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Looking to the associated monoids $A_S^+$ and $A_X^+$, we describe the quasi-centralizer of $A_X^+$ in $A_S^+$ thanks to results in Coxeter groups. These two results generalize earlier results of Paris. Finaly, we compare, in the spherical case, the normalizer of a parabolic subgroup in the Artin-Tits"},"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":"math/0202174","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"2002-02-18T15:14:09Z","cross_cats_sorted":[],"title_canon_sha256":"d45cb5dd3dc22eca71fd38a66f9b2e3b9aabd48efaea8a2cf1380214d963f70b","abstract_canon_sha256":"502851cea806a129330f87e43c96e731a6b0673c834e7f8089e2cafc649bf529"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:29.770620Z","signature_b64":"z86e8vQB5o9Rn4W8Fo02P0dttGiR7RCQ+zwAbc2ajRt3KxiheWk7Eii+Evu7dvXshC/YxPWbYYjZ+aEuyBJwCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3dd79b1691e212625409dc3fc3d0f6b2e0d767ddb5c4831dc68c49ed258b4ed4","last_reissued_at":"2026-05-18T01:05:29.770128Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:29.770128Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Normalisateurs et groupes d'Artin-Tits de type sph\\'erique","license":"","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Eddy Godelle","submitted_at":"2002-02-18T15:14:09Z","abstract_excerpt":"Let $(A_S,S)$ be an Artin-Tits and $X$ a subset of $S$ ; denote by $A_X$ the subgroup of $A_S$ generated by $X$. 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