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We study here this phenomenon, in the general \"quizzy\" framework, which covers the various liberations and twists of $H_N,O_N$. Our results include: (1) an interpretation of the embedding $\\bar{O}_N\\subset S_{2^N}^+$, as corresponding to the antisymmetric representation of $O_N$, (2) a study of the liberations of $H_N$, notably with the result $<H_N^+,\\bar{O"},"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":"1807.07231","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2018-07-19T04:05:46Z","cross_cats_sorted":[],"title_canon_sha256":"7e292038971676992c34b7ede88946ab846f620c2e65f336544661ea2e2587b4","abstract_canon_sha256":"ae5a5a338dc7b94d8ff8d6728b8a84c36ec32aff454144c03bec778196718264"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:58.450914Z","signature_b64":"vPtfr9dblyV6Odm6GV5e0ToSf3pIRf8SVMhplbxLyv5KUvLO81tW2EekvjOYjfRkTXwIW0wYr1lH3W943/ZCBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9306792613d394b05fb14f8956457b20ba544959f32a3eb86d4d908b880e7522","last_reissued_at":"2026-05-17T23:39:58.450525Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:58.450525Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Higher orbitals of quizzy quantum group actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Teodor Banica","submitted_at":"2018-07-19T04:05:46Z","abstract_excerpt":"The hyperoctahedral group $H_N$ is known to have two natural liberations: the \"good\" one $H_N^+$, which is the quantum symmetry group of $N$ segments, and the \"bad\" one $\\bar{O}_N$, which is the quantum symmetry group of the $N$-hypercube. 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