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For $r\\geq 5$ odd, the rational Weyl group elements in $W(D_r)$ are exactly the longest element $w_0$ together with two explicitly described signed cyclic elements $c_I$ and $d_I$ for every non-empty subset $I\\subseteq\\{1,\\ldots,r-1\\}$. Consequently the rationality graph $\\Gamma(D_r)$ is two explicitly labelled Boolean-type halves glued at $w_0$, its number of ve"},"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":"2605.20928","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T09:11:42Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"f666a8a72d7820757e404ffc72aea5894b11692df480cbcf62353fae77eeee28","abstract_canon_sha256":"9e35439bada940c9f35a393d9be0dbc4f5333a5c3d3285948a66051c390019a5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-21T01:05:28.632257Z","signature_b64":"pwxYPhiCXlI2oYlDV+wgzJtcQh759ZUTqpc+WzwaIQoPJyOZ6OhlLxK7TA7yD7hVzHxUuvAvcfu0F/mPEqjcBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5b9d5533dcd23053c037a23b2f138738d0fa1ea1a615adbaf48420fb1f4c94d8","last_reissued_at":"2026-05-21T01:05:28.631720Z","signature_status":"signed_v1","first_computed_at":"2026-05-21T01:05:28.631720Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rational Weyl group elements of odd type D","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Yaoran Yang, Yutong Zhang","submitted_at":"2026-05-20T09:11:42Z","abstract_excerpt":"Voloshyn introduced rational Weyl group elements in connection with rational normal forms on complex reductive groups and conjectured that, in type $D_r$ with $r$ odd, their number is $2^r-1$. We prove a stronger structural statement. For $r\\geq 5$ odd, the rational Weyl group elements in $W(D_r)$ are exactly the longest element $w_0$ together with two explicitly described signed cyclic elements $c_I$ and $d_I$ for every non-empty subset $I\\subseteq\\{1,\\ldots,r-1\\}$. 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