{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:KCXQM4ZOCANI2NEEAUZSLQQBRT","short_pith_number":"pith:KCXQM4ZO","schema_version":"1.0","canonical_sha256":"50af06732e101a8d3484053325c2018cfeed5b8e74491011ac121084571d9928","source":{"kind":"arxiv","id":"1906.06494","version":1},"attestation_state":"computed","paper":{"title":"Finite Reflection Groups: Invariant functions and functions of the Invariants in finite class of differentiability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Gerard Barban\\c{c}on","submitted_at":"2019-06-15T08:18:32Z","abstract_excerpt":"Let $W$ be a finite reflection group. A $W$-invariant function of class~$C^{\\infty}$ may be expressed as a functions of class $C^{\\infty}$ of the basic invariants. In finite class of differentiability, the situation is not this simple. Let~$h$ be the greatest Coxeter number of the irreducible components of $W$ and $P$ be~the Chevalley mapping, if $f$ is an invariant function of class $C^{hr}$, and $F$ is the function of invariants associated by $f=F\\circ P$, then $F$ is of class $C^r$. However if~$F$ is of class $C^r$, in general $f=F\\circ P$ is of class $C^r$ and not of class $C^{hr}$. Here 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":"1906.06494","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-06-15T08:18:32Z","cross_cats_sorted":[],"title_canon_sha256":"0f69e349be1bf846754bb260d5ac77baec9db0bfc89e883d65ae57bf2bb10e51","abstract_canon_sha256":"80513b57c99e4ca772eb8c0bb3e0eec2b835352c8546d241f1bd72fcf65a2654"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:13.537157Z","signature_b64":"hk0nvotOv9ZGpF8BXF6u4Rjr2hNYbQFGZNezYW5QL3B0ZN6YWfP5VzFVAUFTm50unSN4ZL4Dc5QY2tFuO0mcCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50af06732e101a8d3484053325c2018cfeed5b8e74491011ac121084571d9928","last_reissued_at":"2026-05-17T23:43:13.536509Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:13.536509Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Finite Reflection Groups: Invariant functions and functions of the Invariants in finite class of differentiability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Gerard Barban\\c{c}on","submitted_at":"2019-06-15T08:18:32Z","abstract_excerpt":"Let $W$ be a finite reflection group. A $W$-invariant function of class~$C^{\\infty}$ may be expressed as a functions of class $C^{\\infty}$ of the basic invariants. In finite class of differentiability, the situation is not this simple. Let~$h$ be the greatest Coxeter number of the irreducible components of $W$ and $P$ be~the Chevalley mapping, if $f$ is an invariant function of class $C^{hr}$, and $F$ is the function of invariants associated by $f=F\\circ P$, then $F$ is of class $C^r$. However if~$F$ is of class $C^r$, in general $f=F\\circ P$ is of class $C^r$ and not of class $C^{hr}$. Here w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.06494","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":"1906.06494","created_at":"2026-05-17T23:43:13.536632+00:00"},{"alias_kind":"arxiv_version","alias_value":"1906.06494v1","created_at":"2026-05-17T23:43:13.536632+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.06494","created_at":"2026-05-17T23:43:13.536632+00:00"},{"alias_kind":"pith_short_12","alias_value":"KCXQM4ZOCANI","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_16","alias_value":"KCXQM4ZOCANI2NEE","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_8","alias_value":"KCXQM4ZO","created_at":"2026-05-18T12:33:21.387695+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.08725","citing_title":"Harmonic Analysis of the Instanton Prepotential","ref_index":15,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KCXQM4ZOCANI2NEEAUZSLQQBRT","json":"https://pith.science/pith/KCXQM4ZOCANI2NEEAUZSLQQBRT.json","graph_json":"https://pith.science/api/pith-number/KCXQM4ZOCANI2NEEAUZSLQQBRT/graph.json","events_json":"https://pith.science/api/pith-number/KCXQM4ZOCANI2NEEAUZSLQQBRT/events.json","paper":"https://pith.science/paper/KCXQM4ZO"},"agent_actions":{"view_html":"https://pith.science/pith/KCXQM4ZOCANI2NEEAUZSLQQBRT","download_json":"https://pith.science/pith/KCXQM4ZOCANI2NEEAUZSLQQBRT.json","view_paper":"https://pith.science/paper/KCXQM4ZO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1906.06494&json=true","fetch_graph":"https://pith.science/api/pith-number/KCXQM4ZOCANI2NEEAUZSLQQBRT/graph.json","fetch_events":"https://pith.science/api/pith-number/KCXQM4ZOCANI2NEEAUZSLQQBRT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KCXQM4ZOCANI2NEEAUZSLQQBRT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KCXQM4ZOCANI2NEEAUZSLQQBRT/action/storage_attestation","attest_author":"https://pith.science/pith/KCXQM4ZOCANI2NEEAUZSLQQBRT/action/author_attestation","sign_citation":"https://pith.science/pith/KCXQM4ZOCANI2NEEAUZSLQQBRT/action/citation_signature","submit_replication":"https://pith.science/pith/KCXQM4ZOCANI2NEEAUZSLQQBRT/action/replication_record"}},"created_at":"2026-05-17T23:43:13.536632+00:00","updated_at":"2026-05-17T23:43:13.536632+00:00"}