p4Y3rzE5ek/pPGz6tfCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7eea88906223653f83a1630adb7998d0601947be7dac638d88b7a3557b3df823","last_reissued_at":"2026-05-18T01:51:35.869552Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:51:35.869552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"First order deformations of the Fourier matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.OA"],"primary_cat":"math.CO","authors_text":"Teodor Banica","submitted_at":"2013-02-18T03:03:58Z","abstract_excerpt":"The $N\\times N$ complex Hadamard matrices form a real algebraic manifold $C_N$. 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As a consequence, the rationality conjecture $dim_\\mathbb R(\\widetilde{T}_HC_N)=dim_\\mathbb Q(\\widetilde{T}_HC_N\\cap M_N("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.4153","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1302.4153","created_at":"2026-05-18T01:51:35.869614+00:00"},{"alias_kind":"arxiv_version","alias_value":"1302.4153v3","created_at":"2026-05-18T01:51:35.869614+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.4153","created_at":"2026-05-18T01:51:35.869614+00:00"},{"alias_kind":"pith_short_12","alias_value":"P3VIREDCENST","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_16","alias_value":"P3VIREDCENST7A5B","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_8","alias_value":"P3VIREDC","created_at":"2026-05-18T12:27:54.935989+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/P3VIREDCENST7A5BMMFNW6MY2B","json":"https://pith.science/pith/P3VIREDCENST7A5BMMFNW6MY2B.json","graph_json":"https://pith.science/api/pith-number/P3VIREDCENST7A5BMMFNW6MY2B/graph.json","events_json":"https://pith.science/api/pith-number/P3VIREDCENST7A5BMMFNW6MY2B/events.json","paper":"https://pith.science/paper/P3VIREDC"},"agent_actions":{"view_html":"https://pith.science/pith/P3VIREDCENST7A5BMMFNW6MY2B","download_json":"https://pith.science/pith/P3VIREDCENST7A5BMMFNW6MY2B.json","view_paper":"https://pith.science/paper/P3VIREDC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1302.4153&json=true","fetch_graph":"https://pith.science/api/pith-number/P3VIREDCENST7A5BMMFNW6MY2B/graph.json","fetch_events":"https://pith.science/api/pith-number/P3VIREDCENST7A5BMMFNW6MY2B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P3VIREDCENST7A5BMMFNW6MY2B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P3VIREDCENST7A5BMMFNW6MY2B/action/storage_attestation","attest_author":"https://pith.science/pith/P3VIREDCENST7A5BMMFNW6MY2B/action/author_attestation","sign_citation":"https://pith.science/pith/P3VIREDCENST7A5BMMFNW6MY2B/action/citation_signature","submit_replication":"https://pith.science/pith/P3VIREDCENST7A5BMMFNW6MY2B/action/replication_record"}},"created_at":"2026-05-18T01:51:35.869614+00:00","updated_at":"2026-05-18T01:51:35.869614+00:00"}