{"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 $I\\subseteq\\{1,\\ldots,r-1\\}$. Consequently the rationality graph $\\Gamma(D_r)$ is two explicitly labelled Boolean-type halves glued at $w_0$, its number of ve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20928","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.20928/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}