{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2004:BNSGVB42IPMJWR2VY3KRPIXGFB","short_pith_number":"pith:BNSGVB42","schema_version":"1.0","canonical_sha256":"0b646a879a43d89b4755c6d517a2e6287f53d803ab2977657752374f27215dd1","source":{"kind":"arxiv","id":"math/0410575","version":3},"attestation_state":"computed","paper":{"title":"Structure of the unitary valuation algebra","license":"","headline":"","cross_cats":["math.RA"],"primary_cat":"math.DG","authors_text":"Joseph H.G. Fu","submitted_at":"2004-10-27T14:56:44Z","abstract_excerpt":"S. Alesker has shown that if $G$ is a compact subgroup of O(n) acting transitively on the unit sphere $S^{n-1}$ then the vector space $Val^G$ of continuous, translation-invariant, $G$-invariant convex valuations on $R^n$ has the structure of a finite dimensional graded algebra over $R$ satisfying Poincare duality. We show that the kinematic formulas for $G$ are determined by the product pairing. Using this result we then show that the algebra $Val^{U(n) }$ is isomorphic to $R[s,t]/(f_{n+1}, f_{n+2})$, where $s,t$ have degrees 2 and 1 respectively, and the polynomial $f_i$ is the degree $i$ ter"},"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":"math/0410575","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.DG","submitted_at":"2004-10-27T14:56:44Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"adbf8f4b8d1b5ff3e04b9d24068213e21aeebb64ccc491ecff11f25da828b5dc","abstract_canon_sha256":"647747e1fe162c65295867c6a35295d0480d733790d728a0dc9a37524acd9114"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:37.260222Z","signature_b64":"SKqTpAt1IIRfI8nxQUg82I+grPB/eSOBx5EoH1rGCeKxZa4l+gUXTTVVzTh+GSAR2GgMU3wzfojVnJ/ZLYpSAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b646a879a43d89b4755c6d517a2e6287f53d803ab2977657752374f27215dd1","last_reissued_at":"2026-05-18T03:40:37.259637Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:37.259637Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Structure of the unitary valuation algebra","license":"","headline":"","cross_cats":["math.RA"],"primary_cat":"math.DG","authors_text":"Joseph H.G. Fu","submitted_at":"2004-10-27T14:56:44Z","abstract_excerpt":"S. Alesker has shown that if $G$ is a compact subgroup of O(n) acting transitively on the unit sphere $S^{n-1}$ then the vector space $Val^G$ of continuous, translation-invariant, $G$-invariant convex valuations on $R^n$ has the structure of a finite dimensional graded algebra over $R$ satisfying Poincare duality. We show that the kinematic formulas for $G$ are determined by the product pairing. Using this result we then show that the algebra $Val^{U(n) }$ is isomorphic to $R[s,t]/(f_{n+1}, f_{n+2})$, where $s,t$ have degrees 2 and 1 respectively, and the polynomial $f_i$ is the degree $i$ ter"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0410575","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0410575","created_at":"2026-05-18T03:40:37.259741+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0410575v3","created_at":"2026-05-18T03:40:37.259741+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0410575","created_at":"2026-05-18T03:40:37.259741+00:00"},{"alias_kind":"pith_short_12","alias_value":"BNSGVB42IPMJ","created_at":"2026-05-18T12:25:52.051335+00:00"},{"alias_kind":"pith_short_16","alias_value":"BNSGVB42IPMJWR2V","created_at":"2026-05-18T12:25:52.051335+00:00"},{"alias_kind":"pith_short_8","alias_value":"BNSGVB42","created_at":"2026-05-18T12:25:52.051335+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/BNSGVB42IPMJWR2VY3KRPIXGFB","json":"https://pith.science/pith/BNSGVB42IPMJWR2VY3KRPIXGFB.json","graph_json":"https://pith.science/api/pith-number/BNSGVB42IPMJWR2VY3KRPIXGFB/graph.json","events_json":"https://pith.science/api/pith-number/BNSGVB42IPMJWR2VY3KRPIXGFB/events.json","paper":"https://pith.science/paper/BNSGVB42"},"agent_actions":{"view_html":"https://pith.science/pith/BNSGVB42IPMJWR2VY3KRPIXGFB","download_json":"https://pith.science/pith/BNSGVB42IPMJWR2VY3KRPIXGFB.json","view_paper":"https://pith.science/paper/BNSGVB42","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0410575&json=true","fetch_graph":"https://pith.science/api/pith-number/BNSGVB42IPMJWR2VY3KRPIXGFB/graph.json","fetch_events":"https://pith.science/api/pith-number/BNSGVB42IPMJWR2VY3KRPIXGFB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BNSGVB42IPMJWR2VY3KRPIXGFB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BNSGVB42IPMJWR2VY3KRPIXGFB/action/storage_attestation","attest_author":"https://pith.science/pith/BNSGVB42IPMJWR2VY3KRPIXGFB/action/author_attestation","sign_citation":"https://pith.science/pith/BNSGVB42IPMJWR2VY3KRPIXGFB/action/citation_signature","submit_replication":"https://pith.science/pith/BNSGVB42IPMJWR2VY3KRPIXGFB/action/replication_record"}},"created_at":"2026-05-18T03:40:37.259741+00:00","updated_at":"2026-05-18T03:40:37.259741+00:00"}