{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:WNEKXD3QNPUBAWHGXI77EZHH3F","short_pith_number":"pith:WNEKXD3Q","canonical_record":{"source":{"id":"1011.0329","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-01T14:25:13Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"7d7426a78629412ddbf0bbe7cfe0fb0af2849b937323168a318e0ed8c1a828e2","abstract_canon_sha256":"fb4d204171a90b52b6b1ebd6d163fa541277fedf41d226cf171e3c7fe30421f1"},"schema_version":"1.0"},"canonical_sha256":"b348ab8f706be81058e6ba3ff264e7d97851ffa327c8fbe5b69853b995e11194","source":{"kind":"arxiv","id":"1011.0329","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.0329","created_at":"2026-05-18T01:36:39Z"},{"alias_kind":"arxiv_version","alias_value":"1011.0329v3","created_at":"2026-05-18T01:36:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.0329","created_at":"2026-05-18T01:36:39Z"},{"alias_kind":"pith_short_12","alias_value":"WNEKXD3QNPUB","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"WNEKXD3QNPUBAWHG","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"WNEKXD3Q","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:WNEKXD3QNPUBAWHGXI77EZHH3F","target":"record","payload":{"canonical_record":{"source":{"id":"1011.0329","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-01T14:25:13Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"7d7426a78629412ddbf0bbe7cfe0fb0af2849b937323168a318e0ed8c1a828e2","abstract_canon_sha256":"fb4d204171a90b52b6b1ebd6d163fa541277fedf41d226cf171e3c7fe30421f1"},"schema_version":"1.0"},"canonical_sha256":"b348ab8f706be81058e6ba3ff264e7d97851ffa327c8fbe5b69853b995e11194","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:39.289137Z","signature_b64":"h+1npN/hCqiq5X66+7ikuEmc6dlArQLu+6Bw7tSLN/XpenKOf7JzM2Hnv+7eDM1Fhrw0m6x7IvsTOBP99pw2Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b348ab8f706be81058e6ba3ff264e7d97851ffa327c8fbe5b69853b995e11194","last_reissued_at":"2026-05-18T01:36:39.288441Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:39.288441Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1011.0329","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:36:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zGPXQeDCJir+B1wVLFGUL1ifNLKdIT80hY0GF2/RDoUbxDE9rvvdrY5f74wkvw55WNqFiqXshDeDFQ75E3SEBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T18:54:05.389067Z"},"content_sha256":"f708601c7e9ca9b26b7b10dda020097030582fdb18b5cba4f8b54d5d0e071c27","schema_version":"1.0","event_id":"sha256:f708601c7e9ca9b26b7b10dda020097030582fdb18b5cba4f8b54d5d0e071c27"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:WNEKXD3QNPUBAWHGXI77EZHH3F","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Equivariant multiplicities of Coxeter arrangements and invariant bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"Atsushi Wakamiko, Hiroaki Terao, Takuro Abe","submitted_at":"2010-11-01T14:25:13Z","abstract_excerpt":"Let $\\A$ be an irreducible Coxeter arrangement and $W$ be its Coxeter group. Then $W$ naturally acts on $\\A$. A multiplicity $\\bfm : \\A\\rightarrow \\Z$ is said to be equivariant when $\\bfm$ is constant on each $W$-orbit of $\\A$. In this article, we prove that the multi-derivation module $D(\\A, \\bfm)$ is a free module whenever $\\bfm$ is equivariant by explicitly constructing a basis, which generalizes the main theorem of \\cite{T02}. The main tool is a primitive derivation and its covariant derivative. Moreover, we show that the $W$-invariant part $D(\\A, \\bfm)^{W}$ for any multiplicity $\\bfm$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.0329","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:36:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yfscFXXg1wldF0tzg/eNnZhT6/v5YXwV6/KOYwU4ZsK2Xr2BCpEYZ8LsCHBFqlBQGsTZnq/JQzsJF2P9ObuuAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T18:54:05.389423Z"},"content_sha256":"d5602545c4b1a64e570f7e7cfb137ff125708afde9d458044f3837344b10fc06","schema_version":"1.0","event_id":"sha256:d5602545c4b1a64e570f7e7cfb137ff125708afde9d458044f3837344b10fc06"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WNEKXD3QNPUBAWHGXI77EZHH3F/bundle.json","state_url":"https://pith.science/pith/WNEKXD3QNPUBAWHGXI77EZHH3F/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WNEKXD3QNPUBAWHGXI77EZHH3F/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-25T18:54:05Z","links":{"resolver":"https://pith.science/pith/WNEKXD3QNPUBAWHGXI77EZHH3F","bundle":"https://pith.science/pith/WNEKXD3QNPUBAWHGXI77EZHH3F/bundle.json","state":"https://pith.science/pith/WNEKXD3QNPUBAWHGXI77EZHH3F/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WNEKXD3QNPUBAWHGXI77EZHH3F/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:WNEKXD3QNPUBAWHGXI77EZHH3F","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"fb4d204171a90b52b6b1ebd6d163fa541277fedf41d226cf171e3c7fe30421f1","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-01T14:25:13Z","title_canon_sha256":"7d7426a78629412ddbf0bbe7cfe0fb0af2849b937323168a318e0ed8c1a828e2"},"schema_version":"1.0","source":{"id":"1011.0329","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.0329","created_at":"2026-05-18T01:36:39Z"},{"alias_kind":"arxiv_version","alias_value":"1011.0329v3","created_at":"2026-05-18T01:36:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.0329","created_at":"2026-05-18T01:36:39Z"},{"alias_kind":"pith_short_12","alias_value":"WNEKXD3QNPUB","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"WNEKXD3QNPUBAWHG","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"WNEKXD3Q","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:d5602545c4b1a64e570f7e7cfb137ff125708afde9d458044f3837344b10fc06","target":"graph","created_at":"2026-05-18T01:36:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $\\A$ be an irreducible Coxeter arrangement and $W$ be its Coxeter group. Then $W$ naturally acts on $\\A$. A multiplicity $\\bfm : \\A\\rightarrow \\Z$ is said to be equivariant when $\\bfm$ is constant on each $W$-orbit of $\\A$. In this article, we prove that the multi-derivation module $D(\\A, \\bfm)$ is a free module whenever $\\bfm$ is equivariant by explicitly constructing a basis, which generalizes the main theorem of \\cite{T02}. The main tool is a primitive derivation and its covariant derivative. Moreover, we show that the $W$-invariant part $D(\\A, \\bfm)^{W}$ for any multiplicity $\\bfm$ is ","authors_text":"Atsushi Wakamiko, Hiroaki Terao, Takuro Abe","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-01T14:25:13Z","title":"Equivariant multiplicities of Coxeter arrangements and invariant bases"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.0329","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f708601c7e9ca9b26b7b10dda020097030582fdb18b5cba4f8b54d5d0e071c27","target":"record","created_at":"2026-05-18T01:36:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"fb4d204171a90b52b6b1ebd6d163fa541277fedf41d226cf171e3c7fe30421f1","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-11-01T14:25:13Z","title_canon_sha256":"7d7426a78629412ddbf0bbe7cfe0fb0af2849b937323168a318e0ed8c1a828e2"},"schema_version":"1.0","source":{"id":"1011.0329","kind":"arxiv","version":3}},"canonical_sha256":"b348ab8f706be81058e6ba3ff264e7d97851ffa327c8fbe5b69853b995e11194","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b348ab8f706be81058e6ba3ff264e7d97851ffa327c8fbe5b69853b995e11194","first_computed_at":"2026-05-18T01:36:39.288441Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:39.288441Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"h+1npN/hCqiq5X66+7ikuEmc6dlArQLu+6Bw7tSLN/XpenKOf7JzM2Hnv+7eDM1Fhrw0m6x7IvsTOBP99pw2Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:39.289137Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.0329","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f708601c7e9ca9b26b7b10dda020097030582fdb18b5cba4f8b54d5d0e071c27","sha256:d5602545c4b1a64e570f7e7cfb137ff125708afde9d458044f3837344b10fc06"],"state_sha256":"63fb2b5878efb5d79c4141549461a37ee91ac2f7b904845c754b893e33f845ac"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mBu+XGX8CgRUo/CvDm40jXu06qgtYo2DuLXgSyYv0dToFEbrIh9Tmjq+hSy23qp06c5JUxN6ATibR98VrY+sAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T18:54:05.391258Z","bundle_sha256":"d14e312c70facf3f82daa092596e75b6c77cbee3f0e256a4a6897b9ac147128a"}}