{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:G27RJ2MRYHA7HCCTGNZO5AHEL4","short_pith_number":"pith:G27RJ2MR","canonical_record":{"source":{"id":"1011.3253","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-11-14T21:07:12Z","cross_cats_sorted":[],"title_canon_sha256":"62412398f7cc625fd9c2277157ef9063506c3b88c36d6f60b4b25e32963f4937","abstract_canon_sha256":"f89d6c052c74731a2aa1fa0e120cbc747da45cc484a34dbca080074a4622ff95"},"schema_version":"1.0"},"canonical_sha256":"36bf14e991c1c1f388533372ee80e45f28b37113785d114d24c9c5db50e2c102","source":{"kind":"arxiv","id":"1011.3253","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.3253","created_at":"2026-05-18T00:29:06Z"},{"alias_kind":"arxiv_version","alias_value":"1011.3253v1","created_at":"2026-05-18T00:29:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.3253","created_at":"2026-05-18T00:29:06Z"},{"alias_kind":"pith_short_12","alias_value":"G27RJ2MRYHA7","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"G27RJ2MRYHA7HCCT","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"G27RJ2MR","created_at":"2026-05-18T12:26:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:G27RJ2MRYHA7HCCTGNZO5AHEL4","target":"record","payload":{"canonical_record":{"source":{"id":"1011.3253","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-11-14T21:07:12Z","cross_cats_sorted":[],"title_canon_sha256":"62412398f7cc625fd9c2277157ef9063506c3b88c36d6f60b4b25e32963f4937","abstract_canon_sha256":"f89d6c052c74731a2aa1fa0e120cbc747da45cc484a34dbca080074a4622ff95"},"schema_version":"1.0"},"canonical_sha256":"36bf14e991c1c1f388533372ee80e45f28b37113785d114d24c9c5db50e2c102","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:06.168852Z","signature_b64":"4eWZcj8+bVRxKOVnYA23JH8DCXrYAAJ+YGrRHw0vjwrKlmgxJbtXaJ1fN20R6aZ3cm98juznlnFYSN2mjG2zDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"36bf14e991c1c1f388533372ee80e45f28b37113785d114d24c9c5db50e2c102","last_reissued_at":"2026-05-18T00:29:06.168068Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:06.168068Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1011.3253","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:29:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r36hrUFLXOOTPVb1pLZBbAyo4QKjH3H3mHcubP/NHfjnR22MpdbrLnzhDNJhKKTnkto+Hmpiv2RDbEUSwkBEBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T02:55:21.400427Z"},"content_sha256":"d2efa1c88cfba80372eb54270fa099c243e6d149f265a2f4def49014e9a929aa","schema_version":"1.0","event_id":"sha256:d2efa1c88cfba80372eb54270fa099c243e6d149f265a2f4def49014e9a929aa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:G27RJ2MRYHA7HCCTGNZO5AHEL4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hilbert series of PI relatively free G-graded algebras are rational functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Alexei Kanel-Belov, Eli Aljadeff","submitted_at":"2010-11-14T21:07:12Z","abstract_excerpt":"Let G be a finite group, (g_{1},...,g_{r}) an (unordered) r-tuple of G^{(r)} and x_{i,g_i}'s variables that correspond to the g_i's, i=1,...,r. Let F<x_{1,g_1},...,x_{r,g_r}> be the corresponding free G-graded algebra where F is a field of zero characteristic. Here the degree of a monomial is determined by the product of the indices in G. Let I be a G-graded T-ideal of F<x_{1,g_1},...,x_{r,g_r}> which is PI (e.g. any ideal of identities of a G-graded finite dimensional algebra is of this type). We prove that the Hilbert series of F<x_{1,g_1},...,x_{r,g_r}>/I is a rational function. More genera"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.3253","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:29:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fWxHAEyoTKhR0i5rtbKdGF1lOHvXL+P5jdfDpVJmQri28CUisl5IA9gHoe1WV+85Cot5PTmXbuXRHKbSYdCPCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T02:55:21.400778Z"},"content_sha256":"1a5853dead1f86aa032df3119baff369436d158419f72694e0ecbf162c9ad72d","schema_version":"1.0","event_id":"sha256:1a5853dead1f86aa032df3119baff369436d158419f72694e0ecbf162c9ad72d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G27RJ2MRYHA7HCCTGNZO5AHEL4/bundle.json","state_url":"https://pith.science/pith/G27RJ2MRYHA7HCCTGNZO5AHEL4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G27RJ2MRYHA7HCCTGNZO5AHEL4/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":"2026-06-02T02:55:21Z","links":{"resolver":"https://pith.science/pith/G27RJ2MRYHA7HCCTGNZO5AHEL4","bundle":"https://pith.science/pith/G27RJ2MRYHA7HCCTGNZO5AHEL4/bundle.json","state":"https://pith.science/pith/G27RJ2MRYHA7HCCTGNZO5AHEL4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G27RJ2MRYHA7HCCTGNZO5AHEL4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:G27RJ2MRYHA7HCCTGNZO5AHEL4","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":"f89d6c052c74731a2aa1fa0e120cbc747da45cc484a34dbca080074a4622ff95","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-11-14T21:07:12Z","title_canon_sha256":"62412398f7cc625fd9c2277157ef9063506c3b88c36d6f60b4b25e32963f4937"},"schema_version":"1.0","source":{"id":"1011.3253","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.3253","created_at":"2026-05-18T00:29:06Z"},{"alias_kind":"arxiv_version","alias_value":"1011.3253v1","created_at":"2026-05-18T00:29:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.3253","created_at":"2026-05-18T00:29:06Z"},{"alias_kind":"pith_short_12","alias_value":"G27RJ2MRYHA7","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"G27RJ2MRYHA7HCCT","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"G27RJ2MR","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:1a5853dead1f86aa032df3119baff369436d158419f72694e0ecbf162c9ad72d","target":"graph","created_at":"2026-05-18T00:29:06Z","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 G be a finite group, (g_{1},...,g_{r}) an (unordered) r-tuple of G^{(r)} and x_{i,g_i}'s variables that correspond to the g_i's, i=1,...,r. Let F<x_{1,g_1},...,x_{r,g_r}> be the corresponding free G-graded algebra where F is a field of zero characteristic. Here the degree of a monomial is determined by the product of the indices in G. Let I be a G-graded T-ideal of F<x_{1,g_1},...,x_{r,g_r}> which is PI (e.g. any ideal of identities of a G-graded finite dimensional algebra is of this type). We prove that the Hilbert series of F<x_{1,g_1},...,x_{r,g_r}>/I is a rational function. More genera","authors_text":"Alexei Kanel-Belov, Eli Aljadeff","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-11-14T21:07:12Z","title":"Hilbert series of PI relatively free G-graded algebras are rational functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.3253","kind":"arxiv","version":1},"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:d2efa1c88cfba80372eb54270fa099c243e6d149f265a2f4def49014e9a929aa","target":"record","created_at":"2026-05-18T00:29:06Z","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":"f89d6c052c74731a2aa1fa0e120cbc747da45cc484a34dbca080074a4622ff95","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-11-14T21:07:12Z","title_canon_sha256":"62412398f7cc625fd9c2277157ef9063506c3b88c36d6f60b4b25e32963f4937"},"schema_version":"1.0","source":{"id":"1011.3253","kind":"arxiv","version":1}},"canonical_sha256":"36bf14e991c1c1f388533372ee80e45f28b37113785d114d24c9c5db50e2c102","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"36bf14e991c1c1f388533372ee80e45f28b37113785d114d24c9c5db50e2c102","first_computed_at":"2026-05-18T00:29:06.168068Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:29:06.168068Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4eWZcj8+bVRxKOVnYA23JH8DCXrYAAJ+YGrRHw0vjwrKlmgxJbtXaJ1fN20R6aZ3cm98juznlnFYSN2mjG2zDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:29:06.168852Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.3253","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d2efa1c88cfba80372eb54270fa099c243e6d149f265a2f4def49014e9a929aa","sha256:1a5853dead1f86aa032df3119baff369436d158419f72694e0ecbf162c9ad72d"],"state_sha256":"7e9a9c9e24f85601157c991e36ece0660c4c8e308744cf926916da03f45a3474"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PinAxWsB8rn+6V5NDEqkcL2lSK3xcycYnMw57pwmGD571VNT7/pmE8lwQB2pIVlTvBBnax+tWUUYZmsSlv6JCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T02:55:21.402780Z","bundle_sha256":"52b5698c72bd108b21347cf05be5b854cabd2eb7db752149595caffd421a5d93"}}