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The action of $\\phi(W)$ on $F_2^n$ by left multiplication is corresponding to a combinatorial structure extracted and generalized from Vogan diagrams. In each case W of types A, D and E, we determine the orbits of $F_2^n$ under the action of $\\phi(W)$, and find that the kernel of $\\phi$ is the center $Z(W)$ of $W.$"},"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":"0804.2150","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2008-04-14T11:47:51Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d59ec12804b25514379620b487ee121512a7deb51099f7fdebd5a130d896fbf3","abstract_canon_sha256":"1b77af24c9fe9c60089642e896cb20c048c3347b2777176aec021df8b9f87707"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:44.868020Z","signature_b64":"vvKHAJGIVUCLUOGaik1XRpqfEffOJPwu9j67WJocxnDogsUOx5qnI2NfOpFOxi/K4xt9gYTN/y3loItGvWhqAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2217657ddaf81e83c5c9c2fad82b29ccc8f3a348fffe552f7bd204de04ce3f33","last_reissued_at":"2026-05-18T04:42:44.867522Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:44.867522Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Combinatorial representations of Coxeter groups over a field of two elements","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Chih-wen Weng, Hau-wen Huang","submitted_at":"2008-04-14T11:47:51Z","abstract_excerpt":"Let $W$ denote a simply-laced Coxeter group with $n$ generators. 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