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We determine a transversal $T$ of $H$ in $G$ such that each $\\theta \\in T$ is of minimal length in its right-coset, $H \\theta$, for the word length with respect to $\\mathcal{S}$. We show that there exists a regular language $L$ on $\\mathcal{S} \\cup \\mathcal{S}^{-1}$ and a bijection $\\mathrm{ev} : L \\to T$ satisfying $\\mathrm{lg} (U) = \\mathrm{lg}_\\mathcal{S}( \\mathrm{ev}(U))$ for all $U \\in L$. From this we deduce that the coset growth series of $"},"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":"1902.10207","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-02-26T20:35:16Z","cross_cats_sorted":[],"title_canon_sha256":"3eb53fadb3f41b0c050d3487c23c9ef8c03eb219ead40e4c6a296e92c1e491be","abstract_canon_sha256":"fff536db9784ba982ae677e1b3646eeaaa4b1f2c486203ca90c49a940cdd66c6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:52:31.486571Z","signature_b64":"PjDZ5odQuqODZcwaMlCLIw6jsHOqD6/OasNrFJ0K5ZHL+6lG8l+DihENWynmDPUuDY6sH/pELqceDEW2ONGmAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f1df0056428730bf87b0e2b248386cb0c8f6597c813d0777671bbd0a0597a530","last_reissued_at":"2026-05-17T23:52:31.485919Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:52:31.485919Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Transverse properties of parabolic subgroups of Garside groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Luis Paris, Yago Antol\\'in","submitted_at":"2019-02-26T20:35:16Z","abstract_excerpt":"Let $G$ be a Garside group endowed with the generating set $\\mathcal{S}$ of non-trivial simple elements, and let $H$ be a parabolic subgroup of $G$. We determine a transversal $T$ of $H$ in $G$ such that each $\\theta \\in T$ is of minimal length in its right-coset, $H \\theta$, for the word length with respect to $\\mathcal{S}$. We show that there exists a regular language $L$ on $\\mathcal{S} \\cup \\mathcal{S}^{-1}$ and a bijection $\\mathrm{ev} : L \\to T$ satisfying $\\mathrm{lg} (U) = \\mathrm{lg}_\\mathcal{S}( \\mathrm{ev}(U))$ for all $U \\in L$. From this we deduce that the coset growth series of $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.10207","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1902.10207","created_at":"2026-05-17T23:52:31.486013+00:00"},{"alias_kind":"arxiv_version","alias_value":"1902.10207v1","created_at":"2026-05-17T23:52:31.486013+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.10207","created_at":"2026-05-17T23:52:31.486013+00:00"},{"alias_kind":"pith_short_12","alias_value":"6HPQAVSCQ4YL","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_16","alias_value":"6HPQAVSCQ4YL7B5Q","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_8","alias_value":"6HPQAVSC","created_at":"2026-05-18T12:33:10.108867+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6HPQAVSCQ4YL7B5Q4KZEQODMWD","json":"https://pith.science/pith/6HPQAVSCQ4YL7B5Q4KZEQODMWD.json","graph_json":"https://pith.science/api/pith-number/6HPQAVSCQ4YL7B5Q4KZEQODMWD/graph.json","events_json":"https://pith.science/api/pith-number/6HPQAVSCQ4YL7B5Q4KZEQODMWD/events.json","paper":"https://pith.science/paper/6HPQAVSC"},"agent_actions":{"view_html":"https://pith.science/pith/6HPQAVSCQ4YL7B5Q4KZEQODMWD","download_json":"https://pith.science/pith/6HPQAVSCQ4YL7B5Q4KZEQODMWD.json","view_paper":"https://pith.science/paper/6HPQAVSC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1902.10207&json=true","fetch_graph":"https://pith.science/api/pith-number/6HPQAVSCQ4YL7B5Q4KZEQODMWD/graph.json","fetch_events":"https://pith.science/api/pith-number/6HPQAVSCQ4YL7B5Q4KZEQODMWD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6HPQAVSCQ4YL7B5Q4KZEQODMWD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6HPQAVSCQ4YL7B5Q4KZEQODMWD/action/storage_attestation","attest_author":"https://pith.science/pith/6HPQAVSCQ4YL7B5Q4KZEQODMWD/action/author_attestation","sign_citation":"https://pith.science/pith/6HPQAVSCQ4YL7B5Q4KZEQODMWD/action/citation_signature","submit_replication":"https://pith.science/pith/6HPQAVSCQ4YL7B5Q4KZEQODMWD/action/replication_record"}},"created_at":"2026-05-17T23:52:31.486013+00:00","updated_at":"2026-05-17T23:52:31.486013+00:00"}