{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:WTNQIV3RU42TOCDDIDUEDST6RH","short_pith_number":"pith:WTNQIV3R","canonical_record":{"source":{"id":"1810.00409","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-09-30T15:49:40Z","cross_cats_sorted":[],"title_canon_sha256":"1057d5c2d37f379b19a88710faa387185f2d50c24a94eb2c02068557ad89067e","abstract_canon_sha256":"3b1a164ab21d5643a3c44b292ac38258c1a3df738c03b8dd72f5cb90fe66a11b"},"schema_version":"1.0"},"canonical_sha256":"b4db045771a73537086340e841ca7e89d7a7e6dd07c5d7b7eeac464366c8eb18","source":{"kind":"arxiv","id":"1810.00409","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.00409","created_at":"2026-05-18T00:04:27Z"},{"alias_kind":"arxiv_version","alias_value":"1810.00409v1","created_at":"2026-05-18T00:04:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.00409","created_at":"2026-05-18T00:04:27Z"},{"alias_kind":"pith_short_12","alias_value":"WTNQIV3RU42T","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"WTNQIV3RU42TOCDD","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"WTNQIV3R","created_at":"2026-05-18T12:33:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:WTNQIV3RU42TOCDDIDUEDST6RH","target":"record","payload":{"canonical_record":{"source":{"id":"1810.00409","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-09-30T15:49:40Z","cross_cats_sorted":[],"title_canon_sha256":"1057d5c2d37f379b19a88710faa387185f2d50c24a94eb2c02068557ad89067e","abstract_canon_sha256":"3b1a164ab21d5643a3c44b292ac38258c1a3df738c03b8dd72f5cb90fe66a11b"},"schema_version":"1.0"},"canonical_sha256":"b4db045771a73537086340e841ca7e89d7a7e6dd07c5d7b7eeac464366c8eb18","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:27.952695Z","signature_b64":"zx8Bw47xzO3nd2NiduFTlRRBnMEhn1lKppM0/ACS7mymu+zwjClomRbRc10NQXdeb4Z5TJPHwAMTcWxtxj4iCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b4db045771a73537086340e841ca7e89d7a7e6dd07c5d7b7eeac464366c8eb18","last_reissued_at":"2026-05-18T00:04:27.952224Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:27.952224Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1810.00409","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":"2026-05-18T00:04:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T8udAVXI0kzJO2f+CAwou10n1Tm1kJvi6WgAsJYD3dBrdb+VCvA9F8DU4yEz+mPXndzeAiItxgHsjyPFqx4CAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T10:59:06.023971Z"},"content_sha256":"47cc1a555b52095c5cf5edb6de19d100dfda47a57703fc5a5dc3112f522e8a0c","schema_version":"1.0","event_id":"sha256:47cc1a555b52095c5cf5edb6de19d100dfda47a57703fc5a5dc3112f522e8a0c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:WTNQIV3RU42TOCDDIDUEDST6RH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tensor Product Markov Chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Georgia Benkart, Martin W. Liebeck, Persi Diaconis, Pham Huu Tiep","submitted_at":"2018-09-30T15:49:40Z","abstract_excerpt":"We analyze families of Markov chains that arise from decomposing tensor products of irreducible representations. This illuminates the Burnside-Brauer Theorem for building irreducible representations, the McKay Correspondence, and Pitman's 2M-X Theorem. The chains are explicitly diagonalizable, and we use the eigenvalues/eigenvectors to give sharp rates of convergence for the associated random walks. For modular representations, the chains are not reversible, and the analytical details are surprisingly intricate. In the quantum group case, the chains fail to be diagonalizable, but a novel analy"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.00409","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":"2026-05-18T00:04:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FJE+AkEbkLNpIjc0sdcSbj5E+Gilca6dilfs9FAdAkfT1UMxfx7yfknPdJtlCBz5YLWXykM1Y8Mz8+4OTxQECw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T10:59:06.024368Z"},"content_sha256":"ff68be059d36f5f14ff36df6e9502c7d74b7a8fadae9291c0fd68c5f251c7be8","schema_version":"1.0","event_id":"sha256:ff68be059d36f5f14ff36df6e9502c7d74b7a8fadae9291c0fd68c5f251c7be8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WTNQIV3RU42TOCDDIDUEDST6RH/bundle.json","state_url":"https://pith.science/pith/WTNQIV3RU42TOCDDIDUEDST6RH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WTNQIV3RU42TOCDDIDUEDST6RH/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":"2026-06-03T10:59:06Z","links":{"resolver":"https://pith.science/pith/WTNQIV3RU42TOCDDIDUEDST6RH","bundle":"https://pith.science/pith/WTNQIV3RU42TOCDDIDUEDST6RH/bundle.json","state":"https://pith.science/pith/WTNQIV3RU42TOCDDIDUEDST6RH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WTNQIV3RU42TOCDDIDUEDST6RH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:WTNQIV3RU42TOCDDIDUEDST6RH","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":"3b1a164ab21d5643a3c44b292ac38258c1a3df738c03b8dd72f5cb90fe66a11b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-09-30T15:49:40Z","title_canon_sha256":"1057d5c2d37f379b19a88710faa387185f2d50c24a94eb2c02068557ad89067e"},"schema_version":"1.0","source":{"id":"1810.00409","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.00409","created_at":"2026-05-18T00:04:27Z"},{"alias_kind":"arxiv_version","alias_value":"1810.00409v1","created_at":"2026-05-18T00:04:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.00409","created_at":"2026-05-18T00:04:27Z"},{"alias_kind":"pith_short_12","alias_value":"WTNQIV3RU42T","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"WTNQIV3RU42TOCDD","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"WTNQIV3R","created_at":"2026-05-18T12:33:01Z"}],"graph_snapshots":[{"event_id":"sha256:ff68be059d36f5f14ff36df6e9502c7d74b7a8fadae9291c0fd68c5f251c7be8","target":"graph","created_at":"2026-05-18T00:04:27Z","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":"We analyze families of Markov chains that arise from decomposing tensor products of irreducible representations. This illuminates the Burnside-Brauer Theorem for building irreducible representations, the McKay Correspondence, and Pitman's 2M-X Theorem. The chains are explicitly diagonalizable, and we use the eigenvalues/eigenvectors to give sharp rates of convergence for the associated random walks. For modular representations, the chains are not reversible, and the analytical details are surprisingly intricate. In the quantum group case, the chains fail to be diagonalizable, but a novel analy","authors_text":"Georgia Benkart, Martin W. Liebeck, Persi Diaconis, Pham Huu Tiep","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-09-30T15:49:40Z","title":"Tensor Product Markov Chains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.00409","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":"sha256:47cc1a555b52095c5cf5edb6de19d100dfda47a57703fc5a5dc3112f522e8a0c","target":"record","created_at":"2026-05-18T00:04:27Z","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":"3b1a164ab21d5643a3c44b292ac38258c1a3df738c03b8dd72f5cb90fe66a11b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-09-30T15:49:40Z","title_canon_sha256":"1057d5c2d37f379b19a88710faa387185f2d50c24a94eb2c02068557ad89067e"},"schema_version":"1.0","source":{"id":"1810.00409","kind":"arxiv","version":1}},"canonical_sha256":"b4db045771a73537086340e841ca7e89d7a7e6dd07c5d7b7eeac464366c8eb18","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b4db045771a73537086340e841ca7e89d7a7e6dd07c5d7b7eeac464366c8eb18","first_computed_at":"2026-05-18T00:04:27.952224Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:04:27.952224Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zx8Bw47xzO3nd2NiduFTlRRBnMEhn1lKppM0/ACS7mymu+zwjClomRbRc10NQXdeb4Z5TJPHwAMTcWxtxj4iCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:04:27.952695Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.00409","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:47cc1a555b52095c5cf5edb6de19d100dfda47a57703fc5a5dc3112f522e8a0c","sha256:ff68be059d36f5f14ff36df6e9502c7d74b7a8fadae9291c0fd68c5f251c7be8"],"state_sha256":"ab97f6a610d9ec707d539b208e4e6cac07be5eee35019cac103cf7a5035b3483"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QkXqMge0JWhtSeQiPVmfpvJUEyUBpnY/dT4tQw5cxPogY+6pR2+7Wh4oRtwLVqpVLBnz+NBZ2qLjxgMxD2l8CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T10:59:06.026309Z","bundle_sha256":"556d8c0f900551c2ddbb1fb91330fe147b69c6f9050a5f8bf89d86cc18adf620"}}