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We prove that if the compact flat manifolds $\\Gamma_1\\backslash\\mathbb{R}^n$ and $\\Gamma_2\\backslash\\mathbb{R}^n$ are strongly isospectral then the Bieberbach groups $\\Gamma_1$ and $\\Gamma_2$ are representation equivalent, that is, the right regular representations $L^2(\\Gamma_1\\backslash G)$ and $L^2(\\Gamma_2\\backslash G)$ are unitarily equivalent."},"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":"1210.0894","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-10-02T19:48:30Z","cross_cats_sorted":["math.RT","math.SP"],"title_canon_sha256":"0473a657273efe0eb0986e741d43be39d614b0376e3a13d1138f1ea68aa5195f","abstract_canon_sha256":"5abf8dad8c1cb8740b1f0b7048dcda8d67605274d02d58f61c3a677bff531087"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:53:54.140383Z","signature_b64":"snI/0skm5SiAKHLVbxsaMjyu8/sZPfrTk/JGKFr3J5KFZJb3SKaNEPHkthlwRV5dVq0+gNY9Hek3RTG4TU7nDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"25f50d760044aff0d9c4ab67ccf146b847acfab12edd0b7a5291e30df6ece7b6","last_reissued_at":"2026-05-18T02:53:54.139431Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:53:54.139431Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Representation equivalent Bieberbach groups and strongly isospectral flat manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT","math.SP"],"primary_cat":"math.DG","authors_text":"Emilio A. 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